10th Innovations in Theoretical Computer Science Conference (ITCS 2019), ITCS 2019, January 10-12, 2019, San Diego, California, USA
ITCS 2019
January 10-12, 2019
San Diego, California, USA
Innovations in Theoretical Computer Science Conference
ITCS
http://itcs-conf.org/
https://dblp.org/db/conf/innovations
Leibniz International Proceedings in Informatics
LIPIcs
https://www.dagstuhl.de/dagpub/1868-8969
https://dblp.org/db/series/lipics
1868-8969
Avrim
Blum
Avrim Blum
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
124
2019
978-3-95977-095-8
https://www.dagstuhl.de/dagpub/978-3-95977-095-8
Front Matter, Table of Contents, Preface, Conference Organization
Front Matter, Table of Contents, Preface, Conference Organization
Front Matter
Table of Contents
Preface
Conference Organization
0:i-0:xii
Front Matter
Avrim
Blum
Avrim Blum
10.4230/LIPIcs.ITCS.2019.0
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Submodular Secretary Problem with Shortlists
In submodular k-secretary problem, the goal is to select k items in a randomly ordered input so as to maximize the expected value of a given monotone submodular function on the set of selected items. In this paper, we introduce a relaxation of this problem, which we refer to as submodular k-secretary problem with shortlists. In the proposed problem setting, the algorithm is allowed to choose more than k items as part of a shortlist. Then, after seeing the entire input, the algorithm can choose a subset of size k from the bigger set of items in the shortlist. We are interested in understanding to what extent this relaxation can improve the achievable competitive ratio for the submodular k-secretary problem. In particular, using an O(k) sized shortlist, can an online algorithm achieve a competitive ratio close to the best achievable offline approximation factor for this problem? We answer this question affirmatively by giving a polynomial time algorithm that achieves a 1-1/e-epsilon-O(k^{-1}) competitive ratio for any constant epsilon>0, using a shortlist of size eta_epsilon(k)=O(k). This is especially surprising considering that the best known competitive ratio (in polynomial time) for the submodular k-secretary problem is (1/e-O(k^{-1/2}))(1-1/e) [Thomas Kesselheim and Andreas Tönnis, 2017].
The proposed algorithm also has significant implications for another important problem of submodular function maximization under random order streaming model and k-cardinality constraint. We show that our algorithm can be implemented in the streaming setting using a memory buffer of size eta_epsilon(k)=O(k) to achieve a 1-1/e-epsilon-O(k^{-1}) approximation. This result substantially improves upon [Norouzi-Fard et al., 2018], which achieved the previously best known approximation factor of 1/2 + 8 x 10^{-14} using O(k log k) memory; and closely matches the known upper bound for this problem [McGregor and Vu, 2017].
Submodular Optimization
Secretary Problem
Streaming Algorithms
Mathematics of computing~Submodular optimization and polymatroids
Theory of computation~Online algorithms
Theory of computation~Streaming, sublinear and near linear time algorithms
1:1-1:19
Regular Paper
A full version of the paper is available as [Shipra Agrawal et al., 2018], https://arxiv.org/abs/1809.05082.
Shipra
Agrawal
Shipra Agrawal
Columbia University, New York, NY, USA, 10027
Research supported in part by Google Faculty Research Awards 2017 and Amazon Research Awards 2017.
Mohammad
Shadravan
Mohammad Shadravan
Columbia University, New York, NY, USA, 10027
Cliff
Stein
Cliff Stein
Columbia University, New York, NY, USA, 10027
Research supported in part by NSF grants CCF-1421161 and CCF-1714818.
10.4230/LIPIcs.ITCS.2019.1
Shipra Agrawal, Mohammad Shadravan, and Cliff Stein. Submodular Secretary Problem with Shortlists, 2018. URL: http://arxiv.org/abs/1809.05082.
http://arxiv.org/abs/1809.05082
Shipra Agrawal, Zizhuo Wang, and Yinyu Ye. A Dynamic Near-Optimal Algorithm for Online Linear Programming. Operations Research, 62(4):876-890, 2014.
Miklos Ajtai, Nimrod Megiddo, and Orli Waarts. Improved Algorithms and Analysis for Secretary Problems and Generalizations. SIAM J. Discret. Math., 14(1):1-27, January 2001.
Moshe Babaioff, Nicole Immorlica, David Kempe, and Robert Kleinberg. Online Auctions and Generalized Secretary Problems. SIGecom Exch., 7(2):7:1-7:11, June 2008.
Ashwinkumar Badanidiyuru, Baharan Mirzasoleiman, Amin Karbasi, and Andreas Krause. Streaming Submodular Maximization: Massive Data Summarization on the Fly. In Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD '14, pages 671-680, New York, NY, USA, 2014. ACM.
Mohammadhossein Bateni, Mohammadtaghi Hajiaghayi, and Morteza Zadimoghaddam. Submodular Secretary Problem and Extensions. ACM Trans. Algorithms, 9(4):32:1-32:23, October 2013.
Niv Buchbinder, Moran Feldman, and Mohit Garg. Online Submodular Maximization: Beating 1/2 Made Simple. arXiv preprint, 2018. URL: http://arxiv.org/abs/1807.05529.
http://arxiv.org/abs/1807.05529
Niv Buchbinder, Moran Feldman, Joseph (Seffi) Naor, and Roy Schwartz. Submodular Maximization with Cardinality Constraints. In Proceedings of the Twenty-fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '14, pages 1433-1452. Society for Industrial and Applied Mathematics, 2014.
Amit Chakrabarti and Sagar Kale. Submodular maximization meets streaming: matchings, matroids, and more. Mathematical Programming, 154(1):225-247, December 2015.
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Tom Hess and Sivan Sabato. The submodular secretary problem under a cardinality constraint and with limited resources. CoRR, abs/1702.03989, 2017. URL: http://arxiv.org/abs/1702.03989.
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Thomas Kesselheim and Andreas Tönnis. Submodular Secretary Problems: Cardinality, Matching, and Linear Constraints. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017), Leibniz International Proceedings in Informatics (LIPIcs), pages 16:1-16:22, 2017.
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Shipra Agrawal, Mohammad Shadravan, and Cliff Stein
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Hamiltonian Sparsification and Gap-Simulation
Analog quantum simulation - simulation of one Hamiltonian by another - is one of the major goals in the noisy intermediate-scale quantum computation (NISQ) era, and has many applications in quantum complexity. We initiate the rigorous study of the physical resources required for such simulations, where we focus on the task of Hamiltonian sparsification. The goal is to find a simulating Hamiltonian H~ whose underlying interaction graph has bounded degree (this is called degree-reduction) or much fewer edges than that of the original Hamiltonian H (this is called dilution). We set this study in a relaxed framework for analog simulations that we call gap-simulation, where H~ is only required to simulate the groundstate(s) and spectral gap of H instead of its full spectrum, and we believe it is of independent interest.
Our main result is a proof that in stark contrast to the classical setting, general degree-reduction is impossible in the quantum world, even under our relaxed notion of gap-simulation. The impossibility proof relies on devising counterexample Hamiltonians and applying a strengthened variant of Hastings-Koma decay of correlations theorem. We also show a complementary result where degree-reduction is possible when the strength of interactions is allowed to grow polynomially. Furthermore, we prove the impossibility of the related sparsification task of generic Hamiltonian dilution, under a computational hardness assumption. We also clarify the (currently weak) implications of our results to the question of quantum PCP. Our work provides basic answers to many of the "first questions" one would ask about Hamiltonian sparsification and gap-simulation; we hope this serves as a good starting point for future research of these topics.
quantum simulation
quantum Hamiltonian complexity
sparsification
quantum PCP
Theory of computation~Quantum computation theory
2:1-2:21
Regular Paper
Extended version including all proofs is hosted on https://arxiv.org/abs/1804.11084.
Dorit
Aharonov
Dorit Aharonov
School of Computer Science and Engineering, The Hebrew University, Jerusalem 91904, Israel
Supported by ERC grant 280157, ISF grant No. 039-9494.
Leo
Zhou
Leo Zhou
Department of Physics, Harvard University, Cambridge, MA 02138, USA
https://orcid.org/0000-0001-7598-8621
Supported by ERC grant 280157, MIT MISTI.
10.4230/LIPIcs.ITCS.2019.2
See extended version at URL: http://arxiv.org/abs/1804.11084.
http://arxiv.org/abs/1804.11084
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Dorit Aharonov and Leo Zhou
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
On Solving Linear Systems in Sublinear Time
We study sublinear algorithms that solve linear systems locally. In the classical version of this problem the input is a matrix S in R^{n x n} and a vector b in R^n in the range of S, and the goal is to output x in R^n satisfying Sx=b. For the case when the matrix S is symmetric diagonally dominant (SDD), the breakthrough algorithm of Spielman and Teng [STOC 2004] approximately solves this problem in near-linear time (in the input size which is the number of non-zeros in S), and subsequent papers have further simplified, improved, and generalized the algorithms for this setting.
Here we focus on computing one (or a few) coordinates of x, which potentially allows for sublinear algorithms. Formally, given an index u in [n] together with S and b as above, the goal is to output an approximation x^_u for x^*_u, where x^* is a fixed solution to Sx=b.
Our results show that there is a qualitative gap between SDD matrices and the more general class of positive semidefinite (PSD) matrices. For SDD matrices, we develop an algorithm that approximates a single coordinate x_{u} in time that is polylogarithmic in n, provided that S is sparse and has a small condition number (e.g., Laplacian of an expander graph). The approximation guarantee is additive | x^_u-x^*_u | <=epsilon | x^* |_infty for accuracy parameter epsilon>0. We further prove that the condition-number assumption is necessary and tight.
In contrast to the SDD matrices, we prove that for certain PSD matrices S, the running time must be at least polynomial in n (for the same additive approximation), even if S has bounded sparsity and condition number.
Linear systems
Laplacian solver
Sublinear time
Randomized linear algebra
Theory of computation~Streaming, sublinear and near linear time algorithms
3:1-3:19
Regular Paper
Full version at https://arxiv.org/abs/1809.02995.
Alexandr
Andoni
Alexandr Andoni
Columbia University, New York, NY, USA
Work supported in part by Simons Foundation (#491119), NSF grants CCF-1617955 and CCF-1740833.
Robert
Krauthgamer
Robert Krauthgamer
Weizmann Institute of Science, Rehovot, Israel
Work supported in part by ONR Award N00014-18-1-2364, the Israel Science Foundation grant #1086/18, a Minerva Foundation grant, and a Google Faculty Research Award.
Yosef
Pogrow
Yosef Pogrow
Weizmann Institute of Science, Rehovot, Israel
10.4230/LIPIcs.ITCS.2019.3
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Alexandr Andoni, Robert Krauthgamer, and Yosef Pogrow
Creative Commons Attribution 3.0 Unported license
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Placing Conditional Disclosure of Secrets in the Communication Complexity Universe
In the conditional disclosure of secrets (CDS) problem (Gertner et al., J. Comput. Syst. Sci., 2000) Alice and Bob, who hold n-bit inputs x and y respectively, wish to release a common secret z to Carol (who knows both x and y) if and only if the input (x,y) satisfies some predefined predicate f. Alice and Bob are allowed to send a single message to Carol which may depend on their inputs and some shared randomness, and the goal is to minimize the communication complexity while providing information-theoretic security.
Despite the growing interest in this model, very few lower-bounds are known. In this paper, we relate the CDS complexity of a predicate f to its communication complexity under various communication games. For several basic predicates our results yield tight, or almost tight, lower-bounds of Omega(n) or Omega(n^{1-epsilon}), providing an exponential improvement over previous logarithmic lower-bounds.
We also define new communication complexity classes that correspond to different variants of the CDS model and study the relations between them and their complements. Notably, we show that allowing for imperfect correctness can significantly reduce communication - a seemingly new phenomenon in the context of information-theoretic cryptography. Finally, our results show that proving explicit super-logarithmic lower-bounds for imperfect CDS protocols is a necessary step towards proving explicit lower-bounds against the class AM, or even AM cap coAM - a well known open problem in the theory of communication complexity. Thus imperfect CDS forms a new minimal class which is placed just beyond the boundaries of the "civilized" part of the communication complexity world for which explicit lower-bounds are known.
Conditional Disclosure of Secrets
Information-Theoretic Security
Theory of computation~Communication complexity
Theory of computation~Cryptographic protocols
4:1-4:14
Regular Paper
The full version of this paper is available as https://www.eng.tau.ac.il/~bennyap/publications.html.
Benny
Applebaum
Benny Applebaum
Tel Aviv University, Tel Aviv, Israel, https://www.eng.tau.ac.il/~bennyap/
Supported by the European Union’s Horizon 2020 Programme (ERC-StG-2014-2020) under grant agreement no. 639813 ERC-CLC, by an ICRC grant and by the Check Point Institute for Information Security.
Prashant Nalini
Vasudevan
Prashant Nalini Vasudevan
UC Berkeley, Berkeley, USA, http://people.eecs.berkeley.edu/~prashvas
This work was done in part while the author was visiting Tel Aviv University. Supported in part by NSF Grant CNS-1350619, and by the Defense Advanced Research Projects Agency (DARPA) and the U.S. Army Research Office under contracts W911NF-15-C-0226 and W911NF-15-C-0236.
10.4230/LIPIcs.ITCS.2019.4
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Benny Applebaum and Prashant N. Vasudevan
Creative Commons Attribution 3.0 Unported license
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Bitcoin: A Natural Oligopoly
Although Bitcoin was intended to be a decentralized digital currency, in practice, mining power is quite concentrated. This fact is a persistent source of concern for the Bitcoin community.
We provide an explanation using a simple model to capture miners' incentives to invest in equipment. In our model, n miners compete for a prize of fixed size. Each miner chooses an investment q_i, incurring cost c_i q_i, and then receives reward (q_i^alpha)/(sum_j q_j^alpha), for some alpha >= 1. When c_i = c_j for all i,j, and alpha = 1, there is a unique equilibrium where all miners invest equally. However, we prove that under seemingly mild deviations from this model, equilibrium outcomes become drastically more centralized. In particular,
- When costs are asymmetric, if miner i chooses to invest, then miner j has market share at least 1-c_j/c_i. That is, if miner j has costs that are (e.g.) 20% lower than those of miner i, then miner j must control at least 20% of the total mining power.
- In the presence of economies of scale (alpha > 1), every market participant has a market share of at least 1-1/(alpha), implying that the market features at most alpha/(alpha - 1) miners in total.
We discuss the implications of our results for the future design of cryptocurrencies. In particular, our work further motivates the study of protocols that minimize "orphaned" blocks, proof-of-stake protocols, and incentive compatible protocols.
Bitcoin
Cryptocurrencies
Rent-Seeking Competition
Theory of computation~Quality of equilibria
5:1-5:1
Regular Paper
Full version available on arXiv at https://arxiv.org/abs/1811.08572.
Nick
Arnosti
Nick Arnosti
Columbia University, New York City, NY, USA
S. Matthew
Weinberg
S. Matthew Weinberg
Princeton University, Princeton, NJ, USA
Supported by NSF CCF-1717899.
10.4230/LIPIcs.ITCS.2019.5
Nicholas Arnosti and S. Matthew Weinberg
Creative Commons Attribution 3.0 Unported license
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A Simple Sublinear-Time Algorithm for Counting Arbitrary Subgraphs via Edge Sampling
In the subgraph counting problem, we are given a (large) input graph G(V, E) and a (small) target graph H (e.g., a triangle); the goal is to estimate the number of occurrences of H in G. Our focus here is on designing sublinear-time algorithms for approximately computing number of occurrences of H in G in the setting where the algorithm is given query access to G. This problem has been studied in several recent papers which primarily focused on specific families of graphs H such as triangles, cliques, and stars. However, not much is known about approximate counting of arbitrary graphs H in the literature. This is in sharp contrast to the closely related subgraph enumeration problem that has received significant attention in the database community as the database join problem. The AGM bound shows that the maximum number of occurrences of any arbitrary subgraph H in a graph G with m edges is O(m^{rho(H)}), where rho(H) is the fractional edge-cover of H, and enumeration algorithms with matching runtime are known for any H.
We bridge this gap between subgraph counting and subgraph enumeration by designing a simple sublinear-time algorithm that can estimate the number of occurrences of any arbitrary graph H in G, denoted by #H, to within a (1 +/- epsilon)-approximation with high probability in O(m^{rho(H)}/#H) * poly(log(n),1/epsilon) time. Our algorithm is allowed the standard set of queries for general graphs, namely degree queries, pair queries and neighbor queries, plus an additional edge-sample query that returns an edge chosen uniformly at random. The performance of our algorithm matches those of Eden et al. [FOCS 2015, STOC 2018] for counting triangles and cliques and extend them to all choices of subgraph H under the additional assumption of edge-sample queries.
Sublinear-time algorithms
Subgraph counting
AGM bound
Theory of computation~Streaming, sublinear and near linear time algorithms
6:1-6:20
Regular Paper
A full version is available on arXiv [Sepehr Assadi et al., 2018], https://arxiv.org/abs/1811.07780.
Sepehr
Assadi
Sepehr Assadi
Department of Computer and Information Science, University of Pennsylvania, Philadelphia, PA, USA
Supported in part by the National Science Foundation grant CCF-1617851.
Michael
Kapralov
Michael Kapralov
School of Computer and Communication Sciences, EPFL, Lausanne, Switzerland
Supported in part by ERC Starting Grant 759471.
Sanjeev
Khanna
Sanjeev Khanna
Department of Computer and Information Science, University of Pennsylvania, Philadelphia, PA, USA
Supported in part by the National Science Foundation grants CCF-1617851 and CCF-1763514.
10.4230/LIPIcs.ITCS.2019.6
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Sepehr Assadi, Michael Kapralov, and Sanjeev Khanna
Creative Commons Attribution 3.0 Unported license
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Tensor Network Complexity of Multilinear Maps
We study tensor networks as a model of arithmetic computation for evaluating multilinear maps. These capture any algorithm based on low border rank tensor decompositions, such as O(n^{omega+epsilon}) time matrix multiplication, and in addition many other algorithms such as O(n log n) time discrete Fourier transform and O^*(2^n) time for computing the permanent of a matrix. However tensor networks sometimes yield faster algorithms than those that follow from low-rank decompositions. For instance the fastest known O(n^{(omega +epsilon)t}) time algorithms for counting 3t-cliques can be implemented with tensor networks, even though the underlying tensor has border rank n^{3t} for all t >= 2. For counting homomorphisms of a general pattern graph P into a host graph on n vertices we obtain an upper bound of O(n^{(omega+epsilon)bw(P)/2}) where bw(P) is the branchwidth of P. This essentially matches the bound for counting cliques, and yields small improvements over previous algorithms for many choices of P.
While powerful, the model still has limitations, and we are able to show a number of unconditional lower bounds for various multilinear maps, including:
b) an Omega(n^{bw(P)}) time lower bound for counting homomorphisms from P to an n-vertex graph, matching the upper bound if omega = 2. In particular for P a v-clique this yields an Omega(n^{ceil[2v/3]}) time lower bound for counting v-cliques, and for P a k-uniform v-hyperclique we obtain an Omega(n^v) time lower bound for k >= 3, ruling out tensor networks as an approach to obtaining non-trivial algorithms for hyperclique counting and the Max-3-CSP problem.
c) an Omega(2^{0.918n}) time lower bound for the permanent of an n x n matrix.
arithmetic complexity
lower bound
multilinear map
tensor network
Theory of computation~Models of computation
Theory of computation~Computational complexity and cryptography
Theory of computation~Design and analysis of algorithms
7:1-7:21
Regular Paper
Per Austrin was funded by the Swedish Research Council, Grant 621-2012-4546. Petteri Kaski has received funding from the European Research Council, Grant 338077. Kaie Kubjas was supported by Marie Skłodowska-Curie Grant 748354.
A full version of this paper appears at https://arxiv.org/abs/1712.09630.
Per
Austrin
Per Austrin
School of Computer Science and Communication, KTH Royal Institute of Technology, Stockholm, Sweden
Petteri
Kaski
Petteri Kaski
Department of Computer Science, Aalto University, Helsinki, Finland
Kaie
Kubjas
Kaie Kubjas
Department of Mathematics and Systems Analysis, Aalto University, Helsinki, Finland, and, Laboratoire d'Informatique de Paris 6, Sorbonne Université, Paris, France
10.4230/LIPIcs.ITCS.2019.7
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http://dx.doi.org/10.1137/1.9781611973730.111
Per Austrin, Petteri Kaski, and Kaie Kubjas
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
A #SAT Algorithm for Small Constant-Depth Circuits with PTF Gates
We show that there is a zero-error randomized algorithm that, when given a small constant-depth Boolean circuit C made up of gates that compute constant-degree Polynomial Threshold functions or PTFs (i.e., Boolean functions that compute signs of constant-degree polynomials), counts the number of satisfying assignments to C in significantly better than brute-force time.
Formally, for any constants d,k, there is an epsilon > 0 such that the zero-error randomized algorithm counts the number of satisfying assignments to a given depth-d circuit C made up of k-PTF gates such that C has size at most n^{1+epsilon}. The algorithm runs in time 2^{n-n^{Omega(epsilon)}}.
Before our result, no algorithm for beating brute-force search was known for counting the number of satisfying assignments even for a single degree-k PTF (which is a depth-1 circuit of linear size).
The main new tool is the use of a learning algorithm for learning degree-1 PTFs (or Linear Threshold Functions) using comparison queries due to Kane, Lovett, Moran and Zhang (FOCS 2017). We show that their ideas fit nicely into a memoization approach that yields the #SAT algorithms.
SAT
Polynomial Threshold Functions
Constant-depth Boolean Circuits
Linear Decision Trees
Zero-error randomized algorithms
Theory of computation~Circuit complexity
8:1-8:20
Regular Paper
Swapnam
Bajpai
Swapnam Bajpai
Indian Institute of Technology, Bombay, Mumbai, India
Vaibhav
Krishan
Vaibhav Krishan
Indian Institute of Technology, Bombay, Mumbai, India
Deepanshu
Kush
Deepanshu Kush
Indian Institute of Technology, Bombay, Mumbai, India
Nutan
Limaye
Nutan Limaye
Indian Institute of Technology, Bombay, Mumbai, India
Funded by Mathematical Research Impact Centric Support (MATRICS), SERB.
Srikanth
Srinivasan
Srikanth Srinivasan
Department of Mathematics, Indian Institute of Technology Bombay, Mumbai, India
Funded by Mathematical Research Impact Centric Support (MATRICS), SERB. Work partially done while visiting the Simons Institute for the Theory of Computing, Berkeley.
10.4230/LIPIcs.ITCS.2019.8
Amir Abboud and Karl Bringmann. Tighter Connections Between Formula-SAT and Shaving Logs. In 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, pages 8:1-8:18, 2018.
Amir Abboud, Thomas Dueholm Hansen, Virginia Vassilevska Williams, and Ryan Williams. Simulating branching programs with edit distance and friends: or: a polylog shaved is a lower bound made. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 375-388, 2016.
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Swapnam Bajpai, Vaibhav Krishan, Deepanshu Kush, Nutan Limaye, and Srikanth Srinivasan
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Small-Set Expansion in Shortcode Graph and the 2-to-2 Conjecture
Dinur, Khot, Kindler, Minzer and Safra (2016) recently showed that the (imperfect completeness variant of) Khot's 2 to 2 games conjecture follows from a combinatorial hypothesis about the soundness of a certain "Grassmanian agreement tester". In this work, we show that soundness of Grassmannian agreement tester follows from a conjecture we call the "Shortcode Expansion Hypothesis" characterizing the non-expanding sets of the degree-two Short code graph. We also show the latter conjecture is equivalent to a characterization of the non-expanding sets in the Grassman graph, as hypothesized by a follow-up paper of Dinur et al. (2017).
Following our work, Khot, Minzer and Safra (2018) proved the "Shortcode Expansion Hypothesis". Combining their proof with our result and the reduction of Dinur et al. (2016), completes the proof of the 2 to 2 conjecture with imperfect completeness. We believe that the Shortcode graph provides a useful view of both the hypothesis and the reduction, and might be suitable for obtaining new hardness reductions.
Unique Games Conjecture
Small-Set Expansion
Grassmann Graph
Shortcode
Theory of computation~Problems, reductions and completeness
9:1-9:12
Regular Paper
A full version of the paper is available at https://arxiv.org/abs/1804.08662.
Boaz
Barak
Boaz Barak
Harvard University School of Engineering and Applied Sciences, Cambridge, USA
Supported by NSF awards CCF 1565264 and CNS 1618026 and a Simons Investigator Fellowship.
Pravesh K.
Kothari
Pravesh K. Kothari
Princeton University and IAS Princeton, USA
Supported by Schmidt foundation fellowship and NSF CCF-1412958.
David
Steurer
David Steurer
ETH Zurich, Zurich, Switzerland
10.4230/LIPIcs.ITCS.2019.9
Mitali Bafna, Chi-Ning Chou, and Zhao Song. An Exposition of Dinur-Khot-Kindler-Minzer-Safra Proof for the 2-to-2 Games Conjecture, 2018. URL: http://boazbarak.org/dkkmsnotes.pdf.
http://boazbarak.org/dkkmsnotes.pdf
Boaz Barak, Parikshit Gopalan, Johan Håstad, Raghu Meka, Prasad Raghavendra, and David Steurer. Making the long code shorter. SIAM J. Comput., 44(5):1287-1324, 2015. URL: http://dx.doi.org/10.1137/130929394.
http://dx.doi.org/10.1137/130929394
Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. Towards a Proof of the 2-to-1 Games Conjecture? Electronic Colloquium on Computational Complexity (ECCC), 23:198, 2016. URL: http://eccc.hpi-web.de/report/2016/198.
http://eccc.hpi-web.de/report/2016/198
Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. On Non-Optimally Expanding Sets in Grassmann Graphs. STOC, 24:94, 2018.
Subhash Khot. On the Power of Unique 2-Prover 1-Round Games. In Proceedings of the 17th Annual IEEE Conference on Computational Complexity, Montréal, Québec, Canada, May 21-24, 2002, page 25, 2002. URL: http://dx.doi.org/10.1109/CCC.2002.1004334.
http://dx.doi.org/10.1109/CCC.2002.1004334
Subhash Khot, Dor Minzer, and Muli Safra. On independent sets, 2-to-2 games, and Grassmann graphs. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 576-589, 2017. URL: http://dx.doi.org/10.1145/3055399.3055432.
http://dx.doi.org/10.1145/3055399.3055432
Subhash Khot, Dor Minzer, and Muli Safra. Pseudorandom Sets in Grassmann Graph have Near-Perfect Expansion. Electronic Colloquium on Computational Complexity (ECCC), 25:6, 2018. URL: https://eccc.weizmann.ac.il/report/2018/006.
https://eccc.weizmann.ac.il/report/2018/006
Boaz Barak, Pravesh K. Kothari, and David Steurer
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Algorithms, Bounds, and Strategies for Entangled XOR Games
Entangled games are a quantum analog of constraint satisfaction problems and have had important applications to quantum complexity theory, quantum cryptography, and the foundations of quantum mechanics. Given a game, the basic computational problem is to compute its entangled value: the supremum success probability attainable by a quantum strategy. We study the complexity of computing the (commuting-operator) entangled value omega^* of entangled XOR games with any number of players. Based on a duality theory for systems of operator equations, we introduce necessary and sufficient criteria for an XOR game to have omega^* = 1, and use these criteria to derive the following results:
1) An algorithm for symmetric games that decides in polynomial time whether omega^* = 1 or omega^* < 1, a task that was not previously known to be decidable, together with a simple tensor-product strategy that achieves value 1 in the former case. The only previous candidate algorithm for this problem was the Navascués-Pironio-Acín (also known as noncommutative Sum of Squares or ncSoS) hierarchy, but no convergence bounds were known.
2) A family of games with three players and with omega^* < 1, where it takes doubly exponential time for the ncSoS algorithm to witness this. By contrast, our algorithm runs in polynomial time.
3) Existence of an unsatisfiable phase for random (non-symmetric) XOR games. We show that there exists a constant C_k^{unsat} depending only on the number k of players, such that a random k-XOR game over an alphabet of size n has omega^* < 1 with high probability when the number of clauses is above C_k^{unsat} n.
4) A lower bound of Omega(n log(n)/log log(n)) on the number of levels in the ncSoS hierarchy required to detect unsatisfiability for most random 3-XOR games. This is in contrast with the classical case where the (3n)^{th} level of the sum-of-squares hierarchy is equivalent to brute-force enumeration of all possible solutions.
Nonlocal games
XOR Games
Pseudotelepathy games
Multipartite entanglement
Theory of computation~Quantum complexity theory
10:1-10:18
Regular Paper
A full version of the paper is available at https://arxiv.org/abs/1801.00821.
Adam
Bene Watts
Adam Bene Watts
MIT Center for Theoretical Physics, 77 Massachusetts Ave, 6-304, Cambridge, MA, USA
ABW was supported by NSF grant CCF-1729369.
Aram W.
Harrow
Aram W. Harrow
MIT Center for Theoretical Physics, 77 Massachusetts Ave, 6-304, Cambridge, MA, USA
AWH was funded by NSF grants CCF-1452616, CCF-1729369, ARO contract W911NF-17-1-0433 and the MIT-IBM Watson AI Lab under the project Machine Learning in Hilbert space.
Gurtej
Kanwar
Gurtej Kanwar
MIT Center for Theoretical Physics, 77 Massachusetts Ave, 6-304, Cambridge, MA, USA
GK was supported by DOE grant DE-SC0011090.
Anand
Natarajan
Anand Natarajan
California Institute of Technology, 1200 E. California Blvd, Pasadena, CA, USA
AN was supported by NSF grant CCF-1452616.
10.4230/LIPIcs.ITCS.2019.10
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http://arxiv.org/abs/0911.4007
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http://dx.doi.org/10.1016/S0304-3975(00)00157-2
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http://arxiv.org/abs/1610.03133
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http://arxiv.org/abs/1801.03821v2
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http://arxiv.org/abs/0803.4290
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http://arxiv.org/abs/1609.01652
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http://dx.doi.org/10.1063/1.4938052
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http://arxiv.org/abs/quant-ph/0702189
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http://arxiv.org/abs/1606.03140
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Adam Bene Watts, Aram W. Harrow, Gurtej Kanwar, and Anand Natarajan
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Testing Local Properties of Arrays
We study testing of local properties in one-dimensional and multi-dimensional arrays. A property of d-dimensional arrays f:[n]^d -> Sigma is k-local if it can be defined by a family of k x ... x k forbidden consecutive patterns. This definition captures numerous interesting properties. For example, monotonicity, Lipschitz continuity and submodularity are 2-local; convexity is (usually) 3-local; and many typical problems in computational biology and computer vision involve o(n)-local properties.
In this work, we present a generic approach to test all local properties of arrays over any finite (and not necessarily bounded size) alphabet. We show that any k-local property of d-dimensional arrays is testable by a simple canonical one-sided error non-adaptive epsilon-test, whose query complexity is O(epsilon^{-1}k log{(epsilon n)/k}) for d = 1 and O(c_d epsilon^{-1/d} k * n^{d-1}) for d > 1. The queries made by the canonical test constitute sphere-like structures of varying sizes, and are completely independent of the property and the alphabet Sigma. The query complexity is optimal for a wide range of parameters: For d=1, this matches the query complexity of many previously investigated local properties, while for d > 1 we design and analyze new constructions of k-local properties whose one-sided non-adaptive query complexity matches our upper bounds. For some previously studied properties, our method provides the first known sublinear upper bound on the query complexity.
Property Testing
Local Properties
Monotonicity Testing
Hypergrid
Pattern Matching
Theory of computation~Sketching and sampling
11:1-11:20
Regular Paper
A full version of the paper is available at [Ben-Eliezer, 2018], https://eccc.weizmann.ac.il/report/2018/196/.
Omri
Ben-Eliezer
Omri Ben-Eliezer
Tel Aviv University, Tel Aviv 69978, Israel
10.4230/LIPIcs.ITCS.2019.11
N. Alon, O. Ben-Eliezer, and E. Fischer. Testing Hereditary Properties of Ordered Graphs and Matrices. In Proceedings of the 58th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 848-858, 2017.
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https://eccc.weizmann.ac.il/report/2018/196/
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Omri Ben-Eliezer
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The Complexity of User Retention
This paper studies families of distributions T that are amenable to retentive learning, meaning that an expert can retain users that seek to predict their future, assuming user attributes are sampled from T and exposed gradually over time. Limited attention span is the main problem experts face in our model. We make two contributions.
First, we formally define the notions of retentively learnable distributions and properties. Along the way, we define a retention complexity measure of distributions and a natural class of retentive scoring rules that model the way users evaluate experts they interact with. These rules are shown to be tightly connected to truth-eliciting "proper scoring rules" studied in Decision Theory since the 1950's [McCarthy, PNAS 1956].
Second, we take a first step towards relating retention complexity to other measures of significance in computational complexity. In particular, we show that linear properties (over the binary field) are retentively learnable, whereas random Low Density Parity Check (LDPC) codes have, with high probability, maximal retention complexity. Intriguingly, these results resemble known results from the field of property testing and suggest that deeper connections between retentive distributions and locally testable properties may exist.
retentive learning
retention complexity
information elicitation
proper scoring rules
Theory of computation~Models of computation
12:1-12:30
Regular Paper
This work was supported by the Israeli Science Foundation under grant number 1501/14.
https://eccc.weizmann.ac.il/report/2017/160/
Eli
Ben-Sasson
Eli Ben-Sasson
Department of Computer Science, Technion, Haifa, Israel
https://orcid.org/0000-0002-0708-0483
Eden
Saig
Eden Saig
Department of Computer Science, Technion, Haifa, Israel
https://orcid.org/0000-0002-0810-2218
10.4230/LIPIcs.ITCS.2019.12
Sanjeev Arora and Boaz Barak. Computational complexity: a modern approach. Cambridge University Press, 2009.
Amir Ban and Nati Linial. The dynamics of reputation systems. In Proceedings of the 13th Conference on Theoretical Aspects of Rationality and Knowledge, pages 91-100. ACM, 2011.
Ayelet Ben-Sasson, Eli Ben-Sasson, Kayla Jacobs, and Eden Saig. Baby CROINC: An Online, Crowd-based, Expert-curated System for Monitoring Child Development. In Proceedings of the 11th EAI International Conference on Pervasive Computing Technologies for Healthcare, PervasiveHealth '17, pages 110-119, New York, NY, USA, 2017. ACM. URL: http://dx.doi.org/10.1145/3154862.3154887.
http://dx.doi.org/10.1145/3154862.3154887
Eli Ben-Sasson, Prahladh Harsha, and Sofya Raskhodnikova. Some 3CNF properties are hard to test. SIAM Journal on Computing, 35(1):1-21, 2005.
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http://dx.doi.org/10.1145/100216.100225
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Eli Ben-Sasson and Eden Saig
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Torus Polynomials: An Algebraic Approach to ACC Lower Bounds
We propose an algebraic approach to proving circuit lower bounds for ACC^0 by defining and studying the notion of torus polynomials. We show how currently known polynomial-based approximation results for AC^0 and ACC^0 can be reformulated in this framework, implying that ACC^0 can be approximated by low-degree torus polynomials. Furthermore, as a step towards proving ACC^0 lower bounds for the majority function via our approach, we show that MAJORITY cannot be approximated by low-degree symmetric torus polynomials. We also pose several open problems related to our framework.
Circuit complexity
ACC
lower bounds
polynomials
Theory of computation~Circuit complexity
13:1-13:16
Regular Paper
Supported by NSF grant CCF-1614023.
https://arxiv.org/abs/1804.08176
Abhishek
Bhrushundi
Abhishek Bhrushundi
Rutgers University, New Brunswick, USA
Part of this work was done when the author was visiting the University of California, San Diego. Research supported in part by Rutgers AAUP-AFT TA-GA Professional Development Fund.
Kaave
Hosseini
Kaave Hosseini
University of California, San Diego, USA
Shachar
Lovett
Shachar Lovett
University of California, San Diego, USA
Sankeerth
Rao
Sankeerth Rao
University of California, San Diego, USA
10.4230/LIPIcs.ITCS.2019.13
Richard Beigel and Jun Tarui. On ACC (circuit complexity). In Proceedings of the 32nd Annual Symposium on Foundations of Computer Science, FOCS 1991, pages 783-792. IEEE, 1991.
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Johan Håstad. The shrinkage exponent of De Morgan formulas is 2. SIAM Journal on Computing, 27(1):48-64, 1998.
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Ryan Williams. Nonuniform ACC circuit lower bounds. Journal of the ACM (JACM), 61(1):2, 2014.
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Abhishek Bhrushundi, Kaave Hosseini, Shachar Lovett, and Sankeerth Rao
Creative Commons Attribution 3.0 Unported license
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Almost Envy-Free Allocations with Connected Bundles
We study the existence of allocations of indivisible goods that are envy-free up to one good (EF1), under the additional constraint that each bundle needs to be connected in an underlying item graph G. When the items are arranged in a path, we show that EF1 allocations are guaranteed to exist for arbitrary monotonic utility functions over bundles, provided that either there are at most four agents, or there are any number of agents but they all have identical utility functions. Our existence proofs are based on classical arguments from the divisible cake-cutting setting, and involve discrete analogues of cut-and-choose, of Stromquist's moving-knife protocol, and of the Su-Simmons argument based on Sperner's lemma. Sperner's lemma can also be used to show that on a path, an EF2 allocation exists for any number of agents. Except for the results using Sperner's lemma, all of our procedures can be implemented by efficient algorithms. Our positive results for paths imply the existence of connected EF1 or EF2 allocations whenever G is traceable, i.e., contains a Hamiltonian path. For the case of two agents, we completely characterize the class of graphs G that guarantee the existence of EF1 allocations as the class of graphs whose biconnected components are arranged in a path. This class is strictly larger than the class of traceable graphs; one can check in linear time whether a graph belongs to this class, and if so return an EF1 allocation.
Envy-free Division
Cake-cutting
Resource Allocation
Algorithmic Game Theory
Theory of computation~Algorithmic game theory
Mathematics of computing~Combinatoric problems
Mathematics of computing~Graph theory
14:1-14:21
Regular Paper
D. Peters is supported by ERC grant 639945 (ACCORD). A. Igarashi is supported by KAKENHI (Grant-in-Aid for JSPS Fellows, No. 18J00997) Japan. While working on this paper, W. S. Zwicker was supported by the Oliver Smithies Visiting Fellowship at Balliol College, Oxford.
[V. Bilò et al., 2018], https://arxiv.org/abs/1808.09406 with omitted proofs, and additional results
Vittorio
Bilò
Vittorio Bilò
University of Salento, Lecce, Italy
Ioannis
Caragiannis
Ioannis Caragiannis
University of Patras, Rion-Patras, Greece
Michele
Flammini
Michele Flammini
Gran Sasso Science Institute and University of L'Aquila, L'Aquila, Italy
Ayumi
Igarashi
Ayumi Igarashi
Kyushu University, Fukuoka, Japan
Gianpiero
Monaco
Gianpiero Monaco
University of L'Aquila, L'Aquila, Italy
Dominik
Peters
Dominik Peters
University of Oxford, Oxford, U.K.
Cosimo
Vinci
Cosimo Vinci
University of L'Aquila, L'Aquila, Italy
William S.
Zwicker
William S. Zwicker
Union College, Schenectady, USA
10.4230/LIPIcs.ITCS.2019.14
H. Aziz, I. Caragiannis, and A. Igarashi. Fair allocation of combinations of indivisible goods and chores. CoRR, 2018. arXiv:1807.10684.
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Vittorio Bilò, Ioannis Caragiannis, Michele Flammini, Ayumi Igarashi, Gianpiero Monaco, Dominik Peters, Cosimo Vinci, and William S. Zwicker
Creative Commons Attribution 3.0 Unported license
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"Quantum Supremacy" and the Complexity of Random Circuit Sampling
A critical goal for the field of quantum computation is quantum supremacy - a demonstration of any quantum computation that is prohibitively hard for classical computers. It is both a necessary milestone on the path to useful quantum computers as well as a test of quantum theory in the realm of high complexity. A leading near-term candidate, put forth by the Google/UCSB team, is sampling from the probability distributions of randomly chosen quantum circuits, called Random Circuit Sampling (RCS).
While RCS was defined with experimental realization in mind, we give strong complexity-theoretic evidence for the classical hardness of RCS, placing it on par with the best theoretical proposals for supremacy. Specifically, we show that RCS satisfies an average-case hardness condition - computing output probabilities of typical quantum circuits is as hard as computing them in the worst-case, and therefore #P-hard. Our reduction exploits the polynomial structure in the output amplitudes of random quantum circuits, enabled by the Feynman path integral. In addition, it follows from known results that RCS also satisfies an anti-concentration property, namely that errors in estimating output probabilities are small with respect to the probabilities themselves. This makes RCS the first proposal for quantum supremacy with both of these properties. We also give a natural condition under which an existing statistical measure, cross-entropy, verifies RCS, as well as describe a new verification measure which in some formal sense maximizes the information gained from experimental samples.
quantum supremacy
average-case hardness
verification
Theory of computation~Quantum complexity theory
15:1-15:2
Regular Paper
A.B., B.F., C.N. and U.V. were supported by ARO grant W911NF-12-1-0541 and NSF grant CCF-1410022 and a Vannevar Bush faculty fellowship. B.F. is supported in part by an Air Force Office of Scientific Research Young Investigator Program award number FA9550-18-1-0148. Parts of this work were done at the Kavli Institute for Theoretical Physics. Portions of this paper are a contribution of NIST, an agency of the US government, and are not subject to US copyright.
A full version of this paper, including all proofs, is available at [Bouland et al., 2018], https://doi.org/10.1038/s41567-018-0318-2.
Adam
Bouland
Adam Bouland
Electrical Engineering and Computer Sciences, University of California, Berkeley, 387 Soda Hall Berkeley, CA 94720, U.S.A.
https://orcid.org/0000-0002-8556-8337
Bill
Fefferman
Bill Fefferman
Electrical Engineering and Computer Sciences, University of California, Berkeley, 387 Soda Hall Berkeley, CA 94720, U.S.A. , Joint Center for Quantum Information and Computer Science (QuICS), University of Maryland/NIST, Bldg 224 Stadium Dr Room 3100, College Park, MD 20742, U.S.A.
https://orcid.org/0000-0002-9627-0210
Chinmay
Nirkhe
Chinmay Nirkhe
Electrical Engineering and Computer Sciences, University of California, Berkeley, 387 Soda Hall Berkeley, CA 94720, U.S.A.
https://orcid.org/0000-0002-5808-4994
Umesh
Vazirani
Umesh Vazirani
Electrical Engineering and Computer Sciences, University of California, Berkeley, 387 Soda Hall Berkeley, CA 94720, U.S.A.
10.4230/LIPIcs.ITCS.2019.15
Adam Bouland, Bill Fefferman, Chinmay Nirkhe, and Umesh Vazirani. On the complexity and verification of quantum random circuit sampling. Nature Physics, 2018. URL: http://dx.doi.org/10.1038/s41567-018-0318-2.
http://dx.doi.org/10.1038/s41567-018-0318-2
Adam Bouland, Bill Fefferman, Chinmay Nirkhe, and Umesh Vazirani
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Adversarially Robust Property-Preserving Hash Functions
Property-preserving hashing is a method of compressing a large input x into a short hash h(x) in such a way that given h(x) and h(y), one can compute a property P(x, y) of the original inputs. The idea of property-preserving hash functions underlies sketching, compressed sensing and locality-sensitive hashing.
Property-preserving hash functions are usually probabilistic: they use the random choice of a hash function from a family to achieve compression, and as a consequence, err on some inputs. Traditionally, the notion of correctness for these hash functions requires that for every two inputs x and y, the probability that h(x) and h(y) mislead us into a wrong prediction of P(x, y) is negligible. As observed in many recent works (incl. Mironov, Naor and Segev, STOC 2008; Hardt and Woodruff, STOC 2013; Naor and Yogev, CRYPTO 2015), such a correctness guarantee assumes that the adversary (who produces the offending inputs) has no information about the hash function, and is too weak in many scenarios.
We initiate the study of adversarial robustness for property-preserving hash functions, provide definitions, derive broad lower bounds due to a simple connection with communication complexity, and show the necessity of computational assumptions to construct such functions. Our main positive results are two candidate constructions of property-preserving hash functions (achieving different parameters) for the (promise) gap-Hamming property which checks if x and y are "too far" or "too close". Our first construction relies on generic collision-resistant hash functions, and our second on a variant of the syndrome decoding assumption on low-density parity check codes.
Hash function
compression
property-preserving
one-way communication
Theory of computation~Cryptographic primitives
16:1-16:20
Regular Paper
A full version of the paper is available at https://eprint.iacr.org/2018/1158.pdf.
Elette
Boyle
Elette Boyle
IDC Herzliya, Kanfei Nesharim Herzliya, Israel
Supported in part by ISF grant 1861/16, AFOSR Award FA9550-17-1-0069, and ERC grant 742754 (project NTSC).
Rio
LaVigne
Rio LaVigne
MIT CSAIL, 32 Vassar Street, Cambridge MA, 02139 USA
This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. 1122374. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors(s) and do not necessarily reflect the views of the National Science Foundation. This research was also supported in part by grants of the third author.
Vinod
Vaikuntanathan
Vinod Vaikuntanathan
MIT CSAIL, 32 Vassar Street, Cambridge MA, 02139 USA
Research supported in part by NSF Grants CNS-1350619 and CNS-1414119, an MIT-IBM grant and a DARPA Young Faculty Award.
10.4230/LIPIcs.ITCS.2019.16
Noga Alon, Yossi Matias, and Mario Szegedy. The Space Complexity of Approximating the Frequency Moments. In Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, Philadelphia, Pennsylvania, USA, May 22-24, 1996, pages 20-29, 1996.
Benny Applebaum, Naama Haramaty, Yuval Ishai, Eyal Kushilevitz, and Vinod Vaikuntanathan. Low-Complexity Cryptographic Hash Functions. In 8th Innovations in Theoretical Computer Science Conference, ITCS 2017, January 9-11, 2017, Berkeley, CA, USA, pages 7:1-7:31, 2017.
László Babai, Anna Gál, Peter G. Kimmel, and Satyanarayana V. Lokam. Communication Complexity of Simultaneous Messages. SIAM J. Comput., 33(1):137-166, 2003.
Michael Capalbo, Omer Reingold, Salil Vadhan, and Avi Wigderson. Randomness Conductors and Constant-degree Lossless Expanders. In Proceedings of the Thiry-fourth Annual ACM Symposium on Theory of Computing, STOC '02, pages 659-668, New York, NY, USA, 2002. ACM. URL: http://dx.doi.org/10.1145/509907.510003.
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Larry Carter and Mark N. Wegman. Universal Classes of Hash Functions (Extended Abstract). In Proceedings of the 9th Annual ACM Symposium on Theory of Computing, May 4-6, 1977, Boulder, Colorado, USA, pages 106-112, 1977.
Moses Charikar, Kevin C. Chen, and Martin Farach-Colton. Finding frequent items in data streams. Theor. Comput. Sci., 312(1):3-15, 2004.
Scott Shaobing Chen, David L. Donoho, and Michael A. Saunders. Atomic Decomposition by Basis Pursuit. SIAM Rev., 43(1):129-159, 2001.
Graham Cormode and S. Muthukrishnan. An improved data stream summary: the count-min sketch and its applications. J. Algorithms, 55(1):58-75, 2005. URL: http://dx.doi.org/10.1016/j.jalgor.2003.12.001.
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Bella Dubrov and Yuval Ishai. On the randomness complexity of efficient sampling. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing, Seattle, WA, USA, May 21-23, 2006, pages 711-720, 2006.
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Elette Boyle, Rio LaVigne, and Vinod Vaikuntanathan
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
On Closest Pair in Euclidean Metric: Monochromatic is as Hard as Bichromatic
Given a set of n points in R^d, the (monochromatic) Closest Pair problem asks to find a pair of distinct points in the set that are closest in the l_p-metric. Closest Pair is a fundamental problem in Computational Geometry and understanding its fine-grained complexity in the Euclidean metric when d=omega(log n) was raised as an open question in recent works (Abboud-Rubinstein-Williams [FOCS'17], Williams [SODA'18], David-Karthik-Laekhanukit [SoCG'18]).
In this paper, we show that for every p in R_{>= 1} cup {0}, under the Strong Exponential Time Hypothesis (SETH), for every epsilon>0, the following holds:
- No algorithm running in time O(n^{2-epsilon}) can solve the Closest Pair problem in d=(log n)^{Omega_{epsilon}(1)} dimensions in the l_p-metric.
- There exists delta = delta(epsilon)>0 and c = c(epsilon)>= 1 such that no algorithm running in time O(n^{1.5-epsilon}) can approximate Closest Pair problem to a factor of (1+delta) in d >= c log n dimensions in the l_p-metric.
In particular, our first result is shown by establishing the computational equivalence of the bichromatic Closest Pair problem and the (monochromatic) Closest Pair problem (up to n^{epsilon} factor in the running time) for d=(log n)^{Omega_epsilon(1)} dimensions.
Additionally, under SETH, we rule out nearly-polynomial factor approximation algorithms running in subquadratic time for the (monochromatic) Maximum Inner Product problem where we are given a set of n points in n^{o(1)}-dimensional Euclidean space and are required to find a pair of distinct points in the set that maximize the inner product.
At the heart of all our proofs is the construction of a dense bipartite graph with low contact dimension, i.e., we construct a balanced bipartite graph on n vertices with n^{2-epsilon} edges whose vertices can be realized as points in a (log n)^{Omega_epsilon(1)}-dimensional Euclidean space such that every pair of vertices which have an edge in the graph are at distance exactly 1 and every other pair of vertices are at distance greater than 1. This graph construction is inspired by the construction of locally dense codes introduced by Dumer-Miccancio-Sudan [IEEE Trans. Inf. Theory'03].
Closest Pair
Bichromatic Closest Pair
Contact Dimension
Fine-Grained Complexity
Theory of computation~Computational geometry
Theory of computation~Problems, reductions and completeness
17:1-17:16
Regular Paper
A full version of the paper is available at https://arxiv.org/abs/1812.00901.
Karthik
C. S.
Karthik C. S.
Weizmann Institute of Science, Rehovot, Israel
https://orcid.org/0000-0001-9105-364X
Supported by Irit Dinur’s ERC-CoG grant 772839 and BSF grant 2014371.
Pasin
Manurangsi
Pasin Manurangsi
University of California, Berkeley, USA
Supported by NSF under Grants No. CCF 1655215 and CCF 1815434.
10.4230/LIPIcs.ITCS.2019.17
Amir Abboud, Aviad Rubinstein, and Ryan Williams. Distributed PCP Theorems for Hardness of Approximation in P. CoRR, abs/1706.06407, 2017. URL: http://arxiv.org/abs/1706.06407.
http://arxiv.org/abs/1706.06407
Amir Abboud, Aviad Rubinstein, and Ryan Williams. Distributed PCP Theorems for Hardness of Approximation in P. In FOCS, pages 25-36, 2017.
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Karthik C. S. and Pasin Manurangsi
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Expander-Based Cryptography Meets Natural Proofs
We introduce new forms of attack on expander-based cryptography, and in particular on Goldreich's pseudorandom generator and one-way function. Our attacks exploit low circuit complexity of the underlying expander's neighbor function and/or of the local predicate. Our two key conceptual contributions are:
1) We put forward the possibility that the choice of expander matters in expander-based cryptography. In particular, using expanders whose neighbour function has low circuit complexity might compromise the security of Goldreich's PRG and OWF in certain settings.
2) We show that the security of Goldreich's PRG and OWF is closely related to two other long-standing problems: Specifically, to the existence of unbalanced lossless expanders with low-complexity neighbor function, and to limitations on circuit lower bounds (i.e., natural proofs). In particular, our results further motivate the investigation of affine/local unbalanced lossless expanders and of average-case lower bounds against DNF-XOR circuits.
We prove two types of technical results that support the above conceptual messages. First, we unconditionally break Goldreich's PRG when instantiated with a specific expander (whose existence we prove), for a class of predicates that match the parameters of the currently-best "hard" candidates, in the regime of quasi-polynomial stretch. Secondly, conditioned on the existence of expanders whose neighbor functions have extremely low circuit complexity, we present attacks on Goldreich's generator in the regime of polynomial stretch. As one corollary, conditioned on the existence of the foregoing expanders, we show that either the parameters of natural properties for several constant-depth circuit classes cannot be improved, even mildly; or Goldreich's generator is insecure in the regime of a large polynomial stretch, regardless of the predicate used.
Pseudorandom Generators
One-Way Functions
Expanders
Circuit Complexity
Theory of computation~Circuit complexity
Theory of computation~Cryptographic primitives
Theory of computation~Pseudorandomness and derandomization
Theory of computation~Expander graphs and randomness extractors
18:1-18:14
Regular Paper
For a full online version see [Igor Carboni Oliveira et al., 2018], https://eccc.weizmann.ac.il/report/2018/159/.
Igor
Carboni Oliveira
Igor Carboni Oliveira
Department of Computer Science, University of Oxford, UK
Rahul
Santhanam
Rahul Santhanam
Department of Computer Science, University of Oxford, UK
Roei
Tell
Roei Tell
Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel
10.4230/LIPIcs.ITCS.2019.18
Michael Alekhnovich. More on average case vs approximation complexity. ç, 20(4):755-786, 2011.
Benny Applebaum. Cryptography in Constant Parallel Time. Information Security and Cryptography. Springer, 2014.
Benny Applebaum. Cryptographic Hardness of Random Local Functions. ç, 25(3):667-722, 2016.
Benny Applebaum, Boaz Barak, and Avi Wigderson. Public-key cryptography from different assumptions. In \stoc42nd, pages 171-180. ACM, 2010.
Benny Applebaum, Andrej Bogdanov, and Alon Rosen. A dichotomy for local small-bias generators. Journal of Cryptology, 29(3):577-596, 2016.
Benny Applebaum, Yuval Ishai, and Eyal Kushilevitz. Cryptography in NC^0. \siamj, 36(4):845-888, 2006.
Benny Applebaum and Shachar Lovett. Algebraic attacks against random local functions and their countermeasures. \siamj, 47:52-79, 2018.
Benny Applebaum and Pavel Raykov. Fast pseudorandom functions based on expander graphs. In Theory of cryptography. Part I, volume 9985 of Lecture Notes in Comput. Sci., pages 27-56. Springer, Berlin, 2016.
Andrej Bogdanov and Youming Qiao. On the security of Goldreich’s one-way function. ç, 21(1):83-127, 2012.
Andrej Bogdanov and Alon Rosen. Input locality and hardness amplification. Journal of Cryptology, 26(1):144-171, 2013.
Michael Capalbo, Omer Reingold, Salil Vadhan, and Avi Wigderson. Randomness Conductors and Constant-degree Lossless Expanders. In \stoc34th, pages 659-668, 2002.
Marco L. Carmosino, Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova. Learning algorithms from natural proofs. In çc31st, page 10 (24), 2016.
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James Cook, Omid Etesami, Rachel Miller, and Luca Trevisan. On the one-way function candidate proposed by Goldreich. \toct, 6(3):Art. 14, 35, 2014.
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Ryan O'Donnell and David Witmer. Goldreich’s PRG: evidence for near-optimal polynomial stretch. In çc29th, pages 1-12. IEEE, 2014.
Igor Carboni Oliveira, Rahul Santhanam, and Roei Tell. Expander-Based Cryptography Meets Natural Proofs. \eccc, 25:159, 2018. URL: https://eccc.weizmann.ac.il/report/2018/159/.
https://eccc.weizmann.ac.il/report/2018/159/
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https://www.math.ias.edu/avi/book
Igor C. Oliveira, Rahul Santhanam, and Roei Tell
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
A Note on the Quantum Query Complexity of Permutation Symmetric Functions
It is known since the work of [Aaronson and Ambainis, 2014] that for any permutation symmetric function f, the quantum query complexity is at most polynomially smaller than the classical randomized query complexity, more precisely that R(f) = O~(Q^7(f)). In this paper, we improve this result and show that R(f) = O(Q^3(f)) for a more general class of symmetric functions. Our proof is constructive and relies largely on the quantum hardness of distinguishing a random permutation from a random function with small range from Zhandry [Zhandry, 2015].
quantum query complexity
permutation symmetric functions
Theory of computation~Quantum query complexity
19:1-19:7
Regular Paper
André
Chailloux
André Chailloux
Inria de Paris, EPI SECRET, Paris, France
10.4230/LIPIcs.ITCS.2019.19
Scott Aaronson and Andris Ambainis. The Need for Structure in Quantum Speedups. Theory of Computing, 10(6):133-166, 2014. URL: http://dx.doi.org/10.4086/toc.2014.v010a006.
http://dx.doi.org/10.4086/toc.2014.v010a006
Scott Aaronson and Yaoyun Shi. Quantum Lower Bounds for the Collision and the Element Distinctness Problems. J. ACM, 51(4):595-605, July 2004. URL: http://dx.doi.org/10.1145/1008731.1008735.
http://dx.doi.org/10.1145/1008731.1008735
A. Ambainis. Understanding Quantum Algorithms via Query Complexity. ArXiv e-prints, December 2017. URL: http://arxiv.org/abs/1712.06349.
http://arxiv.org/abs/1712.06349
Andris Ambainis. Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range. Theory of Computing, 1(3):37-46, 2005. URL: http://dx.doi.org/10.4086/toc.2005.v001a003.
http://dx.doi.org/10.4086/toc.2005.v001a003
Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf. Quantum Lower Bounds by Polynomials. J. ACM, 48(4):778-797, July 2001. URL: http://dx.doi.org/10.1145/502090.502097.
http://dx.doi.org/10.1145/502090.502097
Charles H. Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani. Strengths and Weaknesses of Quantum Computing. SIAM Journal of Computing, 26(5):1510-1523, October 1997. URL: http://dx.doi.org/10.1137/S0097539796300933.
http://dx.doi.org/10.1137/S0097539796300933
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http://dx.doi.org/10.1098/rspa.1992.0167
Samuel Kutin. Quantum Lower Bound for the Collision Problem with Small Range. Theory of Computing, 1(2):29-36, 2005. URL: http://dx.doi.org/10.4086/toc.2005.v001a002.
http://dx.doi.org/10.4086/toc.2005.v001a002
Peter W. Shor. Algorithms for Quantum Computation: Discrete Logarithms and Factoring. In IEEE Symposium on Foundations of Computer Science, pages 124-134, 1994. URL: https://citeseer.ist.psu.edu/14533.html.
https://citeseer.ist.psu.edu/14533.html
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http://dx.doi.org/10.1109/SFCS.1994.365701
Mark Zhandry. A Note on the Quantum Collision and Set Equality Problems. Quantum Info. Comput., 15(7-8):557-567, May 2015. URL: http://dl.acm.org/citation.cfm?id=2871411.2871413.
http://dl.acm.org/citation.cfm?id=2871411.2871413
Mark Zhandry. How to Record Quantum Queries, and Applications to Quantum Indifferentiability. IACR Cryptology ePrint Archive, 2018:276, 2018. URL: https://eprint.iacr.org/2018/276.
https://eprint.iacr.org/2018/276
André Chailloux
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Adaptive Boolean Monotonicity Testing in Total Influence Time
Testing monotonicity of a Boolean function f:{0,1}^n -> {0,1} is an important problem in the field of property testing. It has led to connections with many interesting combinatorial questions on the directed hypercube: routing, random walks, and new isoperimetric theorems. Denoting the proximity parameter by epsilon, the best tester is the non-adaptive O~(epsilon^{-2}sqrt{n}) tester of Khot-Minzer-Safra (FOCS 2015). A series of recent results by Belovs-Blais (STOC 2016) and Chen-Waingarten-Xie (STOC 2017) have led to Omega~(n^{1/3}) lower bounds for adaptive testers. Reducing this gap is a significant question, that touches on the role of adaptivity in monotonicity testing of Boolean functions.
We approach this question from the perspective of parametrized property testing, a concept recently introduced by Pallavoor-Raskhodnikova-Varma (ACM TOCT 2017), where one seeks to understand performance of testers with respect to parameters other than just the size. Our result is an adaptive monotonicity tester with one-sided error whose query complexity is O(epsilon^{-2}I(f)log^5 n), where I(f) is the total influence of the function. Therefore, adaptivity provably helps monotonicity testing for low influence functions.
Property Testing
Monotonicity Testing
Influence of Boolean Functions
Theory of computation~Streaming, sublinear and near linear time algorithms
20:1-20:7
Regular Paper
Deeparnab
Chakrabarty
Deeparnab Chakrabarty
Dartmouth College, Hanover, NH 03755, USA
Supported by NSF CCF-1813165.
C.
Seshadhri
C. Seshadhri
University of California, Santa Cruz, CA 95064, USA
Supported by NSF TRIPODS CCF-1740850 and NSF CCF-1813165.
10.4230/LIPIcs.ITCS.2019.20
Aleksandrs Belovs and Eric Blais. A Polynomial Lower Bound for Testing Monotonicity. In Proceedings, ACM Symposium on Theory of Computing (STOC), 2016.
Deeparnab Chakrabarty and C. Seshadhri. An o(n) Monotonicity Tester for Boolean Functions over the Hypercube. SIAM Journal on Computing (SICOMP), 45(2):461-472, 2014.
Xi Chen, Anindya De, Rocco A. Servedio, and Li-Yang Tan. Boolean Function Monotonicity Testing Requires (Almost) O(n^1/2) Non-adaptive Queries. In Proceedings, ACM Symposium on Theory of Computing (STOC), 2015.
Xi Chen, Rocco A. Servedio, and Li-Yang. Tan. New Algorithms and Lower Bounds for Monotonicity Testing. In Proceedings, IEEE Symposium on Foundations of Computer Science (FOCS), 2014.
Xi Chen, Erik Waingarten, and Jinyu Xie. Beyond Talagrand: New Lower Bounds for Testing Monotonicity and Unateness. In Proceedings, ACM Symposium on Theory of Computing (STOC), 2017.
Yevgeny Dodis, Oded Goldreich, Eric Lehman, Sofya Raskhodnikova, Dana Ron, and Alex Samorodnitsky. Improved Testing Algorithms for Monotonicity. Proceedings, International Workshop on Randomization and Computation (RANDOM), 1999.
Eldar Fischer, Eric Lehman, Ilan Newman, Sofya Raskhodnikova, and Ronitt Rubinfeld. Monotonicity Testing over General Poset Domains. Proceedings, ACM Symposium on Theory of Computing (STOC), 2002.
Oded Goldreich, Shafi Goldwasser, Eric Lehman, Dana Ron, and Alex Samordinsky. Testing Monotonicity. Combinatorica, 20:301-337, 2000.
Subhash Khot, Dor Minzer, and Muli Safra. On Monotonicity Testing and Boolean Isoperimetric Type Theorems. In Proceedings, IEEE Symposium on Foundations of Computer Science (FOCS), 2015.
Ramesh Krishnan S. Pallavoor, Sofya Raskhodnikova, and Nithin Varma. Parameterized Property Testing of Functions. ACM Transactions on Computation Theory (TOCT), 9(4), 2017.
Sofya Raskhodnikova. Monotonicity Testing. Masters Thesis, MIT, 1999.
Michel Talagrand. Isoperimetry, Logarithmic Sobolev inequalities on the Discrete Cube, and Margulis’ Graph Connectivity Theorem. Geom. Func. Anal., 3(3):295-314, 1993.
Deeparnab Chakrabarty and C. Seshadhri
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
On Locality-Sensitive Orderings and Their Applications
For any constant d and parameter epsilon > 0, we show the existence of (roughly) 1/epsilon^d orderings on the unit cube [0,1)^d, such that any two points p, q in [0,1)^d that are close together under the Euclidean metric are "close together" in one of these linear orderings in the following sense: the only points that could lie between p and q in the ordering are points with Euclidean distance at most epsilon | p - q | from p or q. These orderings are extensions of the Z-order, and they can be efficiently computed.
Functionally, the orderings can be thought of as a replacement to quadtrees and related structures (like well-separated pair decompositions). We use such orderings to obtain surprisingly simple algorithms for a number of basic problems in low-dimensional computational geometry, including (i) dynamic approximate bichromatic closest pair, (ii) dynamic spanners, (iii) dynamic approximate minimum spanning trees, (iv) static and dynamic fault-tolerant spanners, and (v) approximate nearest neighbor search.
Approximation algorithms
Data structures
Computational geometry
Theory of computation~Computational geometry
21:1-21:17
Regular Paper
Timothy M.
Chan
Timothy M. Chan
Department of Computer Science, University of Illinois at Urbana-Champaign, USA
Sariel
Har-Peled
Sariel Har-Peled
Department of Computer Science, University of Illinois at Urbana-Champaign, USA
Supported in part by NSF AF award CCF-1421231.
Mitchell
Jones
Mitchell Jones
Department of Computer Science, University of Illinois at Urbana-Champaign, USA
10.4230/LIPIcs.ITCS.2019.21
Brian Alspach. The wonderful Walecki construction. Bull. Inst. Combin. Appl., 52:7-20, 2008.
Sanjeev Arora. Polynomial time approximation schemes for Euclidean TSP and other geometric problems. J. Assoc. Comput. Mach., 45(5):753-782, September 1998. URL: http://www.cs.princeton.edu/~arora/pubs/tsp.ps.
http://www.cs.princeton.edu/~arora/pubs/tsp.ps
Marshall W. Bern. Approximate Closest-Point Queries in High Dimensions. Inform. Process. Lett., 45(2):95-99, 1993. URL: http://dx.doi.org/10.1016/0020-0190(93)90222-U.
http://dx.doi.org/10.1016/0020-0190(93)90222-U
Paul B. Callahan and S. Rao Kosaraju. Faster Algorithms for Some Geometric Graph Problems in Higher Dimensions. In Vijaya Ramachandran, editor, Proc. 4th ACM-SIAM Sympos. Discrete Alg. (SODA), pages 291-300. ACM/SIAM, 1993. URL: http://dl.acm.org/citation.cfm?id=313559.313777.
http://dl.acm.org/citation.cfm?id=313559.313777
T.-H. Hubert Chan, Mingfei Li, Li Ning, and Shay Solomon. New Doubling Spanners: Better and Simpler. SIAM J. Comput., 44(1):37-53, 2015. URL: http://dx.doi.org/10.1137/130930984.
http://dx.doi.org/10.1137/130930984
Timothy M. Chan. Approximate nearest neighbor queries revisited. Discrete Comput. Geom., 20(3):359-373, 1998. URL: http://dx.doi.org/10.1007/PL00009390.
http://dx.doi.org/10.1007/PL00009390
Timothy M. Chan. Closest-point problems simplified on the RAM. In Proc. 13th ACM-SIAM Sympos. Discrete Alg. (SODA), pages 472-473. SIAM, 2002. URL: http://dl.acm.org/citation.cfm?id=545381.545444.
http://dl.acm.org/citation.cfm?id=545381.545444
Timothy M. Chan. A minimalist’s implementation of an approximate nearest neighbor algorithm in fixed dimensions, 2006. URL: http://tmc.web.engr.illinois.edu/sss.ps.
http://tmc.web.engr.illinois.edu/sss.ps
Timothy M. Chan. Well-separated pair decomposition in linear time? Inform. Process. Lett., 107(5):138-141, 2008. URL: http://dx.doi.org/10.1016/j.ipl.2008.02.008.
http://dx.doi.org/10.1016/j.ipl.2008.02.008
Timothy M. Chan and Dimitrios Skrepetos. Dynamic data structures for approximate Hausdorff distance in the word RAM. Comput. Geom. Theory Appl., 60:37-44, 2017. URL: http://dx.doi.org/10.1016/j.comgeo.2016.08.002.
http://dx.doi.org/10.1016/j.comgeo.2016.08.002
Artur Czumaj and Hairong Zhao. Fault-Tolerant Geometric Spanners. Discrete Comput. Geom., 32(2):207-230, 2004. URL: http://www.springerlink.com/index/10.1007/s00454-004-1121-7.
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http://dx.doi.org/10.1007/s00454-004-2916-2
Timothy M. Chan, Sariel Har-Peled, and Mitchell Jones
Creative Commons Attribution 3.0 Unported license
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Pseudorandom Generators from the Second Fourier Level and Applications to AC0 with Parity Gates
A recent work of Chattopadhyay et al. (CCC 2018) introduced a new framework for the design of pseudorandom generators for Boolean functions. It works under the assumption that the Fourier tails of the Boolean functions are uniformly bounded for all levels by an exponential function. In this work, we design an alternative pseudorandom generator that only requires bounds on the second level of the Fourier tails. It is based on a derandomization of the work of Raz and Tal (ECCC 2018) who used the above framework to obtain an oracle separation between BQP and PH.
As an application, we give a concrete conjecture for bounds on the second level of the Fourier tails for low degree polynomials over the finite field F_2. If true, it would imply an efficient pseudorandom generator for AC^0[oplus], a well-known open problem in complexity theory. As a stepping stone towards resolving this conjecture, we prove such bounds for the first level of the Fourier tails.
Derandomization
Pseudorandom generator
Explicit construction
Random walk
Small-depth circuits with parity gates
Theory of computation~Pseudorandomness and derandomization
22:1-22:15
Regular Paper
Eshan
Chattopadhyay
Eshan Chattopadhyay
Department of Computer Science, Cornell University, 107 Hoy Rd, Ithaca, NY, USA
Pooya
Hatami
Pooya Hatami
Department of Computer Science, University of Texas at Austin, 2317 Speedway, Austin, TX, USA
Supported by a Simons Investigator Award #409864, David Zuckerman.
Shachar
Lovett
Shachar Lovett
Department of Computer Science, University of California San Diego, La Jolla, CA, USA
Supported by NSF grant CCF-1614023.
Avishay
Tal
Avishay Tal
Department of Computer Science, Stanford University, 353 Serra Mall, Stanford, CA, USA
Supported by a Motwani Postdoctoral Fellowship and by NSF grant CCF-1763299.
10.4230/LIPIcs.ITCS.2019.22
Boaz Barack and Jarosław Błasiok. On the Raz-Tal oracle separation of BQP and PH, 2018. URL: https://windowsontheory.org/2018/06/17/on-the-raz-tal-oracle-separation-of-bqp-and-ph/.
https://windowsontheory.org/2018/06/17/on-the-raz-tal-oracle-separation-of-bqp-and-ph/
Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, and Shachar Lovett. Pseudorandom Generators from Polarizing Random Walks. In 33rd Computational Complexity Conference, CCC 2018, pages 1:1-1:21, 2018.
Eshan Chattopadhyay, Pooya Hatami, Omer Reingold, and Avishay Tal. Improved pseudorandomness for unordered branching programs through local monotonicity. In STOC, pages 363-375. ACM, 2018.
Bill Fefferman, Ronen Shaltiel, Christopher Umans, and Emanuele Viola. On Beating the Hybrid Argument. Theory of Computing, 9:809-843, 2013.
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Swastik Kopparty. On the complexity of powering in finite fields. In STOC, pages 489-498. ACM, 2011.
Peter Mörters and Yuval Peres. Brownian motion, volume 30. Cambridge University Press, 2010.
Ran Raz and Avishay Tal. Oracle Separation of BQP and PH. Electronic Colloquium on Computational Complexity (ECCC), 25:107, 2018.
Alexander A Razborov. Lower bounds on the size of bounded depth circuits over a complete basis with logical addition. Mathematical Notes of the Academy of Sciences of the USSR, 41(4):333-338, 1987.
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Eshan Chattopadhyay, Pooya Hatami, Shachar Lovett, and Avishay Tal
Creative Commons Attribution 3.0 Unported license
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Classical Algorithms from Quantum and Arthur-Merlin Communication Protocols
In recent years, the polynomial method from circuit complexity has been applied to several fundamental problems and obtains the state-of-the-art running times (e.g., R. Williams's n^3 / 2^{Omega(sqrt{log n})} time algorithm for APSP). As observed in [Alman and Williams, STOC 2017], almost all applications of the polynomial method in algorithm design ultimately rely on certain (probabilistic) low-rank decompositions of the computation matrices corresponding to key subroutines. They suggest that making use of low-rank decompositions directly could lead to more powerful algorithms, as the polynomial method is just one way to derive such a decomposition.
Inspired by their observation, in this paper, we study another way of systematically constructing low-rank decompositions of matrices which could be used by algorithms - communication protocols. Since their introduction, it is known that various types of communication protocols lead to certain low-rank decompositions (e.g., P protocols/rank, BQP protocols/approximate rank). These are usually interpreted as approaches for proving communication lower bounds, while in this work we explore the other direction.
We have the following two generic algorithmic applications of communication protocols:
- Quantum Communication Protocols and Deterministic Approximate Counting. Our first connection is that a fast BQP communication protocol for a function f implies a fast deterministic additive approximate counting algorithm for a related pair counting problem. Applying known BQP communication protocols, we get fast deterministic additive approximate counting algorithms for Count-OV (#OV), Sparse Count-OV and Formula of SYM circuits. In particular, our approximate counting algorithm for #OV runs in near-linear time for all dimensions d = o(log^2 n). Previously, even no truly-subquadratic time algorithm was known for d = omega(log n).
- Arthur-Merlin Communication Protocols and Faster Satisfying-Pair Algorithms. Our second connection is that a fast AM^{cc} protocol for a function f implies a faster-than-bruteforce algorithm for f-Satisfying-Pair. Using the classical Goldwasser-Sisper AM protocols for approximating set size, we obtain a new algorithm for approximate Max-IP_{n,c log n} in time n^{2 - 1/O(log c)}, matching the state-of-the-art algorithms in [Chen, CCC 2018].
We also apply our second connection to shed some light on long-standing open problems in communication complexity. We show that if the Longest Common Subsequence (LCS) problem admits a fast (computationally efficient) AM^{cc} protocol (polylog(n) complexity), then polynomial-size Formula-SAT admits a 2^{n - n^{1-delta}} time algorithm for any constant delta > 0, which is conjectured to be unlikely by a recent work [Abboud and Bringmann, ICALP 2018]. The same holds even for a fast (computationally efficient) PH^{cc} protocol.
Quantum communication protocols
Arthur-Merlin communication protocols
approximate counting
approximate rank
Theory of computation~Communication complexity
23:1-23:20
Regular Paper
A full version of the paper is available at https://arxiv.org/abs/1811.07515.
Lijie
Chen
Lijie Chen
Massachusetts Institute of Technology, Cambridge, MA, USA
Ruosong
Wang
Ruosong Wang
Carnegie Mellon University, Pittsburgh, PA, USA
10.4230/LIPIcs.ITCS.2019.23
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Josh Alman. An Illuminating Algorithm for the Light Bulb Problem. In SOSA, 2019.
Josh Alman, Timothy M. Chan, and R. Ryan Williams. Polynomial Representations of Threshold Functions and Algorithmic Applications. In IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 467-476, 2016.
Josh Alman and R. Ryan Williams. Probabilistic rank and matrix rigidity. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 641-652, 2017.
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Alexander A Razborov. Lower bounds on the size of bounded depth circuits over a complete basis with logical addition. Mathematical Notes of the Academy of Sciences of the USSR, 41(4):333-338, 1987.
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http://arxiv.org/abs/1811.02078
Lijie Chen and Ruosong Wang
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Capturing Complementarity in Set Functions by Going Beyond Submodularity/Subadditivity
We introduce two new "degree of complementarity" measures: supermodular width and superadditive width. Both are formulated based on natural witnesses of complementarity. We show that both measures are robust by proving that they, respectively, characterize the gap of monotone set functions from being submodular and subadditive. Thus, they define two new hierarchies over monotone set functions, which we will refer to as Supermodular Width (SMW) hierarchy and Superadditive Width (SAW) hierarchy, with foundations - i.e. level 0 of the hierarchies - resting exactly on submodular and subadditive functions, respectively.
We present a comprehensive comparative analysis of the SMW hierarchy and the Supermodular Degree (SD) hierarchy, defined by Feige and Izsak. We prove that the SMW hierarchy is strictly more expressive than the SD hierarchy: Every monotone set function of supermodular degree d has supermodular width at most d, and there exists a supermodular-width-1 function over a ground set of m elements whose supermodular degree is m-1. We show that previous results regarding approximation guarantees for welfare and constrained maximization as well as regarding the Price of Anarchy (PoA) of simple auctions can be extended without any loss from the supermodular degree to the supermodular width. We also establish almost matching information-theoretical lower bounds for these two well-studied fundamental maximization problems over set functions. The combination of these approximation and hardness results illustrate that the SMW hierarchy provides not only a natural notion of complementarity, but also an accurate characterization of "near submodularity" needed for maximization approximation. While SD and SMW hierarchies support nontrivial bounds on the PoA of simple auctions, we show that our SAW hierarchy seems to capture more intrinsic properties needed to realize the efficiency of simple auctions. So far, the SAW hierarchy provides the best dependency for the PoA of Single-bid Auction, and is nearly as competitive as the Maximum over Positive Hypergraphs (MPH) hierarchy for Simultaneous Item First Price Auction (SIA). We also provide almost tight lower bounds for the PoA of both auctions with respect to the SAW hierarchy.
set functions
measure of complementarity
submodularity
subadditivity
cardinality constrained maximization
welfare maximization
simple auctions
price of anarchy
Theory of computation~Approximation algorithms analysis
Theory of computation~Algorithmic game theory and mechanism design
24:1-24:20
Regular Paper
A full version of the paper is available at [Chen et al., 2018], https://arxiv.org/abs/1805.04436.
Wei
Chen
Wei Chen
Microsoft Research, Beijing, China
Supported in part by the National Natural Science Foundation of China (Grant No. 61433014).
Shang-Hua
Teng
Shang-Hua Teng
USC, Los Angeles, CA, USA
Supported in part by Simons Investigator Award and by NSF grant CCF-1815254.
Hanrui
Zhang
Hanrui Zhang
Duke University, Durham, NC, USA
Supported by NSF awards IIS-1814056 and IIS-1527434.
10.4230/LIPIcs.ITCS.2019.24
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Wei Chen, Shang-Hua Teng, and Hanrui Zhang. Capturing Complementarity in Set Functions by Going Beyond Submodularity/Subadditivity. arXiv preprint, 2018. URL: http://arxiv.org/abs/1805.04436.
http://arxiv.org/abs/1805.04436
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Wei Chen, Shang-Hua Teng, and Hanrui Zhang
Creative Commons Attribution 3.0 Unported license
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Probabilistic Checking Against Non-Signaling Strategies from Linearity Testing
Non-signaling strategies are a generalization of quantum strategies that have been studied in physics over the past three decades. Recently, they have found applications in theoretical computer science, including to proving inapproximability results for linear programming and to constructing protocols for delegating computation. A central tool for these applications is probabilistically checkable proofs (PCPs) that are sound against non-signaling strategies.
In this paper we prove that the exponential-length constant-query PCP construction due to Arora et al. (JACM 1998) is sound against non-signaling strategies.
Our result offers a new length-vs-query tradeoff when compared to the non-signaling PCP of Kalai, Raz, and Rothblum (STOC 2013 and 2014) and, moreover, may serve as an intermediate step to a proof of a non-signaling analogue of the PCP Theorem.
probabilistically checkable proofs
linearity testing
non-signaling strategies
Theory of computation~Computational complexity and cryptography
25:1-25:17
Regular Paper
This work was supported by the UC Berkeley Center for Long-Term Cybersecurity.
Full version is available on the Electronic Colloquium on Computational Complexity as TR18-123, https://eccc.weizmann.ac.il/report/2018/123/.
Alessandro
Chiesa
Alessandro Chiesa
UC Berkeley, Berkeley, CA, USA
Peter
Manohar
Peter Manohar
UC Berkeley, Berkeley, CA, USA
Igor
Shinkar
Igor Shinkar
Simon Fraser University, Vancouver, Canada
10.4230/LIPIcs.ITCS.2019.25
Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. Journal of the ACM, 45(3):501-555, 1998. Preliminary version in FOCS '92.
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Eli Ben-Sasson, Oded Goldreich, Prahladh Harsha, Madhu Sudan, and Salil P. Vadhan. Robust PCPs of Proximity, Shorter PCPs, and Applications to Coding. SIAM Journal on Computing, 36(4):889-974, 2006.
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Alessandro Chiesa, Peter Manohar, and Igor Shinkar. Testing Linearity against Non-Signaling Strategies. In Proceedings of the 33rd Annual Conference on Computational Complexity, CCC '18, pages 17:1-17:37, 2018.
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Alessandro Chiesa, Peter Manohar, and Igor Shinkar
Creative Commons Attribution 3.0 Unported license
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On the Algorithmic Power of Spiking Neural Networks
Spiking Neural Networks (SNN) are mathematical models in neuroscience to describe the dynamics among a set of neurons that interact with each other by firing instantaneous signals, a.k.a., spikes. Interestingly, a recent advance in neuroscience [Barrett-Denève-Machens, NIPS 2013] showed that the neurons' firing rate, i.e., the average number of spikes fired per unit of time, can be characterized by the optimal solution of a quadratic program defined by the parameters of the dynamics. This indicated that SNN potentially has the computational power to solve non-trivial quadratic programs. However, the results were justified empirically without rigorous analysis.
We put this into the context of natural algorithms and aim to investigate the algorithmic power of SNN. Especially, we emphasize on giving rigorous asymptotic analysis on the performance of SNN in solving optimization problems. To enforce a theoretical study, we first identify a simplified SNN model that is tractable for analysis. Next, we confirm the empirical observation in the work of Barrett et al. by giving an upper bound on the convergence rate of SNN in solving the quadratic program. Further, we observe that in the case where there are infinitely many optimal solutions, SNN tends to converge to the one with smaller l_1 norm. We give an affirmative answer to our finding by showing that SNN can solve the l_1 minimization problem under some regular conditions.
Our main technical insight is a dual view of the SNN dynamics, under which SNN can be viewed as a new natural primal-dual algorithm for the l_1 minimization problem. We believe that the dual view is of independent interest and may potentially find interesting interpretation in neuroscience.
Spiking Neural Networks
Natural Algorithms
l_1 Minimization
Theory of computation~Models of computation
26:1-26:20
Regular Paper
A full version of the paper is available at https://arxiv.org/abs/1803.10375.
Chi-Ning
Chou
Chi-Ning Chou
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA
Supported by NSF awards CCF 1565264 and CNS 1618026.
Kai-Min
Chung
Kai-Min Chung
Institute of Information Science, Academia Sinica, Taipei, Taiwan
Chi-Jen
Lu
Chi-Jen Lu
Institute of Information Science, Academia Sinica, Taipei, Taiwan
10.4230/LIPIcs.ITCS.2019.26
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Chi-Ning Chou, Kai-Min Chung, and Chi-Jen Lu
Creative Commons Attribution 3.0 Unported license
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Last-Iterate Convergence: Zero-Sum Games and Constrained Min-Max Optimization
Motivated by applications in Game Theory, Optimization, and Generative Adversarial Networks, recent work of Daskalakis et al [Daskalakis et al., ICLR, 2018] and follow-up work of Liang and Stokes [Liang and Stokes, 2018] have established that a variant of the widely used Gradient Descent/Ascent procedure, called "Optimistic Gradient Descent/Ascent (OGDA)", exhibits last-iterate convergence to saddle points in unconstrained convex-concave min-max optimization problems. We show that the same holds true in the more general problem of constrained min-max optimization under a variant of the no-regret Multiplicative-Weights-Update method called "Optimistic Multiplicative-Weights Update (OMWU)". This answers an open question of Syrgkanis et al [Syrgkanis et al., NIPS, 2015].
The proof of our result requires fundamentally different techniques from those that exist in no-regret learning literature and the aforementioned papers. We show that OMWU monotonically improves the Kullback-Leibler divergence of the current iterate to the (appropriately normalized) min-max solution until it enters a neighborhood of the solution. Inside that neighborhood we show that OMWU becomes a contracting map converging to the exact solution. We believe that our techniques will be useful in the analysis of the last iterate of other learning algorithms.
No regret learning
Zero-sum games
Convergence
Dynamical Systems
KL divergence
Theory of computation~Convergence and learning in games
27:1-27:18
Regular Paper
Constantinos
Daskalakis
Constantinos Daskalakis
CSAIL, MIT, Cambridge MA, USA
Supported by NSF awards CCF-1617730 and IIS-1741137, a Simons Investigator Award, a Google Faculty Research Award, and an MIT-IBM Watson AI Lab research grant.
Ioannis
Panageas
Ioannis Panageas
ISTD, SUTD, Singapore
Supported by SRG ISTD 2018 136. Part of this work was done while Ioannis was postdoc at MIT.
10.4230/LIPIcs.ITCS.2019.27
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Constantinos Daskalakis and Ioannis Panageas
Creative Commons Attribution 3.0 Unported license
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Density Estimation for Shift-Invariant Multidimensional Distributions
We study density estimation for classes of shift-invariant distributions over R^d. A multidimensional distribution is "shift-invariant" if, roughly speaking, it is close in total variation distance to a small shift of it in any direction. Shift-invariance relaxes smoothness assumptions commonly used in non-parametric density estimation to allow jump discontinuities. The different classes of distributions that we consider correspond to different rates of tail decay.
For each such class we give an efficient algorithm that learns any distribution in the class from independent samples with respect to total variation distance. As a special case of our general result, we show that d-dimensional shift-invariant distributions which satisfy an exponential tail bound can be learned to total variation distance error epsilon using O~_d(1/ epsilon^{d+2}) examples and O~_d(1/ epsilon^{2d+2}) time. This implies that, for constant d, multivariate log-concave distributions can be learned in O~_d(1/epsilon^{2d+2}) time using O~_d(1/epsilon^{d+2}) samples, answering a question of [Diakonikolas et al., 2016]. All of our results extend to a model of noise-tolerant density estimation using Huber's contamination model, in which the target distribution to be learned is a (1-epsilon,epsilon) mixture of some unknown distribution in the class with some other arbitrary and unknown distribution, and the learning algorithm must output a hypothesis distribution with total variation distance error O(epsilon) from the target distribution. We show that our general results are close to best possible by proving a simple Omega (1/epsilon^d) information-theoretic lower bound on sample complexity even for learning bounded distributions that are shift-invariant.
Density estimation
unsupervised learning
log-concave distributions
non-parametrics
Theory of computation~Unsupervised learning and clustering
28:1-28:20
Regular Paper
A full version of the paper is available at [A. De et al., 2018], https://arxiv.org/abs/1811.03744.
Anindya
De
Anindya De
EECS Department, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA
Supported by NSF grant CCF-1814706.
Philip M.
Long
Philip M. Long
Google, 1600 Amphitheatre Parkway, Mountain View, CA 94043, USA
Rocco A.
Servedio
Rocco A. Servedio
Department of Computer Science, Columbia University, 500 W. 120th Street, Room 450, New York, NY 10027, USA
Supported by NSF grants CCF-1563155, IIS-1838154 and CCF-1814873.
10.4230/LIPIcs.ITCS.2019.28
J. Acharya, C. Daskalakis, and G. Kamath. Optimal Testing for Properties of Distributions. In NIPS, pages 3591-3599, 2015.
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Maria-Florina Balcan and Philip M Long. Active and passive learning of linear separators under log-concave distributions. In Conference on Learning Theory, pages 288-316, 2013.
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A. De, P. M. Long, and R. A. Servedio. Density estimation for shift-invariant multidimensional distributions, 2018. URL: http://arxiv.org/abs/1811.03744.
http://arxiv.org/abs/1811.03744
A. De, P. M. Long, and R. A. Servedio. Learning Sums of Independent Random Variables with Sparse Collective Support. FOCS, 2018.
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I. Diakonikolas, D. M. Kane, and A. Stewart. Efficient Robust Proper Learning of Log-concave Distributions. CoRR, abs/1606.03077, 2016.
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I. Diakonikolas, D. M. Kane, and A. Stewart. The Fourier transform of Poisson multinomial distributions and its algorithmic applications. In STOC, pages 1060-1073, 2016.
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Ilias Diakonikolas, Daniel M Kane, and Alistair Stewart. Learning multivariate log-concave distributions. In Conference on Learning Theory (COLT), pages 711-727, 2016.
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http://arxiv.org/abs/1508.04376
A. T. Kalai, A. Moitra, and G. Valiant. Efficiently learning mixtures of two Gaussians. In STOC, pages 553-562, 2010.
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http://arxiv.org/abs/1404.2298
Jussi Klemelä. Smoothing of Multivariate Data: Density Estimation and Visualization. Wiley Publishing, 2009.
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http://arxiv.org/abs/1404.5886
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Y. G. Yatracos. Rates of convergence of minimum distance estimators and Kolmogorov’s entropy. Annals of Statistics, 13:768-774, 1985.
Anindya De, Philip M. Long, and Rocco A. Servedio
Creative Commons Attribution 3.0 Unported license
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From Local to Robust Testing via Agreement Testing
A local tester for an error-correcting code is a probabilistic procedure that queries a small subset of coordinates, accepts codewords with probability one, and rejects non-codewords with probability proportional to their distance from the code. The local tester is robust if for non-codewords it satisfies the stronger property that the average distance of local views from accepting views is proportional to the distance from the code. Robust testing is an important component in constructions of locally testable codes and probabilistically checkable proofs as it allows for composition of local tests.
In this work we show that for certain codes, any (natural) local tester can be converted to a roubst tester with roughly the same number of queries. Our result holds for the class of affine-invariant lifted codes which is a broad class of codes that includes Reed-Muller codes, as well as recent constructions of high-rate locally testable codes (Guo, Kopparty, and Sudan, ITCS 2013). Instantiating this with known local testing results for lifted codes gives a more direct proof that improves some of the parameters of the main result of Guo, Haramaty, and Sudan (FOCS 2015), showing robustness of lifted codes.
To obtain the above transformation we relate the notions of local testing and robust testing to the notion of agreement testing that attempts to find out whether valid partial assignments can be stitched together to a global codeword. We first show that agreement testing implies robust testing, and then show that local testing implies agreement testing. Our proof is combinatorial, and is based on expansion / sampling properties of the collection of local views of local testers. Thus, it immediately applies to local testers of lifted codes that query random affine subspaces in F_q^m, and moreover seems amenable to extension to other families of locally testable codes with expanding families of local views.
Local testing
Robust testing
Agreement testing
Affine-invariant codes
Lifted codes
Theory of computation~Computational complexity and cryptography
29:1-29:18
Regular Paper
Irit
Dinur
Irit Dinur
Department of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, Israel
Supported by ERC-CoG grant number 772839.
Prahladh
Harsha
Prahladh Harsha
Tata Institute of Fundamental Research, India
Research supported in part by the UGC-ISF grant and the Swarnajayanti Fellowship. Part of the work was done when the author was visiting the Weizmann Institute of Science.
Tali
Kaufman
Tali Kaufman
Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel
Noga
Ron-Zewi
Noga Ron-Zewi
Department of Computer Science, University of Haifa, Haifa, Israel
10.4230/LIPIcs.ITCS.2019.29
Sanjeev Arora. Probabilistic checking of proofs and hardness of approximation problems. PhD thesis, Princeton University, 1994.
Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. Journal of the ACM, 45(3):501-555, May 1998.
Sanjeev Arora and Shmuel Safra. Probabilistic Checking of Proofs: A New Characterization of NP. Journal of the ACM, 45(1):70-122, January 1998.
Sanjeev Arora and Madhu Sudan. Improved Low Degree Testing and its Applications. Combinatorica, 23(3):365-426, 2003.
Eli Ben-Sasson, Oded Goldreich, Prahladh Harsha, Madhu Sudan, and Salil P. Vadhan. Robust PCPs of Proximity, Shorter PCPs, and Applications to Coding. SIAM J. Comput, 36(4):889-974, 2006.
Eli Ben-Sasson, Prahladh Harsha, and Sofya Raskhodnikova. Some 3CNF Properties Are Hard to Test. SICOMP: SIAM Journal on Computing, 35, 2005.
Eli Ben-Sasson, Ghid Maatouk, Amir Shpilka, and Madhu Sudan. Symmetric LDPC Codes are not Necessarily Locally Testable. In IEEE Conference on Computational Complexity, pages 55-65. IEEE Computer Society, 2011. URL: http://dx.doi.org/10.1109/CCC.2011.14.
http://dx.doi.org/10.1109/CCC.2011.14
Eli Ben-Sasson and Madhu Sudan. Robust locally testable codes and products of codes. Random Struct. Algorithms, 28(4):387-402, 2006. URL: http://dx.doi.org/10.1002/rsa.20120.
http://dx.doi.org/10.1002/rsa.20120
Eli Ben-Sasson and Michael Viderman. Composition of semi-LTCs by two-wise tensor products. Computational Complexity, 24(3):601-643, 2015. URL: http://dx.doi.org/10.1007/s00037-013-0074-8.
http://dx.doi.org/10.1007/s00037-013-0074-8
Manuel Blum, Michael Luby, and Ronitt Rubinfeld. Self-Testing/Correcting with Applications to Numerical Problems. In STOC, pages 73-83. ACM, 1990.
A. E. Brouwer, A. M. Cohen, and A. Neumaier. Distance-Regular Graphs. Springer Verlag, 1989.
Irit Dinur and Elazar Goldenberg. Locally Testing Direct Product in the Low Error Range. In FOCS, pages 613-622. IEEE Computer Society, 2008. URL: http://dx.doi.org/10.1109/FOCS.2008.26.
http://dx.doi.org/10.1109/FOCS.2008.26
Irit Dinur and Tali Kaufman. High Dimensional Expanders Imply Agreement Expanders. In Chris Umans, editor, FOCS, pages 974-985. IEEE Computer Society, 2017. URL: http://dx.doi.org/10.1109/FOCS.2017.94.
http://dx.doi.org/10.1109/FOCS.2017.94
Irit Dinur and Inbal Livni-Navon. Exponentially Small Soundness for the Direct Product Z-Test. In Ryan O'Donnell, editor, Computational Complexity Conference, volume 79 of LIPIcs, pages 29:1-29:50. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2017.29.
http://dx.doi.org/10.4230/LIPIcs.CCC.2017.29
Irit Dinur and Omer Reingold. Assignment Testers: Towards a Combinatorial Proof of the PCP Theorem. SIAM J. Comput, 36(4):975-1024, 2006. URL: http://dx.doi.org/10.1137/S0097539705446962.
http://dx.doi.org/10.1137/S0097539705446962
Irit Dinur and David Steurer. Direct Product Testing. In IEEE Conference on Computational Complexity, pages 188-196. IEEE Computer Society, 2014. URL: http://dx.doi.org/10.1109/CCC.2014.27.
http://dx.doi.org/10.1109/CCC.2014.27
Peter Gemmell, Richard J. Lipton, Ronitt Rubinfeld, Madhu Sudan, and Avi Wigderson. Self-Testing/Correcting for Polynomials and for Approximate Functions. In Cris Koutsougeras and Jeffrey Scott Vitter, editors, STOC, pages 32-42. ACM, 1991. URL: http://dx.doi.org/10.1145/103418.103429.
http://dx.doi.org/10.1145/103418.103429
Goldreich and Safra. A Combinatorial Consistency Lemma with Application to Proving the PCP Theorem. SICOMP: SIAM Journal on Computing, 29, 1999.
Alan Guo, Elad Haramaty, and Madhu Sudan. Robust Testing of Lifted Codes with Applications to Low-Degree Testing. In Venkatesan Guruswami, editor, FOCS, pages 825-844. IEEE Computer Society, 2015. URL: http://dx.doi.org/10.1109/FOCS.2015.56.
http://dx.doi.org/10.1109/FOCS.2015.56
Alan Guo, Swastik Kopparty, and Madhu Sudan. New affine-invariant codes from lifting. In ITCS, pages 529-540, 2013.
Elad Haramaty, Noga Ron-Zewi, and Madhu Sudan. Absolutely Sound Testing of Lifted Codes. Theory of Computing, 11:299-338, 2015. URL: http://dx.doi.org/10.4086/toc.2015.v011a012.
http://dx.doi.org/10.4086/toc.2015.v011a012
Russell Impagliazzo, Valentine Kabanets, and Avi Wigderson. New Direct-Product Testers and 2-Query PCPs. SIAM J. Comput, 41(6):1722-1768, 2012. URL: http://dx.doi.org/10.1137/09077299X.
http://dx.doi.org/10.1137/09077299X
Tali Kaufman and Madhu Sudan. Algebraic property testing: the role of invariance. In STOC, pages 403-412. ACM, 2008. URL: http://dx.doi.org/10.1145/1374376.1374434.
http://dx.doi.org/10.1145/1374376.1374434
Swastik Kopparty, Or Meir, Noga Ron-Zewi, and Shubhangi Saraf. High-Rate Locally Correctable and Locally Testable Codes with Sub-Polynomial Query Complexity. Journal of ACM, 64(2):11:1-11:42, 2017. URL: http://dx.doi.org/10.1145/3051093.
http://dx.doi.org/10.1145/3051093
Ran Raz and Shmuel Safra. A Sub-Constant Error-Probability Low-Degree Test, and a Sub-Constant Error-Probability PCP Characterization of NP. In STOC, pages 475-484. ACM, 1997. URL: http://dx.doi.org/10.1145/258533.258641.
http://dx.doi.org/10.1145/258533.258641
Ronitt Rubinfeld and Madhu Sudan. Robust characterizations of polynomials with applications to program testing. SIAM Journal on Computing, 25(2):252-271, April 1996.
Michael Viderman. A combination of testability and decodability by tensor products. Random Struct. Algorithms, 46(3):572-598, 2015.
Irit Dinur, Prahladh Harsha, Tal Kaufman, and Noga Ron-Zewi
Creative Commons Attribution 3.0 Unported license
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Every Set in P Is Strongly Testable Under a Suitable Encoding
We show that every set in P is strongly testable under a suitable encoding. By "strongly testable" we mean having a (proximity oblivious) tester that makes a constant number of queries and rejects with probability that is proportional to the distance of the tested object from the property. By a "suitable encoding" we mean one that is polynomial-time computable and invertible. This result stands in contrast to the known fact that some sets in P are extremely hard to test, providing another demonstration of the crucial role of representation in the context of property testing.
The testing result is proved by showing that any set in P has a strong canonical PCP, where canonical means that (for yes-instances) there exists a single proof that is accepted with probability 1 by the system, whereas all other potential proofs are rejected with probability proportional to their distance from this proof. In fact, we show that UP equals the class of sets having strong canonical PCPs (of logarithmic randomness), whereas the class of sets having strong canonical PCPs with polynomial proof length equals "unambiguous- MA". Actually, for the testing result, we use a PCP-of-Proximity version of the foregoing notion and an analogous positive result (i.e., strong canonical PCPPs of logarithmic randomness for any set in UP).
Probabilistically checkable proofs
property testing
Theory of computation~Probabilistic computation
30:1-30:17
Regular Paper
Full technical report hosted on ECCC [I. Dinur et al., 2018], https://eccc.weizmann.ac.il/report/2018/050/.
Irit
Dinur
Irit Dinur
Weizmann Institute, Rehovot, Israel
Supported by ERC-CoG grant 772839. Weizmann Institute of Science.
Oded
Goldreich
Oded Goldreich
Weizmann Institute, Rehovot, Israel
Supported by the Israel Science Foundation (grants No. 671/13 and 1146/18).
Tom
Gur
Tom Gur
University of Warwick, UK
Supported in part by the UC Berkeley Center for Long-Term Cybersecurity.
10.4230/LIPIcs.ITCS.2019.30
N. Alon, E. Fischer, M. Krivelevich, and M. Szegedy. Efficient testing of large graphs. Combinatorica, 20:451-476, 2000.
N. Alon, E. Fischer, I. Newman, and A. Shapira. A combinatorial characterization of the testable graph properties: It’s all about regularity. STOC, pages 251-260, 2006.
S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof verification and intractability of approximation problems. JACM, 45:501-555, 1998.
S. Arora and S. Safra. Probabilistic checkable proofs: A new characterization of NP. JACM, 45:70-122, 1998.
E. Ben-Sasson, O. Goldreich, P. Harsha, M. Sudan, and S. Vadhan. Robust pcps of proximity, shorter pcps, and applications to coding. SICOMP, 36(4):889-974, 2006.
E. Ben-Sasson and M. Sudan. Short pcps with polylog query complexity. SICOMP, 38(2):551-607, 2008.
M. Blum, M. Luby, and R. Rubinfeld. Self-testing/correcting with applications to numerical problems. JCSS, 47(3):549-595, 1993.
I. Dinur. The pcp theorem by gap amplification. JACM, 54:3, 2007.
I. Dinur, O. Goldreich, and T. Gur. Every set in p is strongly testable under a suitable encoding. Technical report, in ECCC TR18-050, 2018.
I. Dinur and O. Reingold. Assignment-testers: Towards a combinatorial proof of the pcp-theorem. FOCS, 45, 2004.
O. Goldreich. Computational Complexity: A Conceptual Perspective. Cambridge University Press, 2008.
O. Goldreich. Introduction to Property Testing. Cambridge University Press, 2017.
O. Goldreich, S. Goldwasser, and D. Ron. Property testing and its connection to learning and approximation. JACM, pages 653-750, 1998.
O. Goldreich, T. Gur, and I. Komargodski. Strong locally testable codes with relaxed local decoders. CCC, 30:1-41, 2015.
O. Goldreich, M. Krivelevich, I. Newman, and E. Rozenberg. Hierarchy theorems for property testing. Computational Complexity, 21(1):129-192, 2012.
O. Goldreich and D. Ron. Property testing in bounded degree graphs. Algorithmica, pages 302-343, 2002.
O. Goldreich and D. Ron. On proximity oblivious testing. SICOMP, 40(2):534-566, 2011.
O. Goldreich and I. Shinkar. Two-sided error proximity oblivious testing. RSA, 48(2):341-383, 2016.
O. Goldreich and M. Sudan. Locally testable codes and pcps of almost-linear length. JACM, 53(4):558-655, 2006.
T. Gur, G. Ramnarayan, and R. Rothblum. Relaxed Locally Correctable Codes. ITCS. ECCC, 2018.
T. Gur and R. Rothblum. Non-interactive proofs of proximity. Computational Complexity, 2018.
Irit Dinur, Oded Goldreich, and Tom Gur
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Alea Iacta Est: Auctions, Persuasion, Interim Rules, and Dice
To select a subset of samples or "winners" from a population of candidates, order sampling [Rosén, 1997] and the k-unit Myerson auction [Myerson, 1981] share a common scheme: assign a (random) score to each candidate, then select the k candidates with the highest scores. We study a generalization of both order sampling and Myerson's allocation rule, called winner-selecting dice. The setting for winner-selecting dice is similar to auctions with feasibility constraints: candidates have random types drawn from independent prior distributions, and the winner set must be feasible subject to certain constraints. Dice (distributions over scores) are assigned to each type, and winners are selected to maximize the sum of the dice rolls, subject to the feasibility constraints. We examine the existence of winner-selecting dice that implement prescribed probabilities of winning (i.e., an interim rule) for all types.
Our first result shows that when the feasibility constraint is a matroid, then for any feasible interim rule, there always exist winner-selecting dice that implement it. Unfortunately, our proof does not yield an efficient algorithm for constructing the dice. In the special case of a 1-uniform matroid, i.e., only one winner can be selected, we give an efficient algorithm that constructs winner-selecting dice for any feasible interim rule. Furthermore, when the types of the candidates are drawn in an i.i.d. manner and the interim rule is symmetric across candidates, unsurprisingly, an algorithm can efficiently construct symmetric dice that only depend on the type but not the identity of the candidate.
One may ask whether we can extend our result to "second-order" interim rules, which not only specify the winning probability of a type, but also the winning probability conditioning on each other candidate's type. We show that our result does not extend, by exhibiting an instance of Bayesian persuasion whose optimal scheme is equivalent to a second-order interim rule, but which does not admit any dice-based implementation.
Interim rule
order sampling
virtual value function
Border's theorem
Theory of computation~Algorithmic game theory
31:1-31:20
Regular Paper
Work supported in part by NSF Grant CCF-1423618.
A full version of the paper is available at https://arxiv.org/abs/1811.11417.
Shaddin
Dughmi
Shaddin Dughmi
University of Southern California, Los Angeles, CA, USA
David
Kempe
David Kempe
University of Southern California, Los Angeles, CA, USA
Ruixin
Qiang
Ruixin Qiang
University of Southern California, Los Angeles, CA, USA
10.4230/LIPIcs.ITCS.2019.31
Nibia Aires, Johan Jonasson, and Olle Nerman. Order sampling design with prescribed inclusion probabilities. Scandinavian journal of statistics, 29(1):183-187, 2002.
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Shaddin Dughmi, David Kempe, and Ruixin Qiang
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Spanoids - An Abstraction of Spanning Structures, and a Barrier for LCCs
We introduce a simple logical inference structure we call a spanoid (generalizing the notion of a matroid), which captures well-studied problems in several areas. These include combinatorial geometry (point-line incidences), algebra (arrangements of hypersurfaces and ideals), statistical physics (bootstrap percolation), network theory (gossip / infection processes) and coding theory. We initiate a thorough investigation of spanoids, from computational and structural viewpoints, focusing on parameters relevant to the applications areas above and, in particular, to questions regarding Locally Correctable Codes (LCCs).
One central parameter we study is the rank of a spanoid, extending the rank of a matroid and related to the dimension of codes. This leads to one main application of our work, establishing the first known barrier to improving the nearly 20-year old bound of Katz-Trevisan (KT) on the dimension of LCCs. On the one hand, we prove that the KT bound (and its more recent refinements) holds for the much more general setting of spanoid rank. On the other hand we show that there exist (random) spanoids whose rank matches these bounds. Thus, to significantly improve the known bounds one must step out of the spanoid framework.
Another parameter we explore is the functional rank of a spanoid, which captures the possibility of turning a given spanoid into an actual code. The question of the relationship between rank and functional rank is one of the main questions we raise as it may reveal new avenues for constructing new LCCs (perhaps even matching the KT bound). As a first step, we develop an entropy relaxation of functional rank to create a small constant gap and amplify it by tensoring to construct a spanoid whose functional rank is smaller than rank by a polynomial factor. This is evidence that the entropy method we develop can prove polynomially better bounds than KT-type methods on the dimension of LCCs.
To facilitate the above results we also develop some basic structural results on spanoids including an equivalent formulation of spanoids as set systems and properties of spanoid products. We feel that given these initial findings and their motivations, the abstract study of spanoids merits further investigation. We leave plenty of concrete open problems and directions.
Locally correctable codes
spanoids
entropy
bootstrap percolation
gossip spreading
matroid
union-closed family
Theory of computation~Error-correcting codes
32:1-32:20
Regular Paper
Full version at https://arxiv.org/abs/1809.10372.
Zeev
Dvir
Zeev Dvir
Dept of Computer Science and Dept of Mathematics, Princeton University, Princeton, NJ, USA
Research supported by NSF CAREER award DMS-1451191 and NSF grant CCF-1523816.
Sivakanth
Gopi
Sivakanth Gopi
Microsoft Research, Redmond, WA, USA
Research supported by NSF CAREER award DMS-1451191 and NSF grant CCF-1523816.
Yuzhou
Gu
Yuzhou Gu
MIT, Cambridge, MA, USA
Research supported by Jacobs Family Presidential Fellowship.
Avi
Wigderson
Avi Wigderson
Institute of Advanced Study, Princeton, NJ, USA
Research was partially supported by NSF grant CCF-1412958.
10.4230/LIPIcs.ITCS.2019.32
Michael Alekhnovich, Eli Ben-Sasson, Alexander A Razborov, and Avi Wigderson. Pseudorandom generators in propositional proof complexity. SIAM Journal on Computing, 34(1):67-88, 2004.
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Boaz Barak, Zeev Dvir, Avi Wigderson, and Amir Yehudayoff. Fractional Sylvester-Gallai theorems. Proceedings of the National Academy of Sciences, 2012.
Boaz Barak, Zeev Dvir, Amir Yehudayoff, and Avi Wigderson. Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes. In Proceedings of the forty-third annual ACM symposium on Theory of computing, pages 519-528. ACM, 2011.
Eli Ben-Sasson and Avi Wigderson. Short proofs are narrow—resolution made simple. Journal of the ACM (JACM), 48(2):149-169, 2001.
Arnab Bhattacharyya, Sivakanth Gopi, and Avishay Tal. Lower Bounds for 2-Query LCCs over Large Alphabet. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2017, August 16-18, 2017, Berkeley, CA, USA, pages 30:1-30:20, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2017.30.
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http://dx.doi.org/10.1561/0400000056
Zeev Dvir, Shubhangi Saraf, and Avi Wigderson. Breaking the quadratic barrier for 3-LCC’s over the reals. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pages 784-793. ACM, 2014.
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http://arxiv.org/abs/1802.01184
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Zeev Dvir, Sivakanth Gopi, Yuzhou Gu, and Avi Wigderson
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Fairness Under Composition
Algorithmic fairness, and in particular the fairness of scoring and classification algorithms, has become a topic of increasing social concern and has recently witnessed an explosion of research in theoretical computer science, machine learning, statistics, the social sciences, and law. Much of the literature considers the case of a single classifier (or scoring function) used once, in isolation. In this work, we initiate the study of the fairness properties of systems composed of algorithms that are fair in isolation; that is, we study fairness under composition. We identify pitfalls of naïve composition and give general constructions for fair composition, demonstrating both that classifiers that are fair in isolation do not necessarily compose into fair systems and also that seemingly unfair components may be carefully combined to construct fair systems. We focus primarily on the individual fairness setting proposed in [Dwork, Hardt, Pitassi, Reingold, Zemel, 2011], but also extend our results to a large class of group fairness definitions popular in the recent literature, exhibiting several cases in which group fairness definitions give misleading signals under composition.
algorithmic fairness
fairness
fairness under composition
Theory of computation~Computational complexity and cryptography
Theory of computation~Design and analysis of algorithms
Theory of computation~Theory and algorithms for application domains
33:1-33:20
Regular Paper
A full version of the paper is available at https://arxiv.org/abs/1806.06122.
Cynthia
Dwork
Cynthia Dwork
Harvard John A Paulson School of Engineering and Applied Science, Radcliffe Institute for Advanced Study, Cambridge, MA, USA
This work was supported in part by Microsoft Research and the Sloan Foundation.
Christina
Ilvento
Christina Ilvento
Harvard John A Paulson School of Engineering and Applied Science, Cambridge, MA, USA
This work was supported in part by the Smith Family Fellowship and Microsoft Research.
10.4230/LIPIcs.ITCS.2019.33
Amanda Bower, Sarah N. Kitchen, Laura Niss, Martin J. Strauss, Alexander Vargas, and Suresh Venkatasubramanian. Fair Pipelines. CoRR, abs/1707.00391, 2017. URL: http://arxiv.org/abs/1707.00391.
http://arxiv.org/abs/1707.00391
Alexandra Chouldechova. Fair prediction with disparate impact: A study of bias in recidivism prediction instruments. arXiv preprint, 2017. URL: http://arxiv.org/abs/1703.00056.
http://arxiv.org/abs/1703.00056
Amit Datta, Michael Carl Tschantz, and Anupam Datta. Automated experiments on ad privacy settings. Proceedings on Privacy Enhancing Technologies, 2015(1):92-112, 2015.
Cynthia Dwork, Moritz Hardt, Toniann Pitassi, Omer Reingold, and Richard Zemel. Fairness through awareness. In Proceedings of the 3rd innovations in theoretical computer science conference, pages 214-226. ACM, 2012.
Stephen Gillen, Christopher Jung, Michael Kearns, and Aaron Roth. Online Learning with an Unknown Fairness Metric. arXiv preprint, 2018. URL: http://arxiv.org/abs/1802.06936.
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Moritz Hardt, Eric Price, Nati Srebro, et al. Equality of opportunity in supervised learning. In Advances in Neural Information Processing Systems, pages 3315-3323, 2016.
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http://arxiv.org/abs/1711.08513
Lily Hu and Yiling Chen. Fairness at Equilibrium in the Labor Market. CoRR, abs/1707.01590, 2017. URL: http://arxiv.org/abs/1707.01590.
http://arxiv.org/abs/1707.01590
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Toshihiro Kamishima, Shotaro Akaho, and Jun Sakuma. Fairness-aware learning through regularization approach. In Data Mining Workshops (ICDMW), 2011 IEEE 11th International Conference on, pages 643-650. IEEE, 2011.
Michael Kearns, Seth Neel, Aaron Roth, and Zhiwei Steven Wu. Preventing fairness gerrymandering: Auditing and learning for subgroup fairness. arXiv preprint, 2017. URL: http://arxiv.org/abs/1711.05144.
http://arxiv.org/abs/1711.05144
Niki Kilbertus, Mateo Rojas-Carulla, Giambattista Parascandolo, Moritz Hardt, Dominik Janzing, and Bernhard Schölkopf. Avoiding Discrimination through Causal Reasoning. arXiv preprint, 2017. URL: http://arxiv.org/abs/1706.02744.
http://arxiv.org/abs/1706.02744
Michael P Kim, Omer Reingold, and Guy N Rothblum. Fairness Through Computationally-Bounded Awareness. arXiv preprint, 2018. URL: http://arxiv.org/abs/1803.03239.
http://arxiv.org/abs/1803.03239
Jon M. Kleinberg, Sendhil Mullainathan, and Manish Raghavan. Inherent Trade-Offs in the Fair Determination of Risk Scores. CoRR, abs/1609.05807, 2016. URL: http://arxiv.org/abs/1609.05807.
http://arxiv.org/abs/1609.05807
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Matt J Kusner, Joshua R Loftus, Chris Russell, and Ricardo Silva. Counterfactual Fairness. arXiv preprint, 2017. URL: http://arxiv.org/abs/1703.06856.
http://arxiv.org/abs/1703.06856
Anja Lambrecht and Catherine E Tucker. Algorithmic Bias? An Empirical Study into Apparent Gender-Based Discrimination in the Display of STEM Career Ads, 2016.
Lydia T Liu, Sarah Dean, Esther Rolf, Max Simchowitz, and Moritz Hardt. Delayed Impact of Fair Machine Learning. arXiv preprint, 2018. URL: http://arxiv.org/abs/1803.04383.
http://arxiv.org/abs/1803.04383
David Madras, Elliot Creager, Toniann Pitassi, and Richard Zemel. Learning Adversarially Fair and Transferable Representations. arXiv preprint, 2018. URL: http://arxiv.org/abs/1802.06309.
http://arxiv.org/abs/1802.06309
Dino Pedreshi, Salvatore Ruggieri, and Franco Turini. Discrimination-aware data mining. In Proceedings of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 560-568. ACM, 2008.
Ya'acov Ritov, Yuekai Sun, and Ruofei Zhao. On conditional parity as a notion of non-discrimination in machine learning. arXiv preprint, 2017. URL: http://arxiv.org/abs/1706.08519.
http://arxiv.org/abs/1706.08519
Rich Zemel, Yu Wu, Kevin Swersky, Toni Pitassi, and Cynthia Dwork. Learning fair representations. In Proceedings of the 30th International Conference on Machine Learning (ICML-13), pages 325-333, 2013.
Cynthia Dwork and Christina Ilvento
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
A Log-Sobolev Inequality for the Multislice, with Applications
Let kappa in N_+^l satisfy kappa_1 + *s + kappa_l = n, and let U_kappa denote the multislice of all strings u in [l]^n having exactly kappa_i coordinates equal to i, for all i in [l]. Consider the Markov chain on U_kappa where a step is a random transposition of two coordinates of u. We show that the log-Sobolev constant rho_kappa for the chain satisfies rho_kappa^{-1} <= n * sum_{i=1}^l 1/2 log_2(4n/kappa_i), which is sharp up to constants whenever l is constant. From this, we derive some consequences for small-set expansion and isoperimetry in the multislice, including a KKL Theorem, a Kruskal - Katona Theorem for the multislice, a Friedgut Junta Theorem, and a Nisan - Szegedy Theorem.
log-Sobolev inequality
small-set expansion
conductance
hypercontractivity
Fourier analysis
representation theory
Markov chains
combinatorics
Mathematics of computing~Markov processes
34:1-34:12
Regular Paper
Full version at https://arxiv.org/abs/1809.03546
Yuval
Filmus
Yuval Filmus
Department of Computer Science, Technion - Israel Institute of Technology, Haifa, Israel
Taub Fellow - supported by the Taub Foundations. The research was funded by ISF grant 1337/16.
Ryan
O'Donnell
Ryan O'Donnell
Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
Supported by NSF grant CCF-1717606. This material is based upon work supported by the National Science Foundation under grant numbers listed above. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation (NSF).
Xinyu
Wu
Xinyu Wu
Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
10.4230/LIPIcs.ITCS.2019.34
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Marcel-Paul Schützenberger. A characteristic property of certain polynomials of E. F. Moore and C. E. Shannon. Quarterly Progress Report, Research Laboratory of Electronics (RLE), 055.IX:117-118, 1959.
Michel Talagrand. On Russo’s approximate zero-one law. Annals of Probability, 22(3):1576-1587, 1994.
Karl Wimmer. Fourier methods and combinatorics in learning theory. PhD thesis, Carnegie Mellon University, 2009.
Karl Wimmer. Low influence functions over slices of the Boolean hypercube depend on few coordinates. In Proceedings of the 29th Annual Computational Complexity Conference, pages 120-131, 2014.
Yuval Filmus, Ryan O'Donnell, and Xinyu Wu
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Cubic Formula Size Lower Bounds Based on Compositions with Majority
We define new functions based on the Andreev function and prove that they require n^{3}/polylog(n) formula size to compute. The functions we consider are generalizations of the Andreev function using compositions with the majority function. Our arguments apply to composing a hard function with any function that agrees with the majority function (or its negation) on the middle slices of the Boolean cube, as well as iterated compositions of such functions. As a consequence, we obtain n^{3}/polylog(n) lower bounds on the (non-monotone) formula size of an explicit monotone function by combining the monotone address function with the majority function.
formula lower bounds
random restrictions
KRW conjecture
composition
Theory of computation~Computational complexity and cryptography
Theory of computation~Circuit complexity
35:1-35:13
Regular Paper
Anna
Gál
Anna Gál
The University of Texas at Austin, Austin, TX, USA
Part of this work was done while visiting the Simons Institute for the Theory of Computing.
Avishay
Tal
Avishay Tal
Stanford University, Palo Alto, CA, USA
Supported by a Motwani Postdoctoral Fellowship and by NSF grant CCF-1763299. Part of this work was done while visiting the Simons Institute for the Theory of Computing.
Adrian
Trejo Nuñez
Adrian Trejo Nuñez
The University of Texas at Austin, Austin, TX, USA
https://orcid.org/0000-0002-5658-9956
10.4230/LIPIcs.ITCS.2019.35
A. E. Andreev. On a method for obtaining more than quadratic effective lower bounds for the complexity of π-scheme. Moscow University Mathematics Bulletin, 42(1):63-66, 1987.
Andrej Bogdanov. Small Bias Requires Large Formulas. In 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, pages 22:1-22:12, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2018.22.
http://dx.doi.org/10.4230/LIPIcs.ICALP.2018.22
Ruiwen Chen, Valentine Kabanets, Antonina Kolokolova, Ronen Shaltiel, and David Zuckerman. Mining Circuit Lower Bound Proofs for Meta-Algorithms. Computational Complexity, 24(2):333-392, June 2015. URL: http://dx.doi.org/10.1007/s00037-015-0100-0.
http://dx.doi.org/10.1007/s00037-015-0100-0
Irit Dinur and Or Meir. Toward the KRW Composition Conjecture: Cubic Formula Lower Bounds via Communication Complexity. In 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, pages 3:1-3:51, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2016.3.
http://dx.doi.org/10.4230/LIPIcs.CCC.2016.3
Johan Håstad. The Shrinkage Exponent of de Morgan Formulas is 2. SIAM Journal on Computing, 27(1):48-64, 1998. URL: http://dx.doi.org/10.1137/S0097539794261556.
http://dx.doi.org/10.1137/S0097539794261556
Wassily Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American statistical association, 58(301):13-30, 1963.
Russell Impagliazzo, Raghu Meka, and David Zuckerman. Pseudorandomness from Shrinkage. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 111-119, 2012. URL: http://dx.doi.org/10.1109/FOCS.2012.78.
http://dx.doi.org/10.1109/FOCS.2012.78
Russell Impagliazzo and Noam Nisan. The effect of random restrictions on formula size. Random Structures &Algorithms, 4(2):121-133, 1993.
Mauricio Karchmer, Ran Raz, and Avi Wigderson. Super-logarithmic depth lower bounds via the direct sum in communication complexity. Computational Complexity, 5(3-4):191-204, 1995.
V. M. Khrapchenko. Complexity of the realization of a linear function in the class of Π-circuits. Mathematical notes of the Academy of Sciences of the USSR, 9(1):21-23, January 1971. URL: http://dx.doi.org/10.1007/BF01405045.
http://dx.doi.org/10.1007/BF01405045
Ilan Komargodski and Ran Raz. Average-case lower bounds for formula size. In Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 171-180, 2013. URL: http://dx.doi.org/10.1145/2488608.2488630.
http://dx.doi.org/10.1145/2488608.2488630
Ilan Komargodski, Ran Raz, and Avishay Tal. Improved Average-Case Lower Bounds for De Morgan Formula Size: Matching Worst-Case Lower Bound. SIAM Journal on Computing, 46(1):37-57, 2017. URL: http://dx.doi.org/10.1137/15M1048045.
http://dx.doi.org/10.1137/15M1048045
Michael S Paterson and Uri Zwick. Shrinkage of de Morgan formulae under restriction. Random Structures &Algorithms, 4(2):135-150, 1993.
P. Pudlak. Personal communication, 2018.
John Riordan and Claude E Shannon. The Number of Two-Terminal Series-Parallel Networks. Studies in Applied Mathematics, 21(1-4):83-93, 1942.
I. S. Sergeev. Complexity and depth of formulas for symmetric Boolean functions. Moscow University Mathematics Bulletin, 71(3):127-130, 2016.
Bella Abramovna Subbotovskaya. Realizations of linear functions by formulas using +, ∗, and -. Doklady Akademii Nauk SSSR, 136(3):553-555, 1961.
Avishay Tal. Shrinkage of De Morgan Formulae by Spectral Techniques. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 551-560, 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.65.
http://dx.doi.org/10.1109/FOCS.2014.65
Avishay Tal. Formula lower bounds via the quantum method. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 1256-1268, 2017. URL: http://dx.doi.org/10.1145/3055399.3055472.
http://dx.doi.org/10.1145/3055399.3055472
Leslie G. Valiant. Short monotone formulae for the majority function. Journal of Algorithms, 5(3):363-366, 1984.
I. Wegener. The critical complexity of all (monotone) Boolean functions and monotone graph properties. Information and Control, 67:212-222, 1985.
Anna Gál, Avishay Tal, and Adrian Trejo Nuñez
Creative Commons Attribution 3.0 Unported license
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The Space Complexity of Mirror Games
We consider the following game between two players Alice and Bob, which we call the mirror game. Alice and Bob take turns saying numbers belonging to the set {1, 2, ...,N}. A player loses if they repeat a number that has already been said. Otherwise, after N turns, when all the numbers have been spoken, both players win. When N is even, Bob, who goes second, has a very simple (and memoryless) strategy to avoid losing: whenever Alice says x, respond with N+1-x. The question is: does Alice have a similarly simple strategy to win that avoids remembering all the numbers said by Bob?
The answer is no. We prove a linear lower bound on the space complexity of any deterministic winning strategy of Alice. Interestingly, this follows as a consequence of the Eventown-Oddtown theorem from extremal combinatorics. We additionally demonstrate a randomized strategy for Alice that wins with high probability that requires only O~(sqrt N) space (provided that Alice has access to a random matching on K_N).
We also investigate lower bounds for a generalized mirror game where Alice and Bob alternate saying 1 number and b numbers each turn (respectively). When 1+b is a prime, our linear lower bounds continue to hold, but when 1+b is composite, we show that the existence of a o(N) space strategy for Bob (when N != 0 mod (1+b)) implies the existence of exponential-sized matching vector families over Z^N_{1+b}.
Mirror Games
Space Complexity
Eventown-Oddtown
Theory of computation~Interactive computation
36:1-36:14
Regular Paper
https://arxiv.org/abs/1710.02898
Sumegha
Garg
Sumegha Garg
Princeton University, Princeton, USA
Jon
Schneider
Jon Schneider
Google Research, New York, USA
10.4230/LIPIcs.ITCS.2019.36
Sanjeev Arora and Boaz Barak. Computational complexity: a modern approach. Cambridge University Press, 2009.
Elwyn R Berlekamp, John Horton Conway, and Richard K Guy. Winning ways for your mathematical plays, volume 3. AK Peters Natick, 2003.
ER Berlekamp. On subsets with intersections of even cardinality. Canad. Math. Bull, 12(4):471-477, 1969.
Abhishek Bhowmick, Zeev Dvir, and Shachar Lovett. New bounds for matching vector families. SIAM Journal on Computing, 43(5):1654-1683, 2014.
Zeev Dvir, Parikshit Gopalan, and Sergey Yekhanin. Matching vector codes. SIAM Journal on Computing, 40(4):1154-1178, 2011.
Zeev Dvir and Sivakanth Gopi. 2-Server PIR with Subpolynomial Communication. Journal of the ACM (JACM), 63(4):39, 2016.
Klim Efremenko. 3-query locally decodable codes of subexponential length. SIAM Journal on Computing, 41(6):1694-1703, 2012.
Peter Frankl and Richard M. Wilson. Intersection theorems with geometric consequences. Combinatorica, 1(4):357-368, 1981.
Vince Grolmusz. Superpolynomial size set-systems with restricted intersections mod 6 and explicit Ramsey graphs. Combinatorica, 20(1):71-86, 2000.
S Muthukrishnan. Data Streams: Algorithms and Applications (Foundations and Trends in Theoretical Computer Science). Foundations and Trends in Theoretical Computer Science, 2005.
Noam Nisan. On read-once vs. multiple access to randomness in logspace. In Structure in Complexity Theory Conference, 1990, Proceedings., Fifth Annual, pages 179-184. IEEE, 1990.
Tibor Szabó. http://discretemath.imp.fu-berlin.de/DMII-2011-12/linalgmethod.pdf, 2011.
Sumegha Garg and Jon Schneider
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
The Subgraph Testing Model
We initiate a study of testing properties of graphs that are presented as subgraphs of a fixed (or an explicitly given) graph. The tester is given free access to a base graph G=([n],E), and oracle access to a function f:E -> {0,1} that represents a subgraph of G. The tester is required to distinguish between subgraphs that posses a predetermined property and subgraphs that are far from possessing this property.
We focus on bounded-degree base graphs and on the relation between testing graph properties in the subgraph model and testing the same properties in the bounded-degree graph model. We identify cases in which testing is significantly easier in one model than in the other as well as cases in which testing has approximately the same complexity in both models. Our proofs are based on the design and analysis of efficient testers and on the establishment of query-complexity lower bounds.
Property Testing
Graph Properties
Theory of computation~Streaming, sublinear and near linear time algorithms
37:1-37:19
Regular Paper
This research was partially supported by the Israel Science Foundation (grants No. 671/13 and 1146/18).
Oded
Goldreich
Oded Goldreich
Department of Computer Science, Weizmann Institute of Science, Rehovot, Israel
Dana
Ron
Dana Ron
School of Electrical Engineering, Tel Aviv University, Tel Aviv, Israel
10.4230/LIPIcs.ITCS.2019.37
N. Alon, P.D. Seymour, and R. Thomas. A separator theorem for graphs with an excluded minor and its applications. In Proceedings of the Twenty-Second Annual ACM Symposium on Theory of Computing (STOC), pages 293-299, 1990.
E. Ben-Sasson, P. Harsha, and S. Raskhodnikova. 3CNF properties are hard to test. SIAM Journal on Computing, 35(1):1-21, 2005.
I. Benjamini, O. Schramm, and A. Shapira. Every minor-closed property of sparse graphs is testable. Advances in mathematics, 223:2200-2218, 2010.
A. Bogdanov, K. Obata, and L. Trevisan. A lower bound for testing 3-colorability in bounded-degree graphs. In Proceedings of the Forty-Third Annual Symposium on Foundations of Computer Science (FOCS), pages 93-102, 2002.
A. Czumaj, M. Monemizadeh, K. Onak, and C. Sohler. Planar Graphs: Random Walks and Bipartiteness Testing. In Proceedings of the Forty-Third Annual Symposium on Foundations of Computer Science (FOCS), pages 423-432, 2011.
A. Czumaj, A. Shapira, and C. Sohler. Testing Hereditary Properties of Nonexpanding Bounded-Degree Graphs. SIAM Journal on Computing, 38(6):2499-2510, 2009.
A. Edelman, A. Hassidim, H. N. Nguyen, and K. Onak. An Efficient Partitioning Oracle for Bounded-Treewidth Graphs. In Proceedings of the Fifteenth International Workshop on Randomization and Computation (RANDOM), pages 530-541, 2011.
G. Elek. The combinatorial cost. ArXiv Mathematics e-prints, 2006. URL: http://arxiv.org/abs/math/0608474.
http://arxiv.org/abs/math/0608474
G. Elek. Parameter testing with bounded degree graphs of subexponential growth. Random Structures and Algorithms, 37:248-270, 2010.
P. Erdos and A. Renyi. Asymmetric graphs. Acta Mathematica Hungarica, 14(3):295-315, 1963.
E. Fischer, O. Lachish, A. Matsliah, I. Newman, and O. Yahalom. On the query complexity of testing orientations for being Eulerian. ACM Transactions on Algorithms, 8(2):15:1-15:41, 2012.
O. Goldreich. Computational Complexity: A Conceptual Perspective. Cambridge University Press, 2008.
O. Goldreich, editor. Property Testing: Current Research and Surveys. Springer, 2010. LNCS 6390.
O. Goldreich. Introduction to Property Testing. Cambridge University Press, 2017.
O. Goldreich, S. Goldwasser, and D. Ron. Property Testing and its Connection to Learning and Approximation. Journal of the ACM, 45(4):653-750, 1998.
O. Goldreich and D. Ron. A sublinear bipartite tester for bounded-degree graphs. Combinatorica, 19(3):335-373, 1999.
O. Goldreich and D. Ron. Property Testing in Bounded Degree Graphs. Algorithmica, 32(2):302-343, 2002.
O. Goldreich and D. Ron. On Proximity Oblivious Testing. SIAM Journal on Computing, 40(2):534-566, 2011.
O. Goldreich and D. Ron. The Subgraph Testing Model. Electronic Colloquium on Computational Complexity (ECCC), 25:45, 2018.
M. Gonen and D. Ron. On the Benefit of Adaptivity in Property Testing of Dense Graphs. Algorithmica, 58(4):811-830, 2010. Special issue for APPROX-RANDOM 2007.
S. Halevy, O. Lachish, I. Newman, and D. Tsur. Testing Orientation Properties. Electronic Colloquium on Computational Complexity (ECCC), 153, 2005.
A. Hassidim, J. Kelner, H. Nguyen, and K. Onak. Local Graph Partitions for Approximation and Testing. In Proceedings of the Fiftieth Annual Symposium on Foundations of Computer Science (FOCS), pages 22-31, 2009.
L.S. Heath. Embedding outerplanar graphs in small books. SIAM Journal on Algebraic and Discrete Methods, 8(2):198-218, 1987.
T. Kaufman, M. Krivelevich, and D. Ron. Tight Bounds for Testing Bipartiteness in General Graphs. SIAM Journal on Computing, 33(6):1441-1483, 2004.
J.H. Kim, B. Sudakov, and V.H. Vu. On the asymmetry of random regular graphs and random graphs. Random Structures and Algorithms, 21(3-4):216-224, 2002.
R. Levi and D. Ron. A quasi-polynomial time partition oracle for graphs with an excluded minor. ACM Transactions on Algorithms, 11(3):24:1-24:13, 2015.
R.J. Lipton and R.E. Tarjan. A separator theorem for planar graphs. SIAM Journal on Discrete Math, 1979.
I. Newman. Property Testing of Massively Parametrized Problems - A Survey. In O. Goldreich, editor, Property Testing - Current Research and Surveys, pages 142-157. Springer, 2010. LNCS 6390.
I. Newman and C. Sohler. Every Property of Hyperfinite Graphs Is Testable. SIAM Journal on Computing, 42(3):1095-1112, 2013.
M. Parnas and D. Ron. Testing the Diameter of Graphs. Random Structures and Algorithms, 20(2):165-183, 2002.
D. Ron. Property Testing: A Learning Theory Perspective. Foundations and Trends in Machine Learning, 1(3):307-402, 2008.
D. Ron. Algorithmic and Analysis Techniques in Property Testing. Foundations and Trends in Theoretical Computer Science, 5(2):73-205, 2010.
G. Valiant. Algorithmic Approaches to Statistical Questions. PhD thesis, University of California at Berkeley, 2012.
G. Valiant and P. Valiant. Estimating the Unseen: Improved Estimators for Entropy and Other Properties. Journal of the ACM, 64(6):37:1-37:41, 2017.
Oded Goldreich and Dana Ron
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Adventures in Monotone Complexity and TFNP
Separations: We introduce a monotone variant of Xor-Sat and show it has exponential monotone circuit complexity. Since Xor-Sat is in NC^2, this improves qualitatively on the monotone vs. non-monotone separation of Tardos (1988). We also show that monotone span programs over R can be exponentially more powerful than over finite fields. These results can be interpreted as separating subclasses of TFNP in communication complexity.
Characterizations: We show that the communication (resp. query) analogue of PPA (subclass of TFNP) captures span programs over F_2 (resp. Nullstellensatz degree over F_2). Previously, it was known that communication FP captures formulas (Karchmer - Wigderson, 1988) and that communication PLS captures circuits (Razborov, 1995).
TFNP
Monotone Complexity
Communication Complexity
Proof Complexity
Theory of computation~Communication complexity
Theory of computation~Circuit complexity
Theory of computation~Proof complexity
38:1-38:19
Regular Paper
A full version of the paper is available at [Mika Göös et al., 2018], https://eccc.weizmann.ac.il/report/2018/163/.
Mika
Göös
Mika Göös
Institute for Advanced Study, Princeton, NJ, USA
Work done while at Harvard University; supported by the Michael O. Rabin Postdoctoral Fellowship.
Pritish
Kamath
Pritish Kamath
Massachusetts Institute of Technology, Cambridge, MA, USA
Supported in parts by NSF grants CCF-1650733, CCF-1733808, and IIS-1741137.
Robert
Robere
Robert Robere
Simons Institute, Berkeley, CA, USA
Work done while at University of Toronto; supported by NSERC.
Dmitry
Sokolov
Dmitry Sokolov
KTH Royal Institute of Technology, Stockholm, Sweden
Supported by the Swedish Research Council grant 2016-00782, Knut and Alice Wallenberg Foundation grants KAW 2016.0066 and KAW 2016.0433.
10.4230/LIPIcs.ITCS.2019.38
Michael Alekhnovich, Eli Ben-Sasson, Alexander Razborov, and Avi Wigderson. Pseudorandom Generators in Propositional Proof Complexity. SIAM Journal on Computing, 34(1):67-88, 2004. URL: http://dx.doi.org/10.1137/S0097539701389944.
http://dx.doi.org/10.1137/S0097539701389944
László Babai, Peter Frankl, and Janos Simon. Complexity Classes in Communication Complexity Theory. In Proceedings of the 27th Symposium on Foundations of Computer Science (FOCS), pages 337-347. IEEE, 1986. URL: http://dx.doi.org/10.1109/SFCS.1986.15.
http://dx.doi.org/10.1109/SFCS.1986.15
László Babai, Anna Gál, and Avi Wigderson. Superpolynomial Lower Bounds for Monotone Span Programs. Combinatorica, 19(3):301-319, 1999. URL: http://dx.doi.org/10.1007/s004930050058.
http://dx.doi.org/10.1007/s004930050058
Yakov Babichenko, Shahar Dobzinski, and Noam Nisan. The Communication Complexity of Local Search. Technical report, arXiv, 2018. URL: http://arxiv.org/abs/1804.02676.
http://arxiv.org/abs/1804.02676
Yakov Babichenko and Aviad Rubinstein. Communication Complexity of Approximate Nash Equilibria. In Proceedings of the 49th Symposium on Theory of Computing (STOC), pages 878-889. ACM, 2017. URL: http://dx.doi.org/10.1145/3055399.3055407.
http://dx.doi.org/10.1145/3055399.3055407
Paul Beame, Stephen Cook, Jeff Edmonds, Russell Impagliazzo, and Toniann Pitassi. The Relative Complexity of NP Search Problems. Journal of Computer and System Sciences, 57(1):3-19, 1998. URL: http://dx.doi.org/10.1006/jcss.1998.1575.
http://dx.doi.org/10.1006/jcss.1998.1575
Paul Beame, Toniann Pitassi, and Nathan Segerlind. Lower Bounds for Lovász-Schrijver Systems and Beyond Follow from Multiparty Communication Complexity. SIAM Journal on Computing, 37(3):845-869, 2007. URL: http://dx.doi.org/10.1137/060654645.
http://dx.doi.org/10.1137/060654645
Paul Beame and Søren Riis. More on the Relative Strength of Counting Principles. In Proceedings of the DIMACS Workshop on Proof Complexity and Feasible Arithmetics, volume 39, pages 13-35, 1998.
Aleksandrs Belovs, Gábor Ivanyos, Youming Qiao, Miklos Santha, and Siyi Yang. On the Polynomial Parity Argument Complexity of the Combinatorial Nullstellensatz. In Proceedings of the 32nd Computational Complexity Conference (CCC), volume 79, pages 30:1-30:24. Schloss Dagstuhl, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2017.30.
http://dx.doi.org/10.4230/LIPIcs.CCC.2017.30
Eli Ben-Sasson and Avi Wigderson. Short Proofs Are Narrow - Resolution Made Simple. Journal of the ACM, 48(2):149-169, 2001. URL: http://dx.doi.org/10.1145/375827.375835.
http://dx.doi.org/10.1145/375827.375835
Andrei Bulatov. A Dichotomy Theorem for Nonuniform CSPs. In Proceedings of the 58th Symposium on Foundations of Computer Science (FOCS), pages 319-330, 2017. URL: http://dx.doi.org/10.1109/FOCS.2017.37.
http://dx.doi.org/10.1109/FOCS.2017.37
Joshua Buresh-Oppenheim and Tsuyoshi Morioka. Relativized NP search problems and propositional proof systems. In Proceedings of the 19th Conference on Computational Complexity (CCC), pages 54-67, 2004. URL: http://dx.doi.org/10.1109/CCC.2004.1313795.
http://dx.doi.org/10.1109/CCC.2004.1313795
Sam Buss, Dima Grigoriev, Russell Impagliazzo, and Toniann Pitassi. Linear Gaps between Degrees for the Polynomial Calculus Modulo Distinct Primes. Journal of Computer and System Sciences, 62(2):267-289, 2001. URL: http://dx.doi.org/10.1006/jcss.2000.1726.
http://dx.doi.org/10.1006/jcss.2000.1726
Siu Man Chan. Just a Pebble Game. In Proceedings of the 28th Conference on Computational Complexity (CCC), pages 133-143, 2013. URL: http://dx.doi.org/10.1109/CCC.2013.22.
http://dx.doi.org/10.1109/CCC.2013.22
Siu Man Chan and Aaron Potechin. Tight Bounds for Monotone Switching Networks via Fourier Analysis. Theory of Computing, 10(15):389-419, 2014. URL: http://dx.doi.org/10.4086/toc.2014.v010a015.
http://dx.doi.org/10.4086/toc.2014.v010a015
Stephen Cook, Yuval Filmus, and Dai Tri Man Lê. The Complexity of the Comparator Circuit Value Problem. ACM Transactions on Computation Theory, 6(4):15:1-15:44, 2014. URL: http://dx.doi.org/10.1145/2635822.
http://dx.doi.org/10.1145/2635822
Constantinos Daskalakis and Christos Papadimitriou. Continuous Local Search. In Proceedings of the 22nd Symposium on Discrete Algorithms (SODA), pages 790-804. SIAM, 2011. URL: http://dl.acm.org/citation.cfm?id=2133036.2133098.
http://dl.acm.org/citation.cfm?id=2133036.2133098
Susanna de Rezende, Jakob Nordström, and Marc Vinyals. How Limited Interaction Hinders Real Communication (and What It Means for Proof and Circuit Complexity). In Proceedings of the 57th Symposium on Foundations of Computer Science (FOCS), pages 295-304. IEEE, 2016. URL: http://dx.doi.org/10.1109/FOCS.2016.40.
http://dx.doi.org/10.1109/FOCS.2016.40
John Fearnley, Spencer Gordon, Ruta Mehta, and Rahul Savani. End of Potential Line. Technical report, arXiv, 2018. URL: http://arxiv.org/abs/1804.03450.
http://arxiv.org/abs/1804.03450
Tomás Feder and Moshe Vardi. The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory. SIAM Journal on Computing, 28(1):57-104, 1998. URL: http://dx.doi.org/10.1137/S0097539794266766.
http://dx.doi.org/10.1137/S0097539794266766
Anna Gál. A Characterization of Span Program Size and Improved Lower Bounds for Monotone Span Programs. Computational Complexity, 10(4):277-296, 2001. URL: http://dx.doi.org/10.1007/s000370100001.
http://dx.doi.org/10.1007/s000370100001
Anat Ganor and Karthik C. S. Communication Complexity of Correlated Equilibrium with Small Support. In Proceedings of the 22nd International Conference on Randomization and Computation (RANDOM), volume 116, pages 12:1-12:16. Schloss Dagstuhl, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.12.
http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.12
Ankit Garg, Mika Göös, Pritish Kamath, and Dmitry Sokolov. Monotone Circuit Lower Bounds from Resolution. In Proceedings of the 50th Symposium on Theory of Computing (STOC), pages 902-911. ACM, 2018. URL: http://dx.doi.org/10.1145/3188745.3188838.
http://dx.doi.org/10.1145/3188745.3188838
Mika Göös, Pritish Kamath, Robert Robere, and Dmitry Sokolov. Adventures in Monotone Complexity and TFNP. Technical Report TR18-163, Electronic Colloquium on Computational Complexity (ECCC), 2018. URL: https://eccc.weizmann.ac.il/report/2018/163/.
https://eccc.weizmann.ac.il/report/2018/163/
Mika Göös and Toniann Pitassi. Communication Lower Bounds via Critical Block Sensitivity. In Proceedings of the 46th Symposium on Theory of Computing (STOC), pages 847-856. ACM, 2014. URL: http://dx.doi.org/10.1145/2591796.2591838.
http://dx.doi.org/10.1145/2591796.2591838
Mika Göös, Toniann Pitassi, and Thomas Watson. Deterministic Communication vs. Partition Number. In Proceedings of the 56th Symposium on Foundations of Computer Science (FOCS), pages 1077-1088. IEEE, 2015. URL: http://dx.doi.org/10.1109/FOCS.2015.70.
http://dx.doi.org/10.1109/FOCS.2015.70
Mika Göös, Toniann Pitassi, and Thomas Watson. The Landscape of Communication Complexity Classes. In Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP), pages 86:1-86:15. Schloss Dagstuhl, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.86.
http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.86
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Mika Göös, Pritish Kamath, Robert Robere, and Dmitry Sokolov
Creative Commons Attribution 3.0 Unported license
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Algorithmic Polarization for Hidden Markov Models
Using a mild variant of polar codes we design linear compression schemes compressing Hidden Markov sources (where the source is a Markov chain, but whose state is not necessarily observable from its output), and to decode from Hidden Markov channels (where the channel has a state and the error introduced depends on the state). We give the first polynomial time algorithms that manage to compress and decompress (or encode and decode) at input lengths that are polynomial both in the gap to capacity and the mixing time of the Markov chain. Prior work achieved capacity only asymptotically in the limit of large lengths, and polynomial bounds were not available with respect to either the gap to capacity or mixing time. Our results operate in the setting where the source (or the channel) is known. If the source is unknown then compression at such short lengths would lead to effective algorithms for learning parity with noise - thus our results are the first to suggest a separation between the complexity of the problem when the source is known versus when it is unknown.
polar codes
error-correcting codes
compression
hidden markov model
Theory of computation~Error-correcting codes
39:1-39:19
Regular Paper
Venkatesan
Guruswami
Venkatesan Guruswami
Computer Science Department, Carnegie Mellon University, Pittsburgh, PA 15213, USA
Most of this work was done when the author was visiting the Center for Mathematical Sciences and Applications, Harvard University, Cambridge, MA. Research supported in part by NSF grants CCF-1422045 and CCF-1814603.
Preetum
Nakkiran
Preetum Nakkiran
Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, 33 Oxford Street, Cambridge, MA 02138, USA
Work supported in part by the NSF Graduate Research Fellowship Grant No. DGE1144152, and Madhu Sudan’s Simons Investigator Award and NSF Award CCF 1715187.
Madhu
Sudan
Madhu Sudan
Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, 33 Oxford Street, Cambridge, MA 02138, USA
Work supported in part by a Simons Investigator Award and NSF Award CCF 1715187.
10.4230/LIPIcs.ITCS.2019.39
Erdal Arıkan. Channel Polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels. IEEE Transactions on Information Theory, pages 3051-3073, July 2009.
Jarosław Błasiok, Venkatesan Guruswami, Preetum Nakkiran, Atri Rudra, and Madhu Sudan. General strong polarization. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 485-492. ACM, 2018. URL: http://arxiv.org/abs/1802.02718.
http://arxiv.org/abs/1802.02718
Jaroslaw Blasiok, Venkatesan Guruswami, and Madhu Sudan. Polar Codes with Exponentially Small Error at Finite Block Length. In LIPIcs-Leibniz International Proceedings in Informatics, volume 116. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018.
Eren Şaşoğlu. Polar coding theorems for discrete systems. PhD thesis, Ecole Polytechnique Fédérale de Lausanne, 2011.
Venkatesan Guruswami and Patrick Xia. Polar Codes: Speed of Polarization and Polynomial Gap to Capacity. IEEE Trans. Information Theory, 61(1):3-16, 2015. Preliminary version in Proc. of FOCS 2013.
Seyed Hamed Hassani, Kasra Alishahi, and Rüdiger L. Urbanke. Finite-Length Scaling for Polar Codes. IEEE Trans. Information Theory, 60(10):5875-5898, 2014. URL: http://dx.doi.org/10.1109/TIT.2014.2341919.
http://dx.doi.org/10.1109/TIT.2014.2341919
Satish Babu Korada, Eren Sasoglu, and Rüdiger L. Urbanke. Polar Codes: Characterization of Exponent, Bounds, and Constructions. IEEE Transactions on Information Theory, 56(12):6253-6264, 2010. URL: http://dx.doi.org/10.1109/TIT.2010.2080990.
http://dx.doi.org/10.1109/TIT.2010.2080990
Ramtin Pedarsani, Seyed Hamed Hassani, Ido Tal, and Emre Telatar. On the construction of polar codes. In Proceedings of 2011 IEEE International Symposium on Information Theory, pages 11-15, 2011. URL: http://dx.doi.org/10.1109/ISIT.2011.6033724.
http://dx.doi.org/10.1109/ISIT.2011.6033724
Eren Sasoglu and Ido Tal. Polar coding for processes with memory. In Proceedings of the IEEE International Symposium on Information Theory, pages 225-229, 2016.
Boaz Shuval and Ido Tal. Fast Polarization for Processes with Memory. In Proceedings of the IEEE International Symposium on Information Theory, pages 851-855, 2018.
Ido Tal and Alexander Vardy. How to Construct Polar Codes. IEEE Transactions on Information Theory, 59(10):6562-6582, October 2013.
Runxin Wang, Junya Honda, Hirosuke Yamamoto, Rongke Liu, and Yi Hou. Construction of polar codes for channels with memory. In Proceedings of the 2015 IEEE Information Theory Workshop - Fall (ITW), pages 187-191, 2015.
Runxin Wang, Rongke Liu, and Yi Hou. Joint Successive Cancellation Decoding of Polar Codes over Intersymbol Interference Channels. CoRR, 2014. URL: http://arxiv.org/abs/1404.3001.
http://arxiv.org/abs/1404.3001
Venkatesan Guruswami, Preetum Nakkiran, and Madhu Sudan
Creative Commons Attribution 3.0 Unported license
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On the Communication Complexity of Key-Agreement Protocols
Key-agreement protocols whose security is proven in the random oracle model are an important alternative to protocols based on public-key cryptography. In the random oracle model, the parties and the eavesdropper have access to a shared random function (an "oracle"), but the parties are limited in the number of queries they can make to the oracle. The random oracle serves as an abstraction for black-box access to a symmetric cryptographic primitive, such as a collision resistant hash. Unfortunately, as shown by Impagliazzo and Rudich [STOC '89] and Barak and Mahmoody [Crypto '09], such protocols can only guarantee limited secrecy: the key of any l-query protocol can be revealed by an O(l^2)-query adversary. This quadratic gap between the query complexity of the honest parties and the eavesdropper matches the gap obtained by the Merkle's Puzzles protocol of Merkle [CACM '78].
In this work we tackle a new aspect of key-agreement protocols in the random oracle model: their communication complexity. In Merkle's Puzzles, to obtain secrecy against an eavesdropper that makes roughly l^2 queries, the honest parties need to exchange Omega(l) bits. We show that for protocols with certain natural properties, ones that Merkle's Puzzle has, such high communication is unavoidable. Specifically, this is the case if the honest parties' queries are uniformly random, or alternatively if the protocol uses non-adaptive queries and has only two rounds. Our proof for the first setting uses a novel reduction from the set-disjointness problem in two-party communication complexity. For the second setting we prove the lower bound directly, using information-theoretic arguments.
Understanding the communication complexity of protocols whose security is proven (in the random-oracle model) is an important question in the study of practical protocols. Our results and proof techniques are a first step in this direction.
key agreement
random oracle
communication complexity
Merkle's puzzles
Theory of computation~Cryptographic protocols
40:1-40:16
Regular Paper
A full version of the paper is available at [Haitner et al., 2018], https://eccc.weizmann.ac.il/report/2018/031/.
Iftach
Haitner
Iftach Haitner
The Blavatnik school of computer science, Tel Aviv University, Israel
Supported by ERC starting grant 638121. Member of the Check Point Institute for Information Security.
Noam
Mazor
Noam Mazor
The Blavatnik school of computer science, Tel Aviv University, Israel
Supported by ERC starting grant 638121.
Rotem
Oshman
Rotem Oshman
The Blavatnik school of computer science, Tel Aviv University, Israel
Supported by the Israeli Centers of Research Excellence program 4/11 and BSF grant 2014256.
Omer
Reingold
Omer Reingold
Computer Science Department, Stanford University, USA
Supported by NSF grant CCF-1749750.
Amir
Yehudayoff
Amir Yehudayoff
Department of Mathematics, Technion-Israel Institute of Technology, Israel
Supported by ISF grant 1162/15.
10.4230/LIPIcs.ITCS.2019.40
Miklós Ajtai and Cynthia Dwork. A public-key cryptosystem with worst-case/average-case equivalence. In Proceedings of the twenty-ninth annual ACM symposium on Theory of computing, pages 284-293. ACM, 1997.
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Iftach Haitner, Eran Omri, and Hila Zarosim. Limits on the usefulness of random oracles. Journal of Cryptology, 29(2):283-335, 2016.
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Iftach Haitner, Noam Mazor, Rotem Oshman, Omer Reingold, and Amir Yehudayoff
Creative Commons Attribution 3.0 Unported license
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The Paulsen Problem Made Simple
The Paulsen problem is a basic problem in operator theory that was resolved in a recent tour-de-force work of Kwok, Lau, Lee and Ramachandran. In particular, they showed that every epsilon-nearly equal norm Parseval frame in d dimensions is within squared distance O(epsilon d^{13/2}) of an equal norm Parseval frame. We give a dramatically simpler proof based on the notion of radial isotropic position, and along the way show an improved bound of O(epsilon d^2).
radial isotropic position
operator scaling
Paulsen problem
Theory of computation~Design and analysis of algorithms
41:1-41:6
Regular Paper
https://arxiv.org/abs/1809.04726
Linus
Hamilton
Linus Hamilton
Massachusetts Institute of Technology, 77 Massachusetts Ave, USA
This work was supported in part by a Fannie and John Hertz Foundation Fellowship.
Ankur
Moitra
Ankur Moitra
Massachusetts Institute of Technology, 77 Massachusetts Ave, USA
This work was supported in part by NSF CAREER Award CCF-1453261, NSF Large CCF-1565235, a David and Lucile Packard Fellowship and an Alfred P. Sloan Fellowship.
10.4230/LIPIcs.ITCS.2019.41
Keith Ball. Volumes of sections of cubes and related problems. In Geometric aspects of functional analysis, pages 251-260. Springer, 1989.
Franck Barthe. On a reverse form of the Brascamp-Lieb inequality. Inventiones mathematicae, 134(2):335-361, 1998.
Jonathan Bennett, Anthony Carbery, Michael Christ, and Terence Tao. The Brascamp-Lieb inequalities: finiteness, structure and extremals. Geometric and Functional Analysis, 17(5):1343-1415, 2008.
Jameson Cahill and Peter G Casazza. The Paulsen problem in operator theory. submitted to Operators and Matrices, 2011.
Zeev Dvir, Shubhangi Saraf, and Avi Wigderson. Superquadratic Lower Bound for 3-Query Locally Correctable Codes over the Reals. Theory of Computing, 13(1):1-36, 2017.
Jack Edmonds. Submodular functions, matroids, and certain polyhedra. Combinatorial structures and their applications, pages 69-87, 1970.
Jürgen Forster. A linear lower bound on the unbounded error probabilistic communication complexity. Journal of Computer and System Sciences, 65(4):612-625, 2002.
Ankit Garg, Leonid Gurvits, Rafael Oliveira, and Avi Wigderson. A deterministic polynomial time algorithm for non-commutative rational identity testing. In Foundations of Computer Science (FOCS), 2016 IEEE 57th Annual Symposium on, pages 109-117. IEEE, 2016.
Ankit Garg, Leonid Gurvits, Rafael Oliveira, and Avi Wigderson. Algorithmic and optimization aspects of Brascamp-Lieb inequalities, via Operator Scaling. Geometric and Functional Analysis, 28(1):100-145, 2018.
Leonid Gurvits. Classical complexity and quantum entanglement. Journal of Computer and System Sciences, 69(3):448-484, 2004.
Moritz Hardt and Ankur Moitra. Algorithms and hardness for robust subspace recovery. In Conference on Learning Theory, pages 354-375, 2013.
Tsz Chiu Kwok, Lap Chi Lau, Yin Tat Lee, and Akshay Ramachandran. The Paulsen problem, continuous operator scaling, and smoothed analysis. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 182-189. ACM, 2018.
Linus Hamilton and Ankur Moitra
Creative Commons Attribution 3.0 Unported license
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How to Subvert Backdoored Encryption: Security Against Adversaries that Decrypt All Ciphertexts
In this work, we examine the feasibility of secure and undetectable point-to-point communication when an adversary (e.g., a government) can read all encrypted communications of surveillance targets. We consider a model where the only permitted method of communication is via a government-mandated encryption scheme, instantiated with government-mandated keys. Parties cannot simply encrypt ciphertexts of some other encryption scheme, because citizens caught trying to communicate outside the government's knowledge (e.g., by encrypting strings which do not appear to be natural language plaintexts) will be arrested. The one guarantee we suppose is that the government mandates an encryption scheme which is semantically secure against outsiders: a perhaps reasonable supposition when a government might consider it advantageous to secure its people's communication against foreign entities. But then, what good is semantic security against an adversary that holds all the keys and has the power to decrypt?
We show that even in the pessimistic scenario described, citizens can communicate securely and undetectably. In our terminology, this translates to a positive statement: all semantically secure encryption schemes support subliminal communication. Informally, this means that there is a two-party protocol between Alice and Bob where the parties exchange ciphertexts of what appears to be a normal conversation even to someone who knows the secret keys and thus can read the corresponding plaintexts. And yet, at the end of the protocol, Alice will have transmitted her secret message to Bob. Our security definition requires that the adversary not be able to tell whether Alice and Bob are just having a normal conversation using the mandated encryption scheme, or they are using the mandated encryption scheme for subliminal communication.
Our topics may be thought to fall broadly within the realm of steganography. However, we deal with the non-standard setting of an adversarially chosen distribution of cover objects (i.e., a stronger-than-usual adversary), and we take advantage of the fact that our cover objects are ciphertexts of a semantically secure encryption scheme to bypass impossibility results which we show for broader classes of steganographic schemes. We give several constructions of subliminal communication schemes under the assumption that key exchange protocols with pseudorandom messages exist (such as Diffie-Hellman, which in fact has truly random messages).
Backdoored Encryption
Steganography
Theory of computation~Cryptographic protocols
42:1-42:20
Regular Paper
A full version of the paper is available at [Thibaut Horel et al., 2018], https://eprint.iacr.org/2018/212.
Thibaut
Horel
Thibaut Horel
Harvard University, Cambridge, MA, USA
Supported, in part, by the National Science Foundation under grants CAREER IIS-1149662, and CNS-1237235, by the Office of Naval Research under grants YIP N00014-14-1-0485 and N00014-17-1-2131, and by a Google Research Award.
Sunoo
Park
Sunoo Park
MIT, Cambridge, MA, USA
Supported by the Center for Science of Information STC (CSoI), an NSF Science and Technology Center (grant agreement CCF-0939370), MACS project NSF grant CNS-1413920, and a Simons Investigator Award Agreement dated 2012-06-05.
Silas
Richelson
Silas Richelson
University of California, Riverside, CA, USA
Vinod
Vaikuntanathan
Vinod Vaikuntanathan
MIT, Cambridge, MA, USA
Supported in part by NSF Grants CNS-1350619, CNS-1414119 and CNS-1718161, Alfred P. Sloan Research Fellowship, Microsoft Faculty Fellowship and a Steven and Renee Finn Career Development Chair from MIT.
10.4230/LIPIcs.ITCS.2019.42
Per Austrin, Kai-Min Chung, Mohammad Mahmoody, Rafael Pass, and Karn Seth. On the Impossibility of Cryptography with Tamperable Randomness. In CRYPTO 2014, Proceedings, Part I, volume 8616 of Lecture Notes in Computer Science, pages 462-479. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-662-44371-2_26.
http://dx.doi.org/10.1007/978-3-662-44371-2_26
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Mihir Bellare, Kenneth G. Paterson, and Phillip Rogaway. Security of Symmetric Encryption against Mass Surveillance. In CRYPTO 2014, Proceedings, Part I, volume 8616 of Lecture Notes in Computer Science, pages 1-19. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-662-44371-2_1.
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Whitfield Diffie and Martin E. Hellman. New directions in cryptography. IEEE Trans. Information Theory, 22(6):644-654, 1976. URL: http://dx.doi.org/10.1109/TIT.1976.1055638.
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http://dx.doi.org/10.1145/6490.6503
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Jan Zöllner, Hannes Federrath, Herbert Klimant, Andreas Pfitzmann, Rudi Piotraschke, Andreas Westfeld, Guntram Wicke, and Gritta Wolf. Modeling the Security of Steganographic Systems. In David Aucsmith, editor, Information Hiding, Second International Workshop, Portland, Oregon, USA, April 14-17, 1998, Proceedings, volume 1525 of Lecture Notes in Computer Science, pages 344-354. Springer, 1998. URL: http://dx.doi.org/10.1007/3-540-49380-8_24.
http://dx.doi.org/10.1007/3-540-49380-8_24
Thibaut Horel, Sunoo Park, Silas Richelson, and Vinod Vaikuntanathan
Creative Commons Attribution 3.0 Unported license
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On Integer Programming and Convolution
Integer programs with a constant number of constraints are solvable in pseudo-polynomial time. We give a new algorithm with a better pseudo-polynomial running time than previous results. Moreover, we establish a strong connection to the problem (min, +)-convolution. (min, +)-convolution has a trivial quadratic time algorithm and it has been conjectured that this cannot be improved significantly. We show that further improvements to our pseudo-polynomial algorithm for any fixed number of constraints are equivalent to improvements for (min, +)-convolution. This is a strong evidence that our algorithm's running time is the best possible. We also present a faster specialized algorithm for testing feasibility of an integer program with few constraints and for this we also give a tight lower bound, which is based on the SETH.
Integer programming
convolution
dynamic programming
SETH
Theory of computation~Integer programming
Theory of computation~Dynamic programming
43:1-43:17
Regular Paper
Research was supported by German Research Foundation (DFG) projects JA 612/20-1 and JA 612/16-1.
Klaus
Jansen
Klaus Jansen
Department of Computer Science, Kiel University, Kiel, Germany
Lars
Rohwedder
Lars Rohwedder
Department of Computer Science, Kiel University, Kiel, Germany
10.4230/LIPIcs.ITCS.2019.43
Amir Abboud, Karl Bringmann, Danny Hermelin, and Dvir Shabtay. SETH-Based Lower Bounds for Subset Sum and Bicriteria Path. CoRR, abs/1704.04546, 2017. URL: http://arxiv.org/abs/1704.04546.
http://arxiv.org/abs/1704.04546
Kyriakos Axiotis and Christos Tzamos. Capacitated Dynamic Programming: Faster Knapsack and Graph Algorithms. CoRR, abs/1802.06440, 2018. URL: http://arxiv.org/abs/1802.06440.
http://arxiv.org/abs/1802.06440
Arturs Backurs, Piotr Indyk, and Ludwig Schmidt. Better Approximations for Tree Sparsity in Nearly-Linear Time. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 2215-2229, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.145.
http://dx.doi.org/10.1137/1.9781611974782.145
David Bremner, Timothy M. Chan, Erik D. Demaine, Jeff Erickson, Ferran Hurtado, John Iacono, Stefan Langerman, Mihai Patrascu, and Perouz Taslakian. Necklaces, Convolutions, and X+Y. Algorithmica, 69(2):294-314, 2014. URL: http://dx.doi.org/10.1007/s00453-012-9734-3.
http://dx.doi.org/10.1007/s00453-012-9734-3
Karl Bringmann. A Near-Linear Pseudopolynomial Time Algorithm for Subset Sum. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 1073-1084, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.69.
http://dx.doi.org/10.1137/1.9781611974782.69
Timothy M. Chan and Moshe Lewenstein. Clustered Integer 3SUM via Additive Combinatorics. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 31-40, 2015. URL: http://dx.doi.org/10.1145/2746539.2746568.
http://dx.doi.org/10.1145/2746539.2746568
Marek Cygan, Marcin Mucha, Karol Wegrzycki, and Michal Wlodarczyk. On Problems Equivalent to (min, +)-Convolution. In 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, pages 22:1-22:15, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2017.22.
http://dx.doi.org/10.4230/LIPIcs.ICALP.2017.22
Friedrich Eisenbrand and Robert Weismantel. Proximity results and faster algorithms for Integer Programming using the Steinitz Lemma. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 808-816, 2018. URL: http://dx.doi.org/10.1137/1.9781611975031.52.
http://dx.doi.org/10.1137/1.9781611975031.52
Fedor V. Fomin, Fahad Panolan, M. S. Ramanujan, and Saket Saurabh. On the Optimality of Pseudo-polynomial Algorithms for Integer Programming. In 26th Annual European Symposium on Algorithms, ESA 2018, August 20-22, 2018, Helsinki, Finland, pages 31:1-31:13, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.ESA.2018.31.
http://dx.doi.org/10.4230/LIPIcs.ESA.2018.31
Klaus Jansen, Kim-Manuel Klein, and José Verschae. Closing the Gap for Makespan Scheduling via Sparsification Techniques. In 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, pages 72:1-72:13, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.72.
http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.72
Hendrik W. Lenstra Jr. Integer Programming with a Fixed Number of Variables. Math. Oper. Res., 8(4):538-548, 1983. URL: http://dx.doi.org/10.1287/moor.8.4.538.
http://dx.doi.org/10.1287/moor.8.4.538
Ravi Kannan. Minkowski’s Convex Body Theorem and Integer Programming. Math. Oper. Res., 12(3):415-440, 1987. URL: http://dx.doi.org/10.1287/moor.12.3.415.
http://dx.doi.org/10.1287/moor.12.3.415
Hans Kellerer, Ulrich Pferschy, and David Pisinger. Knapsack problems. Springer, 2004.
Eduardo Sany Laber, Wilfredo Bardales Roncalla, and Ferdinando Cicalese. On lower bounds for the Maximum Consecutive Subsums Problem and the (min, +)-convolution. In 2014 IEEE International Symposium on Information Theory, Honolulu, HI, USA, June 29 - July 4, 2014, pages 1807-1811, 2014. URL: http://dx.doi.org/10.1109/ISIT.2014.6875145.
http://dx.doi.org/10.1109/ISIT.2014.6875145
Christos H. Papadimitriou. On the complexity of integer programming. J. ACM, 28(4):765-768, 1981. URL: http://dx.doi.org/10.1145/322276.322287.
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Sergey Vasil'evich Sevast'janov. Approximate solution of some problems in scheduling theory. Metody Diskret. Analiz, 32:66-75, 1978. in Russian.
Ernst Steinitz. Bedingt konvergente Reihen und konvexe Systeme. Journal für die reine und angewandte Mathematik, 143:128-176, 1913.
Klaus Jansen and Lars Rohwedder
Creative Commons Attribution 3.0 Unported license
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Empowering the Configuration-IP - New PTAS Results for Scheduling with Setups Times
Integer linear programs of configurations, or configuration IPs, are a classical tool in the design of algorithms for scheduling and packing problems, where a set of items has to be placed in multiple target locations. Herein a configuration describes a possible placement on one of the target locations, and the IP is used to chose suitable configurations covering the items. We give an augmented IP formulation, which we call the module configuration IP. It can be described within the framework of n-fold integer programming and therefore be solved efficiently. As an application, we consider scheduling problems with setup times, in which a set of jobs has to be scheduled on a set of identical machines, with the objective of minimizing the makespan. For instance, we investigate the case that jobs can be split and scheduled on multiple machines. However, before a part of a job can be processed an uninterrupted setup depending on the job has to be paid. For both of the variants that jobs can be executed in parallel or not, we obtain an efficient polynomial time approximation scheme (EPTAS) of running time f(1/epsilon) x poly(|I|) with a single exponential term in f for the first and a double exponential one for the second case. Previously, only constant factor approximations of 5/3 and 4/3 + epsilon respectively were known. Furthermore, we present an EPTAS for a problem where classes of (non-splittable) jobs are given, and a setup has to be paid for each class of jobs being executed on one machine.
Parallel Machines
Setup Time
EPTAS
n-fold integer programming
Theory of computation~Scheduling algorithms
Theory of computation~Discrete optimization
44:1-44:19
Regular Paper
The long version is hosted on arXiv [Klaus Jansen et al., 2018], https://arxiv.org/abs/1801.06460.
Klaus
Jansen
Klaus Jansen
Department of Computer Science, Kiel University, Kiel, Germany
German Research Foundation (DFG) project JA 612/20-1
Kim-Manuel
Klein
Kim-Manuel Klein
Department of Computer Science, Kiel University, Kiel, Germany
Marten
Maack
Marten Maack
Department of Computer Science, Kiel University, Kiel, Germany
Malin
Rau
Malin Rau
Department of Computer Science, Kiel University, Kiel, Germany
10.4230/LIPIcs.ITCS.2019.44
Ali Allahverdi, Jatinder ND Gupta, and Tariq Aldowaisan. A review of scheduling research involving setup considerations. Omega, 27(2):219-239, 1999.
Ali Allahverdi, CT Ng, TC Edwin Cheng, and Mikhail Y Kovalyov. A survey of scheduling problems with setup times or costs. European journal of operational research, 187(3):985-1032, 2008.
Noga Alon, Yossi Azar, Gerhard J Woeginger, and Tal Yadid. Approximation schemes for scheduling on parallel machines. Journal of Scheduling, 1(1):55-66, 1998.
Bo Chen. A better heuristic for preemptive parallel machine scheduling with batch setup times. SIAM Journal on Computing, 22(6):1303-1318, 1993.
Bo Chen, Yinyu Ye, and Jiawei Zhang. Lot-sizing scheduling with batch setup times. Journal of Scheduling, 9(3):299-310, 2006.
Lin Chen, Dániel Marx, Deshi Ye, and Guochuan Zhang. Parameterized and approximation results for scheduling with a low rank processing time matrix. In LIPIcs-Leibniz International Proceedings in Informatics, volume 66. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2017.
José Correa, Alberto Marchetti-Spaccamela, Jannik Matuschke, Leen Stougie, Ola Svensson, Víctor Verdugo, and José Verschae. Strong LP formulations for scheduling splittable jobs on unrelated machines. Mathematical Programming, 154(1-2):305-328, 2015.
José Correa, Victor Verdugo, and José Verschae. Splitting versus setup trade-offs for scheduling to minimize weighted completion time. Operations Research Letters, 44(4):469-473, 2016.
Friedrich Eisenbrand, Christoph Hunkenschröder, and Kim-Manuel Klein. Faster Algorithms for Integer Programs with Block Structure. In 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, pages 49:1-49:13, 2018.
Paul C Gilmore and Ralph E Gomory. A linear programming approach to the cutting-stock problem. Operations research, 9(6):849-859, 1961.
Michel X Goemans and Thomas Rothvoß. Polynomiality for bin packing with a constant number of item types. In Proceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms, pages 830-839. SIAM, 2014.
Raymond Hemmecke, Shmuel Onn, and Lyubov Romanchuk. N-fold integer programming in cubic time. Mathematical Programming, pages 1-17, 2013.
Dorit S Hochbaum and David B Shmoys. Using dual approximation algorithms for scheduling problems theoretical and practical results. Journal of the ACM (JACM), 34(1):144-162, 1987.
Klaus Jansen, Kim-Manuel Klein, Marten Maack, and Malin Rau. Empowering the Configuration-IP - New PTAS Results for Scheduling with Setups Times. CoRR, abs/ 1801.06460v2, 2018. URL: http://arxiv.org/abs/1801.06460.
http://arxiv.org/abs/1801.06460
Klaus Jansen, Kim-Manuel Klein, and José Verschae. Closing the Gap for Makespan Scheduling via Sparsification Techniques. In 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, pages 72:1-72:13, 2016.
Klaus Jansen and Felix Land. Non-preemptive Scheduling with Setup Times: A PTAS. In European Conference on Parallel Processing, pages 159-170. Springer, 2016.
Ravi Kannan. Minkowski’s convex body theorem and integer programming. Mathematics of operations research, 12(3):415-440, 1987.
Dusan Knop and Martin Koutecký. Scheduling meets n-fold Integer Programming. Journal of Scheduling, pages 1-11, 2017.
Dusan Knop, Martin Koutecký, and Matthias Mnich. Combinatorial n-fold Integer Programming and Applications. In 25th Annual European Symposium on Algorithms, ESA 2017, September 4-6, 2017, Vienna, Austria, pages 54:1-54:14, 2017.
Martin Koutecký, Asaf Levin, and Shmuel Onn. A Parameterized Strongly Polynomial Algorithm for Block Structured Integer Programs. In 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, pages 85:1-85:14, 2018.
Hendrik W Lenstra Jr. Integer programming with a fixed number of variables. Mathematics of operations research, 8(4):538-548, 1983.
Alexander Mäcker, Manuel Malatyali, Friedhelm Meyer auf der Heide, and Sören Riechers. Non-preemptive scheduling on machines with setup times. In Workshop on Algorithms and Data Structures, pages 542-553. Springer, 2015.
Clyde L Monma and Chris N Potts. Analysis of heuristics for preemptive parallel machine scheduling with batch setup times. Operations Research, 41(5):981-993, 1993.
Shmuel Onn. Nonlinear discrete optimization. Zurich Lectures in Advanced Mathematics, European Mathematical Society, 2010.
Frans Schalekamp, René Sitters, Suzanne Van Der Ster, Leen Stougie, Víctor Verdugo, and Anke Van Zuylen. Split scheduling with uniform setup times. Journal of scheduling, 18(2):119-129, 2015.
Petra Schuurman and Gerhard J Woeginger. Preemptive scheduling with job-dependent setup times. In Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms, pages 759-767. Society for Industrial and Applied Mathematics, 1999.
Klaus Jansen, Kim-Manuel Klein, Marten Maack, and Malin Rau
Creative Commons Attribution 3.0 Unported license
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Being Corrupt Requires Being Clever, But Detecting Corruption Doesn't
We consider a variation of the problem of corruption detection on networks posed by Alon, Mossel, and Pemantle '15. In this model, each vertex of a graph can be either truthful or corrupt. Each vertex reports about the types (truthful or corrupt) of all its neighbors to a central agency, where truthful nodes report the true types they see and corrupt nodes report adversarially. The central agency aggregates these reports and attempts to find a single truthful node. Inspired by real auditing networks, we pose our problem for arbitrary graphs and consider corruption through a computational lens. We identify a key combinatorial parameter of the graph m(G), which is the minimal number of corrupted agents needed to prevent the central agency from identifying a single corrupt node. We give an efficient (in fact, linear time) algorithm for the central agency to identify a truthful node that is successful whenever the number of corrupt nodes is less than m(G)/2. On the other hand, we prove that for any constant alpha > 1, it is NP-hard to find a subset of nodes S in G such that corrupting S prevents the central agency from finding one truthful node and |S| <= alpha m(G), assuming the Small Set Expansion Hypothesis (Raghavendra and Steurer, STOC '10). We conclude that being corrupt requires being clever, while detecting corruption does not.
Our main technical insight is a relation between the minimum number of corrupt nodes required to hide all truthful nodes and a certain notion of vertex separability for the underlying graph. Additionally, this insight lets us design an efficient algorithm for a corrupt party to decide which graphs require the fewest corrupted nodes, up to a multiplicative factor of O(log n).
Corruption detection
PMC Model
Small Set Expansion
Hardness of Approximation
Theory of computation~Problems, reductions and completeness
45:1-45:14
Regular Paper
A full version of the paper with all proofs can be found at [Yan Jin et al., 2018], https://arxiv.org/abs/1809.10325.
Yan
Jin
Yan Jin
MIT, 77 Massachusetts Ave, MA, USA
Partially supported by Institute for Data, Systems and Society Fellowship.
Elchanan
Mossel
Elchanan Mossel
MIT, 77 Massachusetts Ave, MA, USA
Partially supported by awards ONR N00014-16-1-2227, NSF CCF1665252 and DMS-1737944.
Govind
Ramnarayan
Govind Ramnarayan
MIT, 77 Massachusetts Ave, MA, USA
Partially supported by awards NSF CCF 1665252 and DMS-1737944.
10.4230/LIPIcs.ITCS.2019.45
Noga Alon, Elchanan Mossel, and Robin Pemantle. Corruption Detection on Networks. CoRR, abs/1505.05637, 2015. URL: http://arxiv.org/abs/1505.05637.
http://arxiv.org/abs/1505.05637
Per Austrin, Toniann Pitassi, and Yu Wu. Inapproximability of Treewidth, One-Shot Pebbling, and Related Layout Problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 15th International Workshop, APPROX 2012, and 16th International Workshop, RANDOM 2012, Cambridge, MA, USA, August 15-17, 2012. Proceedings, pages 13-24, 2012. URL: http://dx.doi.org/10.1007/978-3-642-32512-0_2.
http://dx.doi.org/10.1007/978-3-642-32512-0_2
Walid Ben-Ameur, Mohamed-Ahmed Mohamed-Sidi, and José Neto. The k -separator problem: polyhedra, complexity and approximation results. J. Comb. Optim., 29(1):276-307, 2015. URL: http://dx.doi.org/10.1007/s10878-014-9753-x.
http://dx.doi.org/10.1007/s10878-014-9753-x
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Odd-Helge Fjeldstad. Fighting fiscal corruption: lessons from the Tanzania Revenue Authority. Public Administration and Development: The International Journal of Management Research and Practice, 23(2):165-175, 2003.
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http://dx.doi.org/10.1109/CCC.2012.43
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http://dx.doi.org/10.1109/12.364544
Yan Jin, Elchanan Mossel, and Govind Ramnarayan
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Simulating Random Walks on Graphs in the Streaming Model
We study the problem of approximately simulating a t-step random walk on a graph where the input edges come from a single-pass stream. The straightforward algorithm using reservoir sampling needs O(nt) words of memory. We show that this space complexity is near-optimal for directed graphs. For undirected graphs, we prove an Omega(n sqrt{t})-bit space lower bound, and give a near-optimal algorithm using O(n sqrt{t}) words of space with 2^{-Omega(sqrt{t})} simulation error (defined as the l_1-distance between the output distribution of the simulation algorithm and the distribution of perfect random walks). We also discuss extending the algorithms to the turnstile model, where both insertion and deletion of edges can appear in the input stream.
streaming models
random walks
sampling
Theory of computation~Streaming models
46:1-46:15
Regular Paper
Ce
Jin
Ce Jin
Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China
10.4230/LIPIcs.ITCS.2019.46
Kook Jin Ahn and Sudipto Guha. Linear programming in the semi-streaming model with application to the maximum matching problem. Information and Computation, 222:59-79, 2013. URL: http://dx.doi.org/10.1016/j.ic.2012.10.006.
http://dx.doi.org/10.1016/j.ic.2012.10.006
Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Analyzing graph structure via linear measurements. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 459-467, 2012. URL: http://dx.doi.org/10.1137/1.9781611973099.40.
http://dx.doi.org/10.1137/1.9781611973099.40
Reid Andersen, Fan Chung, and Kevin Lang. Using pagerank to locally partition a graph. Internet Mathematics, 4(1):35-64, 2007. URL: http://dx.doi.org/10.1080/15427951.2007.10129139.
http://dx.doi.org/10.1080/15427951.2007.10129139
Reid Andersen and Yuval Peres. Finding sparse cuts locally using evolving sets. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing (STOC), pages 235-244, 2009. URL: http://dx.doi.org/10.1145/1536414.1536449.
http://dx.doi.org/10.1145/1536414.1536449
Moses Charikar, Liadan O'Callaghan, and Rina Panigrahy. Better streaming algorithms for clustering problems. In Proceedings of the 35th Annual ACM Symposium on Theory of Computing (STOC), pages 30-39, 2003. URL: http://dx.doi.org/10.1145/780542.780548.
http://dx.doi.org/10.1145/780542.780548
Graham Cormode and Shan Muthukrishnan. An improved data stream summary: the count-min sketch and its applications. Journal of Algorithms, 55(1):58-75, 2005. URL: http://dx.doi.org/10.1016/j.jalgor.2003.12.001.
http://dx.doi.org/10.1016/j.jalgor.2003.12.001
Atish Das Sarma, Sreenivas Gollapudi, and Rina Panigrahy. Estimating pagerank on graph streams. Journal of the ACM (JACM), 58(3):13, 2011. URL: http://dx.doi.org/10.1145/1970392.1970397.
http://dx.doi.org/10.1145/1970392.1970397
Leah Epstein, Asaf Levin, Julián Mestre, and Danny Segev. Improved Approximation Guarantees for Weighted Matching in the Semi-streaming Model. SIAM Journal on Discrete Mathematics, 25(3):1251-1265, 2011. URL: http://dx.doi.org/10.1137/100801901.
http://dx.doi.org/10.1137/100801901
Rajesh Jayaram and David P. Woodruff. Perfect Lp Sampling in a Data Stream. In Proceedings of the 59th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 544-555, 2018. URL: http://dx.doi.org/10.1109/FOCS.2018.00058.
http://dx.doi.org/10.1109/FOCS.2018.00058
Mark Jerrum and Alistair Sinclair. Approximating the permanent. SIAM Journal on Computing, 18(6):1149-1178, 1989. URL: http://dx.doi.org/10.1137/0218077.
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http://dx.doi.org/10.1016/0304-3975(86)90174-X
Michael Kapralov. Better bounds for matchings in the streaming model. In Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1679-1697, 2013. URL: http://dx.doi.org/10.1137/1.9781611973105.121.
http://dx.doi.org/10.1137/1.9781611973105.121
Michael Kapralov, Yin Tat Lee, Cameron Musco, Christopher Musco, and Aaron Sidford. Single pass spectral sparsification in dynamic streams. SIAM Journal on Computing, 46(1):456-477, 2017. URL: http://dx.doi.org/10.1137/141002281.
http://dx.doi.org/10.1137/141002281
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http://dx.doi.org/10.1007/s00224-012-9396-1
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http://dx.doi.org/10.1137/080744888
Ce Jin
Creative Commons Attribution 3.0 Unported license
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On the Complexity of Symmetric Polynomials
The fundamental theorem of symmetric polynomials states that for a symmetric polynomial f_{Sym} in C[x_1,x_2,...,x_n], there exists a unique "witness" f in C[y_1,y_2,...,y_n] such that f_{Sym}=f(e_1,e_2,...,e_n), where the e_i's are the elementary symmetric polynomials.
In this paper, we study the arithmetic complexity L(f) of the witness f as a function of the arithmetic complexity L(f_{Sym}) of f_{Sym}. We show that the arithmetic complexity L(f) of f is bounded by poly(L(f_{Sym}),deg(f),n). To the best of our knowledge, prior to this work only exponential upper bounds were known for L(f). The main ingredient in our result is an algebraic analogue of Newton's iteration on power series. As a corollary of this result, we show that if VP != VNP then there exist symmetric polynomial families which have super-polynomial arithmetic complexity.
Furthermore, we study the complexity of testing whether a function is symmetric. For polynomials, this question is equivalent to arithmetic circuit identity testing. In contrast to this, we show that it is hard for Boolean functions.
Symmetric Polynomials
Arithmetic Circuits
Arithmetic Complexity
Power Series
Elementary Symmetric Polynomials
Newton's Iteration
Theory of computation~Algebraic complexity theory
47:1-47:14
Regular Paper
Markus
Bläser
Markus Bläser
Department of Computer Science, Saarland University, Saarland Informatics Campus, Saarbrücken, Germany
Gorav
Jindal
Gorav Jindal
Department of Computer Science, Aalto University, Espoo, Finland
Supported by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No 759557) and by Academy of Finland, under grant number 310415. This work was done while the author was a graduate student at Saarland University and the Max-Planck-Institut für Informatik.
10.4230/LIPIcs.ITCS.2019.47
E. Allender, P. Bürgisser, J. Kjeldgaard-Pedersen, and P. Miltersen. On the Complexity of Numerical Analysis. SIAM Journal on Computing, 38(5):1987-2006, 2009. URL: http://dx.doi.org/10.1137/070697926.
http://dx.doi.org/10.1137/070697926
Garrett Birkhoff and Saunders Mac Lane. A survey of modern algebra. New York : Macmillan, 4th ed edition, 1977. URL: http://www.gbv.de/dms/hbz/toc/ht000038471.pdf.
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Ben Blum-Smith and Samuel Coskey. The Fundamental Theorem on Symmetric Polynomials: History’s First Whiff of Galois Theory. The College Mathematics Journal, 48(1):18-29, 2017. URL: http://www.jstor.org/stable/10.4169/college.math.j.48.1.18.
http://www.jstor.org/stable/10.4169/college.math.j.48.1.18
J. N. Bray, M. D. E. Conder, C. R. Leedham-Green, and E. A. O'Brien. Short presentations for alternating and symmetric groups. Trans. Amer. Math. Soc., 363(6):3277-3285, 2011. URL: http://dx.doi.org/10.1090/S0002-9947-2011-05231-1.
http://dx.doi.org/10.1090/S0002-9947-2011-05231-1
Peter Bürgisser. Completeness and reduction in algebraic complexity theory, volume 7. Springer Science &Business Media, 2013.
Xavier Dahan, Éric Schost, and Jie Wu. Evaluation properties of invariant polynomials. Journal of Symbolic Computation, 44(11):1592-1604, 2009. In Memoriam Karin Gatermann. URL: http://dx.doi.org/10.1016/j.jsc.2008.12.002.
http://dx.doi.org/10.1016/j.jsc.2008.12.002
Pierrick Gaudry, Eric Schost, and Nicolas M. Thiéry. Evaluation properties of symmetric polynomials. International Journal of Algebra and Computation, 16(3):505-523, 2006. URL: http://dx.doi.org/10.1142/S0218196706003128.
http://dx.doi.org/10.1142/S0218196706003128
Oded Goldreich. Computational complexity - a conceptual perspective. Cambridge University Press, 2008.
H. T. Kung and J. F. Traub. All Algebraic Functions Can Be Computed Fast. J. ACM, 25(2):245-260, April 1978. URL: http://dx.doi.org/10.1145/322063.322068.
http://dx.doi.org/10.1145/322063.322068
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https://rjlipton.wordpress.com/2009/07/10/arithmetic-complexity-and-symmetry/
Meena Mahajan. Algebraic Complexity Classes. In Manindra Agrawal and Vikraman Arvind, editors, Perspectives in Computational Complexity: The Somenath Biswas Anniversary Volume, pages 51-75. Springer International Publishing, Cham, 2014. URL: http://dx.doi.org/10.1007/978-3-319-05446-9_4.
http://dx.doi.org/10.1007/978-3-319-05446-9_4
Tateaki Sasaki and Fujio Kako. Solving multivariate algebraic equation by Hensel construction. Japan Journal of Industrial and Applied Mathematics, 16(2):257-285, June 1999. URL: http://dx.doi.org/10.1007/BF03167329.
http://dx.doi.org/10.1007/BF03167329
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http://dx.doi.org/10.1561/0400000039
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L. G. Valiant. Completeness Classes in Algebra. In Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing, STOC '79, pages 249-261, New York, NY, USA, 1979. ACM. URL: http://dx.doi.org/10.1145/800135.804419.
http://dx.doi.org/10.1145/800135.804419
Markus Bläser and Gorav Jindal
Creative Commons Attribution 3.0 Unported license
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The Orthogonal Vectors Conjecture for Branching Programs and Formulas
In the Orthogonal Vectors (OV) problem, we wish to determine if there is an orthogonal pair of vectors among n Boolean vectors in d dimensions. The OV Conjecture (OVC) posits that OV requires n^{2-o(1)} time to solve, for all d=omega(log n). Assuming the OVC, optimal time lower bounds have been proved for many prominent problems in P, such as Edit Distance, Frechet Distance, Longest Common Subsequence, and approximating the diameter of a graph.
We prove that OVC is true in several computational models of interest:
- For all sufficiently large n and d, OV for n vectors in {0,1}^d has branching program complexity Theta~(n * min(n,2^d)). In particular, the lower and upper bounds match up to polylog factors.
- OV has Boolean formula complexity Theta~(n * min(n,2^d)), over all complete bases of O(1) fan-in.
- OV requires Theta~(n * min(n,2^d)) wires, in formulas comprised of gates computing arbitrary symmetric functions of unbounded fan-in.
Our lower bounds basically match the best known (quadratic) lower bounds for any explicit function in those models. Analogous lower bounds hold for many related problems shown to be hard under OVC, such as Batch Partial Match, Batch Subset Queries, and Batch Hamming Nearest Neighbors, all of which have very succinct reductions to OV.
The proofs use a certain kind of input restriction that is different from typical random restrictions where variables are assigned independently. We give a sense in which independent random restrictions cannot be used to show hardness, in that OVC is false in the "average case" even for AC^0 formulas:
For all p in (0,1) there is a delta_p > 0 such that for every n and d, OV instances with input bits independently set to 1 with probability p (and 0 otherwise) can be solved with AC^0 formulas of O(n^{2-delta_p}) size, on all but a o_n(1) fraction of instances. Moreover, lim_{p - > 1}delta_p = 1.
fine-grained complexity
orthogonal vectors
branching programs
symmetric functions
Boolean formulas
Theory of computation~Circuit complexity
48:1-48:15
Regular Paper
Daniel M.
Kane
Daniel M. Kane
CSE and Mathematics, UC San Diego, La Jolla CA, USA
Supported by NSF Award CCF-1553288 (CAREER) and a Sloan Research Fellowship.
Richard Ryan
Williams
Richard Ryan Williams
EECS and CSAIL, MIT, 32 Vassar St., Cambridge MA, USA
https://orcid.org/0000-0003-2326-2233
Supported by NSF CCF-1741615 (CAREER: Common Links in Algorithms and Complexity). Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
10.4230/LIPIcs.ITCS.2019.48
Pairwise comparison of bit vectors, January 20, 2017. URL: https://cstheory.stackexchange.com/questions/37361/pairwise-comparison-of-bit-vectors.
https://cstheory.stackexchange.com/questions/37361/pairwise-comparison-of-bit-vectors
Amir Abboud, Arturs Backurs, Thomas Dueholm Hansen, Virginia Vassilevska Williams, and Or Zamir. Subtree Isomorphism Revisited. In SODA, pages 1256-1271, 2016.
Amir Abboud, Arturs Backurs, and Virginia Vassilevska Williams. Tight Hardness Results for LCS and Other Sequence Similarity Measures. In FOCS, pages 59-78, 2015.
Amir Abboud and Virginia Vassilevska Williams. Popular Conjectures Imply Strong Lower Bounds for Dynamic Problems. In FOCS, pages 434-443, 2014.
Amir Abboud, Virginia Vassilevska Williams, and Joshua Wang. Approximation and fixed parameter subquadratic algorithms for radius and diameter in sparse graphs. In SODA, pages 377-391. Society for Industrial and Applied Mathematics, 2016.
Amir Abboud, Virginia Vassilevska Williams, and Oren Weimann. Consequences of Faster Alignment of Sequences. In ICALP, pages 39-51, 2014.
Amir Abboud, Richard Ryan Williams, and Huacheng Yu. More Applications of the Polynomial Method to Algorithm Design. In SODA, pages 218-230, 2015.
Thomas Dybdahl Ahle, Rasmus Pagh, Ilya P. Razenshteyn, and Francesco Silvestri. On the Complexity of Inner Product Similarity Join. In PODS, pages 151-164, 2016.
Arturs Backurs and Piotr Indyk. Edit Distance Cannot Be Computed in Strongly Subquadratic Time (unless SETH is false). In STOC, pages 51-58, 2015.
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http://eprint.iacr.org/2017/202
Allan Borodin. On Relating Time and Space to Size and Depth. SIAM J. Comput., 6(4):733-744, 1977. URL: http://dx.doi.org/10.1137/0206054.
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Karl Bringmann. Why Walking the Dog Takes Time: Frechet Distance Has No Strongly Subquadratic Algorithms Unless SETH Fails. In FOCS, pages 661-670, 2014.
Karl Bringmann and Marvin Künnemann. Quadratic Conditional Lower Bounds for String Problems and Dynamic Time Warping. In FOCS, pages 79-97, 2015.
Karl Bringmann and Wolfgang Mulzer. Approximability of the discrete Fréchet distance. JoCG, 7(2):46-76, 2016.
Kevin Buchin, Maike Buchin, Maximilian Konzack, Wolfgang Mulzer, and André Schulz. Fine-grained analysis of problems on curves. In EuroCG, Lugano, Switzerland, 2016.
Massimo Cairo, Roberto Grossi, and Romeo Rizzi. New Bounds for Approximating Extremal Distances in Undirected Graphs. In SODA, pages 363-376, 2016.
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http://dx.doi.org/10.1137/1.9781611974331.ch87
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http://arxiv.org/abs/1609.08403
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http://dx.doi.org/10.4230/LIPIcs.CCC.2016.2
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http://dx.doi.org/10.1137/1.9781611973402.135
Daniel M. Kane and R. Ryan Williams
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
SOS Lower Bounds with Hard Constraints: Think Global, Act Local
Many previous Sum-of-Squares (SOS) lower bounds for CSPs had two deficiencies related to global constraints. First, they were not able to support a "cardinality constraint", as in, say, the Min-Bisection problem. Second, while the pseudoexpectation of the objective function was shown to have some value beta, it did not necessarily actually "satisfy" the constraint "objective = beta". In this paper we show how to remedy both deficiencies in the case of random CSPs, by translating global constraints into local constraints. Using these ideas, we also show that degree-Omega(sqrt{n}) SOS does not provide a (4/3 - epsilon)-approximation for Min-Bisection, and degree-Omega(n) SOS does not provide a (11/12 + epsilon)-approximation for Max-Bisection or a (5/4 - epsilon)-approximation for Min-Bisection. No prior SOS lower bounds for these problems were known.
sum-of-squares hierarchy
random constraint satisfaction problems
Theory of computation~Semidefinite programming
Theory of computation~Randomness, geometry and discrete structures
49:1-49:21
Regular Paper
https://arxiv.org/abs/1809.01207
Pravesh K.
Kothari
Pravesh K. Kothari
Department of Computer Science, Princeton University and Institute for Advanced Study, Princeton, USA
Ryan
O'Donnell
Ryan O'Donnell
Department of Computer Science, Carnegie Mellon University, Pittsburgh, USA
Some work performed at the Boğaziçi University Computer Engineering Department, supported by Marie Curie International Incoming Fellowship project number 626373. Also supported by NSF grants CCF-1618679, CCF-1717606. This material is based upon work supported by the National Science Foundation under grant numbers listed above. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation (NSF).
Tselil
Schramm
Tselil Schramm
Department of Computer Science, Harvard and MIT, Cambridge, USA
This work was partly supported by an NSF Graduate Research Fellowship (1106400), and also by a Simons Institute Fellowship.
10.4230/LIPIcs.ITCS.2019.49
Per Austrin and Johan Håstad. Randomly supported independence and resistance. SIAM Journal on Computing, 40(1):1-27, 2011.
Boaz Barak, Fernando Brandão, Aram Harrow, Jonathan Kelner, David Steurer, and Yuan Zhou. Hypercontractivity, Sum-of-Squares Proofs, and their Applications. In Proc. of the 44th Annual ACM Symposium on Theory of Computing, pages 307-326, 2012.
Boaz Barak, Siu On Chan, and Pravesh Kothari. Sum of Squares Lower Bounds from Pairwise Independence. In Proc. of the 47th Annual ACM Symposium on Theory of Computing, pages 97-106, 2015.
Boaz Barak, Samuel Hopkins, Jonathan Kelner, Pravesh Kothari, Ankur Moitra, and Aaron Potechin. A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem. In Proc. of the 57th Annual IEEE Symposium on Foundations of Computer Science, 2016.
Boaz Barak and David Steurer. Proofs, beliefs, and algorithms through the lens of sum-of-squares, 2016. URL: http://sumofsquares.org/public/index.html.
http://sumofsquares.org/public/index.html
Irit Dinur, Ehud Friedgut, Guy Kindler, and Ryan O'Donnell. On the Fourier tails of bounded functions over the discrete cube. Israel Journal of Mathematics, 160(1):389-412, 2007.
Uriel Feige. Relations between average case complexity and approximation complexity. In Proc. of the 34th Annual ACM Symposium on Theory of Computing, pages 543-543, 2002.
Dima Grigoriev. Complexity of Positivstellensatz proofs for the knapsack. Computational Complexity, 10(2):139-154, 2001.
Dima Grigoriev. Linear lower bound on degrees of Positivstellensatz calculus proofs for the parity. Theoretical Computer Science, 259(1-2):613-622, 2001.
Dima Grigoriev, Edward Hirsch, and Dmitrii Pasechnik. Complexity of semialgebraic proofs. Moscow Mathematical Journal, 2(4):647-679, 2002.
Venkatesan Guruswami, Ali Kemal Sinop, and Yuan Zhou. Constant factor Lasserre integrality gaps for graph partitioning problems. SIAM Journal on Optimization, 24(4):1698-1717, 2014.
Johan Håstad. Some optimal inapproximability results. Journal of the ACM, 48(4):798-859, 2001.
Subhash Khot. Ruling out PTAS for Graph Min-Bisection, Dense k-Subgraph, and Bipartite Clique. SIAM Journal on Computing, 36(4):1025-1071, 2006.
Pravesh Kothari, Ryuhei Mori, Ryan O'Donnell, and David Witmer. Sum of squares lower bounds for refuting any CSP. In Proc. of the 49th Annual ACM Symposium on Theory of Computing, pages 132-145, 2017.
Pravesh K. Kothari and Ruta Mehta. Sum-of-squares meets Nash: lower bounds for finding any equilibrium. In Proc. of the 50th Annual ACM Symposium on Theory of Computing, pages 1241-1248, 2018.
Ryan O'Donnell. SOS is not obviously automatizable, even approximately. In Proc. of the 8th Annual Innovations in Theoretical Computer Science conference, 2017.
Ryan O'Donnell and John Wright. A new point of NP-hardness for Unique-Games. In Proc. of the 44th Annual ACM Symposium on Theory of Computing, pages 289-306, 2012.
Prasad Raghavendra and Benjamin Weitz. On the Bit Complexity of Sum-of-Squares Proofs. Technical report, arXiv, 2017. URL: http://arxiv.org/abs/1702.05139.
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Grant Schoenebeck. Linear Level Lasserre Lower Bounds for Certain k-CSPs. In Proc. of the 49th Annual IEEE Symposium on Foundations of Computer Science, pages 593-602, 2008.
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Madhur Tulsiani. CSP gaps and reductions in the Lasserre hierarchy. In Proc. of the 41st Annual ACM Symposium on Theory of Computing, pages 303-312, 2009.
Pravesh Kothari, Ryan O'Donnell, and Tselil Schramm
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Semi-Online Bipartite Matching
In this paper we introduce the semi-online model that generalizes the classical online computational model. The semi-online model postulates that the unknown future has a predictable part and an adversarial part; these parts can be arbitrarily interleaved. An algorithm in this model operates as in the standard online model, i.e., makes an irrevocable decision at each step.
We consider bipartite matching in the semi-online model. Our main contributions are competitive algorithms for this problem and a near-matching hardness bound. The competitive ratio of the algorithms nicely interpolates between the truly offline setting (i.e., no adversarial part) and the truly online setting (i.e., no predictable part).
Semi-Online Algorithms
Bipartite Matching
Theory of computation~Online algorithms
50:1-50:20
Regular Paper
A full version of the paper is available at https://arxiv.org/abs/1812.00134.
Ravi
Kumar
Ravi Kumar
Google, Mountain View, CA, USA
Manish
Purohit
Manish Purohit
Google, Mountain View, CA, USA
Aaron
Schild
Aaron Schild
University of California, Berkeley, CA, USA
Zoya
Svitkina
Zoya Svitkina
Google, Mountain View, CA, USA
Erik
Vee
Erik Vee
Google, Mountain View, CA, USA
10.4230/LIPIcs.ITCS.2019.50
S. Albers and M Hellwig. Semi-online scheduling revisited. Theor. Comput. Sci., 443:1-9, 2012.
J. Balogh and J Békési. Semi-on-line bin packing: a short overview and a new lower bound. Cent. Eur. J. Oper. Res., 21(4):685-698, 2013.
Hans-Joachim Böckenhauer, Dennis Komm, Rastislav Královic, Richard Královic, and Tobias Mömke. On the Advice Complexity of Online Problems. In ISAAC, pages 331-340, 2009.
Allan Borodin and Ran El-Yaniv. Online Computation and Competitive Analysis. Cambridge University Press, 2005.
Sebastien Bubeck and Aleksandrs Slivkins. The best of both worlds: Stochastic and adversarial bandits. In COLT, pages 42.1-42.23, 2012.
Rajiv Gandhi, Samir Khuller, Srinivasan Parthasarathy, and Aravind Srinivasan. Dependent rounding and its applications to approximation algorithms. Journal of the ACM (JACM), 53(3):324-360, 2006.
Ashish Goel, Michael Kapralov, and Sanjeev Khanna. On the communication and streaming complexity of maximum bipartite matching. In SODA, pages 468-485. Society for Industrial and Applied Mathematics, 2012.
Pascal Van Hentenryck and Russell Bent. Online stochastic combinatorial optimization. The MIT Press, 2009.
Bala Kalyanasundaram and Kirk Pruhs. An optimal deterministic algorithm for online b-matching. Theor. Comput. Sci., 233(1-2):319-325, 2000. URL: http://dx.doi.org/10.1016/S0304-3975(99)00140-1.
http://dx.doi.org/10.1016/S0304-3975(99)00140-1
Chinmay Karande, Aranyak Mehta, and Pushkar Tripathi. Online bipartite matching with unknown distributions. In STOC, pages 587-596, 2011.
Anna R. Karlin, Claire Kenyon, and Dana Randall. Dynamic TCP Acknowledgment and Other Stories about e/(e-1). Algorithmica, 36(3):209-224, 2003.
Anna R. Karlin, Mark S. Manasse, Lyle A. McGeoch, and Susan Owicki. Competitive randomized algorithms for nonuniform problems. Algorithmica, 11(6):542-571, 1994.
Richard M Karp, Umesh V Vazirani, and Vijay V Vazirani. An optimal algorithm for on-line bipartite matching. In STOC, pages 352-358, 1990.
Ravi Kumar, Manish Purohit, and Zoya Svitkina. Improving Online Algorithms Using ML Predictions. In NIPS, 2018.
Euiwoong Lee and Sahil Singla. Maximum Matching in the Online Batch-Arrival Model. In IPCO, pages 355-367, 2017.
Thodoris Lykouris and Sergei Vassilvitskii. Competitive caching with machine learned advice. In ICML, pages 3302-3311, 2018.
Mohammad Mahdian, Hamid Nazerzadeh, and Amin Saberi. Online optimization with uncertain information. ACM Trans. Algorithms, 8(1):2:1-2:29, 2012.
Mohammad Mahdian and Qiqi Yan. Online bipartite matching with random arrivals: an approach based on strongly factor-revealing LPs. In STOC, pages 597-606, 2011.
Andres Muñoz Medina and Sergei Vassilvitskii. Revenue Optimization with Approximate Bid Predictions. In NIPS, pages 1856-1864, 2017.
Aranyak Mehta. Online Matching and Ad Allocation. Foundations and Trendsregistered in Theoretical Computer Science, 8(4):265-368, 2013.
Vahab S. Mirrokni, Shayan Oveis Gharan, and Morteza Zadimoghaddam. Simultaneous approximations for adversarial and stochastic online budgeted allocation. In SODA, pages 1690-1701, 2012.
Erik Vee, Sergei Vassilvitskii, and Jayavel Shanmugasundaram. Optimal online assignment with forecasts. In EC, pages 109-118, 2010.
Ravi Kumar, Manish Purohit, Aaron Schild, Zoya Svitkina, and Erik Vee
Creative Commons Attribution 3.0 Unported license
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Strategies for Quantum Races
We initiate the study of quantum races, games where two or more quantum computers compete to solve a computational problem. While the problem of dueling algorithms has been studied for classical deterministic algorithms [Immorlica et al., 2011], the quantum case presents additional sources of uncertainty for the players. The foremost among these is that players do not know if they have solved the problem until they measure their quantum state. This question of "when to measure?" presents a very interesting strategic problem. We develop a game-theoretic model of a multiplayer quantum race, and find an approximate Nash equilibrium where all players play the same strategy. In the two-party case, we further show that this strategy is nearly optimal in terms of payoff among all symmetric Nash equilibria. A key role in our analysis of quantum races is played by a more tractable version of the game where there is no payout on a tie; for such races we completely characterize the Nash equilibria in the two-party case.
One application of our results is to the stability of the Bitcoin protocol when mining is done by quantum computers. Bitcoin mining is a race to solve a computational search problem, with the winner gaining the right to create a new block. Our results inform the strategies that eventual quantum miners should use, and also indicate that the collision probability - the probability that two miners find a new block at the same time - would not be too high in the case of quantum miners. Such collisions are undesirable as they lead to forking of the Bitcoin blockchain.
Game theory
Bitcoin mining
Quantum computing
Convex optimization
Theory of computation~Algorithmic game theory
51:1-51:21
Regular Paper
A full version of the paper is available at [Lee et al., 2018], https://arxiv.org/abs/1809.03671.
Troy
Lee
Troy Lee
Centre for Quantum Software and Information, School of Software, Faculty of Engineering and Information Technology, University of Technology Sydney, Australia
Maharshi
Ray
Maharshi Ray
Centre for Quantum Technologies, National University of Singapore, Singapore
Miklos
Santha
Miklos Santha
IRIF, Univ. Paris Diderot, CNRS, 75205 Paris, France
and, Centre for Quantum Technologies and MajuLab, National University of Singapore, Singapore 117543
10.4230/LIPIcs.ITCS.2019.51
Divesh Aggarwal, Gavin Brennen, Troy Lee, Miklos Santha, and Marco Tomamichel. Quantum attacks on Bitcoin and how to prevent against them. Technical report, arXiv, 2017. URL: http://arxiv.org/abs/1710.10377.
http://arxiv.org/abs/1710.10377
Adam Back. Hashcash - a denial of service counter-measure, 2002. Available at: URL: http://www.hashcash.org/papers/hashcash.pdf.
http://www.hashcash.org/papers/hashcash.pdf
Alex Biryukov and Dmitry Khovratovich. Equihash: Asymmetric proof-of-work based on the generalized birthday problem. Ledger, 2:1-30, 2017.
Bitmain. Bitmain Antminer S9. https://shop.bitmain.com/antminer_s9_asic_bitcoin_miner.html, 2018. Accessed 2018-02-16.
https://shop.bitmain.com/antminer_s9_asic_bitcoin_miner.html
Vitalik Buterin. Bitcoin is not quantum safe, and how we can fix it when needed. https://bitcoinmagazine.com/articles/bitcoin-is-not-quantum-safe-and-how-we-can-fix-1375242150/, 2013.
https://bitcoinmagazine.com/articles/bitcoin-is-not-quantum-safe-and-how-we-can-fix-1375242150/
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Catalin Dohotaru and Peter Høyer. Exact quantum lower bound for Grover’s problem. Technical report, arXiv, 2008. URL: http://arxiv.org/abs/0810.3647.
http://arxiv.org/abs/0810.3647
William S. Dorn. Duality in quadratic programming. Quarterly of applied mathematics, 18(2):155-162, 1960.
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Arthur Gervais, Ghassan Karame, Karl Wüst, Vasileios Glykantzis, Hubert Ritzdorf, and Srdjan Capkun. On the security and performance of proof of work blockchains. In Proceedings of the 2016 ACM SIGSAC Conference on Compute and Communications Security (CCS'16), pages 3-16, 2016.
Lov K. Grover. A Fast Quantum Mechanical Algorithm for Database Search. In Proceedings of the Twenty-eighth Annual ACM Symposium on Theory of Computing, STOC '96, pages 212-219, New York, NY, USA, 1996. ACM. URL: http://dx.doi.org/10.1145/237814.237866.
http://dx.doi.org/10.1145/237814.237866
Nicole Immorlica, Adam Tauman Kalai, Brendan Lucier, Ankur Moitra, Andrew Postlewaite, and Moshe Tennenholtz. Dueling algorithms. In Proceedings of the forty-third annual ACM symposium on theory of computing (STOC'11), pages 215-224, 2011.
Aggelos Kiayias, Alexander Russell, Bernardo David, and Roman Oliynykov. Ouroboros: A provably secure proof-of-stake blockchain protocol. In CRYPTO, pages 357-388, 2017.
T. Lee, M. Ray, and M. Santha. Strategies for quantum races. ArXiv e-prints, September 2018. URL: http://arxiv.org/abs/1809.03671.
http://arxiv.org/abs/1809.03671
Olvi L. Mangasarian and H. Stone. Two-person nonzero-sum games and quadratic programming. Journal of mathematical analysis and applications, 9:348-355, 1964.
Satoshi Nakamoto. Bitcoin: A peer-to-peer electronic cash system, 2009. Available at: URL: http://www.bitcoin.org/pdf.
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John F. Nash. Non-cooperative Games. Annals of Mathematics, 54(2):286-295, 1951.
Or Sattath. On the insecurity of quantum Bitcoin mining. Technical report, arXiv, 2018. Appeared in QCRYPT 2018. URL: http://arxiv.org/abs/1804.08118.
http://arxiv.org/abs/1804.08118
Troy Lee, Maharshi Ray, and Miklos Santha
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Lower Bounds for Tolerant Junta and Unateness Testing via Rejection Sampling of Graphs
We introduce a new model for testing graph properties which we call the rejection sampling model. We show that testing bipartiteness of n-nodes graphs using rejection sampling queries requires complexity Omega~(n^2). Via reductions from the rejection sampling model, we give three new lower bounds for tolerant testing of Boolean functions of the form f : {0,1}^n -> {0,1}:
- Tolerant k-junta testing with non-adaptive queries requires Omega~(k^2) queries.
- Tolerant unateness testing requires Omega~(n) queries.
- Tolerant unateness testing with non-adaptive queries requires Omega~(n^{3/2}) queries.
Given the O~(k^{3/2})-query non-adaptive junta tester of Blais [Eric Blais, 2008], we conclude that non-adaptive tolerant junta testing requires more queries than non-tolerant junta testing. In addition, given the O~(n^{3/4})-query unateness tester of Chen, Waingarten, and Xie [Xi Chen et al., 2017] and the O~(n)-query non-adaptive unateness tester of Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova, and Seshadhri [Roksana Baleshzar et al., 2017], we conclude that tolerant unateness testing requires more queries than non-tolerant unateness testing, in both adaptive and non-adaptive settings. These lower bounds provide the first separation between tolerant and non-tolerant testing for a natural property of Boolean functions.
Property Testing
Juntas
Tolerant Testing
Boolean functions
Theory of computation~Probabilistic computation
52:1-52:20
Regular Paper
A full version of the paper is available at https://arxiv.org/abs/1805.01074v1.
Amit
Levi
Amit Levi
University of Waterloo, Canada
Research supported by NSERC Discovery grant and the David R. Cheriton Graduate Scholarship. Part of this work was done while the author was visiting Columbia University.
Erik
Waingarten
Erik Waingarten
Columbia University, USA
This work is supported in part by the NSF Graduate Research Fellowship under Grant No. DGE-16-44869, CCF-1703925, CCF-1563155, and CCF-1420349.
10.4230/LIPIcs.ITCS.2019.52
Nir Ailon, Bernard Chazelle, Seshadhri Comandur, and Ding Liu. Estimating the distance to a monotone function. Random Structures and Algorithms, 31(3):371-383, 2007.
Roksana Baleshzar, Deeparnab Chakrabarty, Ramesh Krishnan S. Pallavoor, Sofya Raskhodnikova, and C. Seshadhri. A lower bound for nonadaptive, one-sided error testing of unateness of Boolean functions over the hypercube. arXiv preprint, 2017. URL: http://arxiv.org/abs/1706.00053.
http://arxiv.org/abs/1706.00053
Roksana Baleshzar, Deeparnab Chakrabarty, Ramesh Krishnan S. Pallavoor, Sofya Raskhodnikova, and C. Seshadhri. Optimal Unateness Testers for Real-Values Functions: Adaptivity Helps. In Proceedings of the 44th International Colloquium on Automata, Languages and Programming (ICALP '2017), 2017.
Aleksandrs Belovs and Eric Blais. A polynomial lower bound for testing monotonicity. In Proceedings of the 48th ACM Symposium on the Theory of Computing (STOC '2016), pages 1021-1032, 2016.
Piotr Berman, Meiram Murzabulatov, and Sofya Raskhodnikova. Tolerant Testers of Image Properties. In Proceedings of the 43th International Colloquium on Automata, Languages and Programming (ICALP '2016), pages 90:1-90:14, 2016.
Piotr Berman, Sofya Raskhodnikova, and Grigory Yaroslavtsev. L_p-testing. In Proceedings of the 46th ACM Symposium on the Theory of Computing (STOC '2014), 2014.
Eric Blais. Improved bounds for testing juntas. In Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques, pages 317-330. Springer, 2008.
Eric Blais. Testing juntas nearly optimally. In Proceedings of the 41st ACM Symposium on the Theory of Computing (STOC '2009), pages 151-158, 2009.
Eric Blais, Clément L Canonne, Talya Eden, Amit Levi, and Dana Ron. Tolerant junta testing and the connection to submodular optimization and function isomorphism. In Proceedings of the 29th ACM-SIAM Symposium on Discrete Algorithms (SODA '2018), pages 2113-2132, 2018.
Harry Buhrman, David Garcıa-Soriano, Arie Matsliah, and Ronald de Wolf. The non-adaptive query complexity of testing k-parities. Chicago Journal of Theoretical Computer Science, 6:1-11, 2013.
Andrea Campagna, Alan Guo, and Ronitt Rubinfeld. Local reconstructors and tolerant testers for connectivity and diameter. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 411-424. Springer, 2013.
Clément L. Canonne, Dana Ron, and Rocco A. Servedio. Testing probability distributions using conditional samples. SIAM Journal on Computing, 44(3):540-616, 2015.
Deeparnab Chakrabarty and Seshadhri Comandur. An o(n) monotonicity tester for boolean functions over the hypercube. SIAM Journal on Computing, 45(2):461-472, 2016.
Deeparnab Chakrabarty and C. Seshadhri. A Õ(n) non-adaptive tester for unateness. arXiv preprint, 2016. URL: http://arxiv.org/abs/1608.06980.
http://arxiv.org/abs/1608.06980
Sourav Chakraborty, Eldar Fischer, David García-Soriano, and Arie Matsliah. Junto-symmetric functions, hypergraph isomorphism and crunching. In Proceedings of the 27th Conference on Computational Complexity (CCC '2012), pages 148-158. IEEE, 2012.
Sourav Chakraborty, Eldar Fischer, Yonatan Goldhirsh, and Arie Matsliah. On the power of conditional samples in distribution testing. SIAM Journal on Computing, 45(4):1261-1296, 2016.
Xi Chen, Rocco A. Servedio, Li-Yang Tan, Erik Waingarten, and Jinyu Xie. Settling the query complexity of non-adaptive junta testing. In Proceedings of the 32nd Conference on Computational Complexity (CCC '2017), 2017.
Xi Chen, Erik Waingarten, and Jinyu Xie. Beyond Talagrand functions: new lower bounds for testing monotonicity and unateness. In Proceedings of the 49th ACM Symposium on the Theory of Computing (STOC '2017), 2017.
Xi Chen, Erik Waingarten, and Jinyu Xie. Boolean Unateness Testing with Õ(n^3/4) Adaptive Queries. In Proceedings of the 58th Annual IEEE Symposium on Foundations of Computer Science (FOCS '2017), 2017.
Hana Chockler and Dan Gutfreund. A lower bound for testing juntas. Information Processing Letters, pages 301-305, 2004.
Ilias Diakonikolas, Homin K Lee, Kevin Matulef, Krzysztof Onak, Ronitt Rubinfeld, Rocco A Servedio, and Andrew Wan. Testing for concise representations. In Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS '2007), pages 549-558. IEEE, 2007.
Shahar Fattal and Dana Ron. Approximating the distance to monotonicity in high dimensions. ACM Transactions on Algorithms, 6(3):52, 2010.
Eldar Fischer and Lance Fortnow. Tolerant versus intolerant testing for Boolean properties. Theory of Computing, 2(9):173-183, 2006.
Eldar Fischer, Guy Kindler, Dana Ron, Shmuel Safra, and Alex Samorodnitsky. Testing juntas. Journal of Computer and System Sciences, 68(4):753-787, 2004. URL: http://dx.doi.org/10.1016/j.jcss.2003.11.004.
http://dx.doi.org/10.1016/j.jcss.2003.11.004
Eldar Fischer and Ilan Newman. Testing versus estimation of graph properties. SIAM Journal on Computing, 37(2):482-501, 2007.
Oded Goldreich. Introduction to property testing. Cambridge University Press, 2017.
Oded Goldreich, Shafi Goldwasser, Eric Lehman, Dana Ron, and Alex Samordinsky. Testing Monotonicity. Combinatorica, 20(3):301-337, 2000.
Oded Goldreich, Shafi Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. Journal of the ACM, 45(4):653-750, 1998.
Oded Goldreich and Dana Ron. On sample-based testers. ACM Transactions on Computation Theory, 8(2), 2016.
Venkatesan Guruswami and Atri Rudra. Tolerant locally testable codes. In Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques, pages 306-317. Springer, 2005.
Subhash Khot and Igor Shinkar. An Õ(n) queries adaptive tester for unateness. In Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques, pages 37:1-37:7, 2016.
Swastik Kopparty and Shubhangi Saraf. Tolerant linearity testing and locally testable codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 601-614. Springer, 2009.
Sharon Marko and Dana Ron. Approximating the distance to properties in bounded-degree and general sparse graphs. ACM Transactions on Algorithms, 5(2):22, 2009.
Michal Parnas, Dana Ron, and Ronitt Rubinfeld. Tolerant property testing and distance approximation. Journal of Computer and System Sciences, 72(6):1012-1042, 2006.
Dana Ron. Property testing: A learning theory perspective. Foundations and Trendsregistered in Machine Learning, 1(3):307-402, 2008.
Dana Ron. Algorithmic and analysis techniques in property testing. Foundations and Trendsregistered in Theoretical Computer Science, 5(2):73-205, 2010.
Rocco A Servedio, Li-Yang Tan, and John Wright. Adaptivity helps for testing juntas. In Proceedings of the 30th Conference on Computational Complexity (CCC '2015), pages 264-279, 2015.
Roei Tell. A Note on Tolerant Testing with One-Sided Error. In Electronic Colloquium on Computational Complexity (ECCC), volume 23, page 32, 2016.
Amit Levi and Erik Waingarten
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Secret Sharing with Binary Shares
Shamir's celebrated secret sharing scheme provides an efficient method for encoding a secret of arbitrary length l among any N <= 2^l players such that for a threshold parameter t, (i) the knowledge of any t shares does not reveal any information about the secret and, (ii) any choice of t+1 shares fully reveals the secret. It is known that any such threshold secret sharing scheme necessarily requires shares of length l, and in this sense Shamir's scheme is optimal. The more general notion of ramp schemes requires the reconstruction of secret from any t+g shares, for a positive integer gap parameter g. Ramp secret sharing scheme necessarily requires shares of length l/g. Other than the bound related to secret length l, the share lengths of ramp schemes can not go below a quantity that depends only on the gap ratio g/N.
In this work, we study secret sharing in the extremal case of bit-long shares and arbitrarily small gap ratio g/N, where standard ramp secret sharing becomes impossible. We show, however, that a slightly relaxed but equally effective notion of semantic security for the secret, and negligible reconstruction error probability, eliminate the impossibility. Moreover, we provide explicit constructions of such schemes. One of the consequences of our relaxation is that, unlike standard ramp schemes with perfect secrecy, adaptive and non-adaptive adversaries need different analysis and construction. For non-adaptive adversaries, we explicitly construct secret sharing schemes that provide secrecy against any tau fraction of observed shares, and reconstruction from any rho fraction of shares, for any choices of 0 <= tau < rho <= 1. Our construction achieves secret length N(rho-tau-o(1)), which we show to be optimal. For adaptive adversaries, we construct explicit schemes attaining a secret length Omega(N(rho-tau)). We discuss our results and open questions.
Secret sharing scheme
Wiretap channel
Security and privacy~Information-theoretic techniques
Theory of computation~Expander graphs and randomness extractors
Theory of computation~Error-correcting codes
53:1-53:20
Regular Paper
The research of Fuchun Lin and Huaxiong Wang was supported by Singapore Ministry of Education under Research Grant MOE2016-T2-2-014(S) and RG133/17(S). The research of Venkatesan Guruswami was supported in part by United States NSF grants CCF-1422045 and CCF-1563742. The research of Reihaneh Safavi-Naini was in part supported by Natural Sciences and Engineering Research Council of Canada, Discovery Grants Program.
Full version at https://eprint.iacr.org/2018/746.
Fuchun
Lin
Fuchun Lin
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, SG
Mahdi
Cheraghchi
Mahdi Cheraghchi
Department of Computing, Imperial College London, UK
Venkatesan
Guruswami
Venkatesan Guruswami
Computer Science Department, Carnegie Mellon University, USA
Reihaneh
Safavi-Naini
Reihaneh Safavi-Naini
Department of Computer Science, University of Calgary, CA
Huaxiong
Wang
Huaxiong Wang
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, SG
10.4230/LIPIcs.ITCS.2019.53
Vaneet Aggarwal, Lifeng Lai, A. Robert Calderbank, and H. Vincent Poor. Wiretap channel type II with an active eavesdropper. IEEE International Symposium on Information Theory, ISIT 2009, pages 1944-1948, 2009. URL: http://dx.doi.org/10.1109/ISIT.2009.5205631.
http://dx.doi.org/10.1109/ISIT.2009.5205631
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http://dx.doi.org/10.1007/s00493-011-2604-9
Fuchun Lin, Mahdi Cheraghchi, Venkatesan Guruswami, Reihaneh Safavi-Naini, and Huaxiong Wang
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
On the Communication Complexity of High-Dimensional Permutations
We study the multiparty communication complexity of high dimensional permutations in the Number On the Forehead (NOF) model. This model is due to Chandra, Furst and Lipton (CFL) who also gave a nontrivial protocol for the Exactly-n problem where three players receive integer inputs and need to decide if their inputs sum to a given integer n. There is a considerable body of literature dealing with the same problem, where (N,+) is replaced by some other abelian group. Our work can be viewed as a far-reaching extension of this line of research. We show that the known lower bounds for that group-theoretic problem apply to all high dimensional permutations. We introduce new proof techniques that reveal new and unexpected connections between NOF communication complexity of permutations and a variety of well-known problems in combinatorics. We also give a direct algorithmic protocol for Exactly-n. In contrast, all previous constructions relied on large sets of integers without a 3-term arithmetic progression.
High dimensional permutations
Number On the Forehead model
Additive combinatorics
Theory of computation~Communication complexity
54:1-54:20
Regular Paper
A full version of the paper is available at https://arxiv.org/abs/1706.02207.
Nati
Linial
Nati Linial
Hebrew University of Jerusalem, Jerusalem, Israel
Supported in part by ERC grant 339096, High-dimensional combinatorics.
Toniann
Pitassi
Toniann Pitassi
University of Toronto, Toronto, Canada and IAS, Princeton, U.S.A.
Adi
Shraibman
Adi Shraibman
The Academic College of Tel-Aviv-Yaffo, Tel-Aviv, Israel
10.4230/LIPIcs.ITCS.2019.54
A. Ada, A. Chattopadhyay, O. Fawzi, and P. Nguyen. The NOF multiparty communication complexity of composed functions. computational complexity, 24(3):645-694, 2015.
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Nati Linial, Toniann Pitassi, and Adi Shraibman
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Fisher Zeros and Correlation Decay in the Ising Model
The Ising model originated in statistical physics as a means of studying phase transitions in magnets, and has been the object of intensive study for almost a century. Combinatorially, it can be viewed as a natural distribution over cuts in a graph, and it has also been widely studied in computer science, especially in the context of approximate counting and sampling. In this paper, we study the complex zeros of the partition function of the Ising model, viewed as a polynomial in the "interaction parameter"; these are known as Fisher zeros in light of their introduction by Fisher in 1965. While the zeros of the partition function as a polynomial in the "field" parameter have been extensively studied since the classical work of Lee and Yang, comparatively little is known about Fisher zeros. Our main result shows that the zero-field Ising model has no Fisher zeros in a complex neighborhood of the entire region of parameters where the model exhibits correlation decay. In addition to shedding light on Fisher zeros themselves, this result also establishes a formal connection between two distinct notions of phase transition for the Ising model: the absence of complex zeros (analyticity of the free energy, or the logarithm of the partition function) and decay of correlations with distance. We also discuss the consequences of our result for efficient deterministic approximation of the partition function. Our proof relies heavily on algorithmic techniques, notably Weitz's self-avoiding walk tree, and as such belongs to a growing body of work that uses algorithmic methods to resolve classical questions in statistical physics.
Ising model
zeros of polynomials
partition functions
approximate counting
phase transitions
Mathematics of computing~Discrete mathematics
55:1-55:8
Regular Paper
Full version available at [Jingcheng Liu et al., 2018], https://arxiv.org/abs/1807.06577.
Jingcheng
Liu
Jingcheng Liu
Computer Science Division, UC Berkeley, USA
Supported by US NSF grant CCF-1815328.
Alistair
Sinclair
Alistair Sinclair
Computer Science Division, UC Berkeley, USA
Supported by US NSF grant CCF-1815328.
Piyush
Srivastava
Piyush Srivastava
Tata Institute of Fundamental Research, Mumbai, India
Supported by a Ramanujan Fellowship of the Indian Department of Science and Technology.
10.4230/LIPIcs.ITCS.2019.55
Nima Anari and Shayan Oveis Gharan. The Kadison-Singer problem for strongly Rayleigh measures and applications to Asymmetric TSP. In Proc. 56th IEEE Symp. Found. Comp. Sci. (FOCS), October 2015.
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Jingcheng Liu, Alistair Sinclair, and Piyush Srivastava. Fisher zeros and correlation decay in the Ising model, 2018. URL: https://arxiv.org/abs/1807.06577.
https://arxiv.org/abs/1807.06577
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http://dx.doi.org/10.1016/j.ipl.2011.04.012
Jingcheng Liu, Alistair Sinclair, and Piyush Srivastava
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Quadratic Time-Space Lower Bounds for Computing Natural Functions with a Random Oracle
We define a model of size-S R-way branching programs with oracles that can make up to S distinct oracle queries over all of their possible inputs, and generalize a lower bound proof strategy of Beame [SICOMP 1991] to apply in the case of random oracles. Through a series of succinct reductions, we prove that the following problems require randomized algorithms where the product of running time and space usage must be Omega(n^2/poly(log n)) to obtain correct answers with constant nonzero probability, even for algorithms with constant-time access to a uniform random oracle (i.e., a uniform random hash function):
- Given an unordered list L of n elements from [n] (possibly with repeated elements), output [n]-L.
- Counting satisfying assignments to a given 2CNF, and printing any satisfying assignment to a given 3CNF. Note it is a major open problem to prove a time-space product lower bound of n^{2-o(1)} for the decision version of SAT, or even for the decision problem Majority-SAT.
- Printing the truth table of a given CNF formula F with k inputs and n=O(2^k) clauses, with values printed in lexicographical order (i.e., F(0^k), F(0^{k-1}1), ..., F(1^k)). Thus we have a 4^k/poly(k) lower bound in this case.
- Evaluating a circuit with n inputs and O(n) outputs.
As our lower bounds are based on R-way branching programs, they hold for any reasonable model of computation (e.g. log-word RAMs and multitape Turing machines).
branching programs
random oracles
time-space tradeoffs
lower bounds
SAT
counting complexity
Theory of computation~Circuit complexity
Theory of computation~Oracles and decision trees
56:1-56:20
Regular Paper
Supported by NSF CCF-1741615 (CAREER: Common Links in Algorithms and Complexity).
Dylan M.
McKay
Dylan M. McKay
EECS and CSAIL, MIT, 32 Vassar St., Cambridge MA, USA
Richard Ryan
Williams
Richard Ryan Williams
EECS and CSAIL, MIT, 32 Vassar St., Cambridge MA, USA
https://orcid.org/0000-0003-2326-2233
Parts of this work were performed while visiting the Simons Institute for the Theory of Computing and the EECS department at UC Berkeley.
10.4230/LIPIcs.ITCS.2019.56
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Dylan M. McKay and Richard Ryan Williams
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Random Projection in the Brain and Computation with Assemblies of Neurons
It has been recently shown via simulations [Dasgupta et al., 2017] that random projection followed by a cap operation (setting to one the k largest elements of a vector and everything else to zero), a map believed to be an important part of the insect olfactory system, has strong locality sensitivity properties. We calculate the asymptotic law whereby the overlap in the input vectors is conserved, verifying mathematically this empirical finding. We then focus on the far more complex homologous operation in the mammalian brain, the creation through successive projections and caps of an assembly (roughly, a set of excitatory neurons representing a memory or concept) in the presence of recurrent synapses and plasticity. After providing a careful definition of assemblies, we prove that the operation of assembly projection converges with high probability, over the randomness of synaptic connectivity, even if plasticity is relatively small (previous proofs relied on high plasticity). We also show that assembly projection has itself some locality preservation properties. Finally, we propose a large repertoire of assembly operations, including associate, merge, reciprocal project, and append, each of them both biologically plausible and consistent with what we know from experiments, and show that this computational system is capable of simulating, again with high probability, arbitrary computation in a quite natural way. We hope that this novel way of looking at brain computation, open-ended and based on reasonably mainstream ideas in neuroscience, may prove an attractive entry point for computer scientists to work on understanding the brain.
Brain computation
random projection
assemblies
plasticity
memory
association
Theory of computation~Models of computation
Theory of computation~Randomness, geometry and discrete structures
57:1-57:19
Regular Paper
This work was supported by NSF grants CCF-1563838, CCF-1819935, CCF-1763970, and CCF-1717349.
Christos H.
Papadimitriou
Christos H. Papadimitriou
Columbia University, USA
Santosh S.
Vempala
Santosh S. Vempala
Georgia Tech, USA
10.4230/LIPIcs.ITCS.2019.57
Rosa I. Arriaga, David Rutter, Maya Cakmak, and Santosh S. Vempala. Visual Categorization with Random Projection. Neural Computation, 27(10):2132-2147, 2015. URL: http://dx.doi.org/10.1162/NECO_a_00769.
http://dx.doi.org/10.1162/NECO_a_00769
Rosa I. Arriaga and Santosh Vempala. An algorithmic theory of learning: Robust concepts and random projection. Machine Learning, 63(2):161-182, 2006. URL: http://dx.doi.org/10.1007/s10994-006-6265-7.
http://dx.doi.org/10.1007/s10994-006-6265-7
M.F. Balcan, A. Blum, and S. Vempala. Kernels as Features: On Kernels, Margins, and Low-dimensional Mappings. Machine Learning, 65(1):79-94, 2006.
Robert C Berwick and Noam Chomsky. Why only us: Language and evolution. MIT Press, 2016.
G Buzsaki. Neural syntax: cell assemblies, synapsembles, and readers. Neuron, 68(3), 2010.
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Sanjoy Dasgupta, Charles F. Stevens, and Saket Navlakha. A neural algorithm for a fundamental computing problem. Science, 358(6364):793-796, 2017. URL: http://dx.doi.org/10.1126/science.aam9868.
http://dx.doi.org/10.1126/science.aam9868
Emanuela De Falco, Matias J Ison, Itzhak Fried, and Rodrigo Quian Quiroga. Long-term coding of personal and universal associations underlying the memory web in the human brain. Nature Communications, 7:13408, 2016.
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Christos H. Papadimitriou and Santosh S. Vempala
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Local Computation Algorithms for Spanners
A graph spanner is a fundamental graph structure that faithfully preserves the pairwise distances in the input graph up to a small multiplicative stretch. The common objective in the computation of spanners is to achieve the best-known existential size-stretch trade-off efficiently.
Classical models and algorithmic analysis of graph spanners essentially assume that the algorithm can read the input graph, construct the desired spanner, and write the answer to the output tape. However, when considering massive graphs containing millions or even billions of nodes not only the input graph, but also the output spanner might be too large for a single processor to store.
To tackle this challenge, we initiate the study of local computation algorithms (LCAs) for graph spanners in general graphs, where the algorithm should locally decide whether a given edge (u,v) in E belongs to the output (sparse) spanner or not. Such LCAs give the user the "illusion" that a specific sparse spanner for the graph is maintained, without ever fully computing it. We present several results for this setting, including:
- For general n-vertex graphs and for parameter r in {2,3}, there exists an LCA for (2r-1)-spanners with O~(n^{1+1/r}) edges and sublinear probe complexity of O~(n^{1-1/2r}). These size/stretch trade-offs are best possible (up to polylogarithmic factors).
- For every k >= 1 and n-vertex graph with maximum degree Delta, there exists an LCA for O(k^2) spanners with O~(n^{1+1/k}) edges, probe complexity of O~(Delta^4 n^{2/3}), and random seed of size polylog(n). This improves upon, and extends the work of [Lenzen-Levi, ICALP'18].
We also complement these constructions by providing a polynomial lower bound on the probe complexity of LCAs for graph spanners that holds even for the simpler task of computing a sparse connected subgraph with o(m) edges.
To the best of our knowledge, our results on 3 and 5-spanners are the first LCAs with sublinear (in Delta) probe-complexity for Delta = n^{Omega(1)}.
Local Computation Algorithms
Sub-linear Algorithms
Graph Spanners
Theory of computation~Sparsification and spanners
Theory of computation~Sketching and sampling
58:1-58:21
Regular Paper
Merav
Parter
Merav Parter
Weizmann IS, Rehovot, Israel
MP is supported by Minerva Foundation (124042) and ISF-2084/18.
Ronitt
Rubinfeld
Ronitt Rubinfeld
CSAIL, MIT, Cambridge, MA, USA and TAU, Tel Aviv, Israel
RR is supported by the NSF grants CCF-1650733, CCF-1733808, IIS-1741137 and CCF-1740751.
Ali
Vakilian
Ali Vakilian
CSAIL, MIT, Cambridge, MA, USA
AV is supported by the NSF grant CCF-1535851.
Anak
Yodpinyanee
Anak Yodpinyanee
CSAIL, MIT, Cambridge, MA, USA
AY is supported by the NSF grants CCF-1650733, CCF-1733808, IIS-1741137 and the DPST scholarship, Royal Thai Government.
10.4230/LIPIcs.ITCS.2019.58
Noga Alon, Ronitt Rubinfeld, Shai Vardi, and Ning Xie. Space-efficient local computation algorithms. In Proc. 23rd ACM-SIAM Sympos. Discrete Algs. (SODA), pages 1132-1139, 2012.
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Merav Parter, Ronitt Rubinfeld, Ali Vakilian, and Anak Yodpinyanee
Creative Commons Attribution 3.0 Unported license
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Proofs of Catalytic Space
Proofs of space (PoS) [Dziembowski et al., CRYPTO'15] are proof systems where a prover can convince a verifier that he "wastes" disk space. PoS were introduced as a more ecological and economical replacement for proofs of work which are currently used to secure blockchains like Bitcoin. In this work we investigate extensions of PoS which allow the prover to embed useful data into the dedicated space, which later can be recovered.
Our first contribution is a security proof for the original PoS from CRYPTO'15 in the random oracle model (the original proof only applied to a restricted class of adversaries which can store a subset of the data an honest prover would store). When this PoS is instantiated with recent constructions of maximally depth robust graphs, our proof implies basically optimal security.
As a second contribution we show three different extensions of this PoS where useful data can be embedded into the space required by the prover. Our security proof for the PoS extends (non-trivially) to these constructions. We discuss how some of these variants can be used as proofs of catalytic space (PoCS), a notion we put forward in this work, and which basically is a PoS where most of the space required by the prover can be used to backup useful data. Finally we discuss how one of the extensions is a candidate construction for a proof of replication (PoR), a proof system recently suggested in the Filecoin whitepaper.
Proofs of Space
Proofs of Replication
Blockchains
Theory of computation~Interactive proof systems
59:1-59:25
Regular Paper
A full version of the paper is available at [Krzysztof Pietrzak, 2018], https://eprint.iacr.org/2018/194.pdf.
Krzysztof
Pietrzak
Krzysztof Pietrzak
Institute of Science and Technology Austria, Austria
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 682815/TOCNeT).
10.4230/LIPIcs.ITCS.2019.59
Chia Network. https://chia.network/, 2017.
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Burstcoin. URL: http://burstcoin.info.
http://burstcoin.info
Hamza Abusalah, Joël Alwen, Bram Cohen, Danylo Khilko, Krzysztof Pietrzak, and Leonid Reyzin. Beyond Hellman’s Time-Memory Trade-Offs with Applications to Proofs of Space. In Tsuyoshi Takagi and Thomas Peyrin, editors, ASIACRYPT 2017, Part II, volume 10625 of LNCS, pages 357-379. Springer, Heidelberg, December 2017.
Joël Alwen, Jeremiah Blocki, and Ben Harsha. Practical Graphs for Optimal Side-Channel Resistant Memory-Hard Functions. In Bhavani M. Thuraisingham, David Evans, Tal Malkin, and Dongyan Xu, editors, ACM CCS 17, pages 1001-1017. ACM Press, October / November 2017.
Joël Alwen, Jeremiah Blocki, and Krzysztof Pietrzak. Depth-Robust Graphs and Their Cumulative Memory Complexity. In Jean-Sébastien Coron and Jesper Buus Nielsen, editors, EUROCRYPT 2017, Part III, volume 10212 of LNCS, pages 3-32. Springer, Heidelberg, April / May 2017.
Joël Alwen, Jeremiah Blocki, and Krzysztof Pietrzak. Sustained Space Complexity. In Jesper Buus Nielsen and Vincent Rijmen, editors, EUROCRYPT 2018, Part II, volume 10821 of LNCS, pages 99-130. Springer, Heidelberg, April / May 2018. URL: http://dx.doi.org/10.1007/978-3-319-78375-8_4.
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Joël Alwen, Binyi Chen, Krzysztof Pietrzak, Leonid Reyzin, and Stefano Tessaro. Scrypt Is Maximally Memory-Hard. In Jean-Sébastien Coron and Jesper Buus Nielsen, editors, EUROCRYPT 2017, Part III, volume 10212 of LNCS, pages 33-62. Springer, Heidelberg, April / May 2017.
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Krzysztof Pietrzak
Creative Commons Attribution 3.0 Unported license
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Simple Verifiable Delay Functions
We construct a verifiable delay function (VDF) by showing how the Rivest-Shamir-Wagner time-lock puzzle can be made publicly verifiable.
Concretely, we give a statistically sound public-coin protocol to prove that a tuple (N,x,T,y) satisfies y=x^{2^T} mod N where the prover doesn't know the factorization of N and its running time is dominated by solving the puzzle, that is, compute x^{2^T}, which is conjectured to require T sequential squarings. To get a VDF we make this protocol non-interactive using the Fiat-Shamir heuristic.
The motivation for this work comes from the Chia blockchain design, which uses a VDF as a key ingredient. For typical parameters (T <=2^{40},N=2048), our proofs are of size around 10KB, verification cost around three RSA exponentiations and computing the proof is 8000 times faster than solving the puzzle even without any parallelism.
Verifiable delay functions
Time-lock puzzles
Theory of computation~Cryptographic primitives
Theory of computation~Interactive proof systems
60:1-60:15
Regular Paper
https://eprint.iacr.org/2018/627.pdf
Krzysztof
Pietrzak
Krzysztof Pietrzak
Institute of Science and Technology Austria, Austria
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 682815/TOCNeT).
10.4230/LIPIcs.ITCS.2019.60
David Bernhard, Olivier Pereira, and Bogdan Warinschi. How Not to Prove Yourself: Pitfalls of the Fiat-Shamir Heuristic and Applications to Helios. In Xiaoyun Wang and Kazue Sako, editors, ASIACRYPT 2012, volume 7658 of LNCS, pages 626-643. Springer, Heidelberg, December 2012. URL: http://dx.doi.org/10.1007/978-3-642-34961-4_38.
http://dx.doi.org/10.1007/978-3-642-34961-4_38
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https://eprint.iacr.org/2018/712
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http://dx.doi.org/10.1007/978-3-319-78375-8_15
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http://dx.doi.org/10.1515/jmc-2013-5011
Benjamin Wesolowski. Slow-timed hash functions. Cryptology ePrint Archive, Report 2018/623, 2018. URL: https://eprint.iacr.org/2018/623.
https://eprint.iacr.org/2018/623
Krzysztof Pietrzak
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Sum of Squares Lower Bounds from Symmetry and a Good Story
In this paper, we develop machinery which makes it much easier to prove sum of squares lower bounds when the problem is symmetric under permutations of [1,n] and the unsatisfiability of our problem comes from integrality arguments, i.e. arguments that an expression must be an integer. Roughly speaking, to prove SOS lower bounds with our machinery it is sufficient to verify that the answer to the following three questions is yes:
1) Are there natural pseudo-expectation values for the problem?
2) Are these pseudo-expectation values rational functions of the problem parameters?
3) Are there sufficiently many values of the parameters for which these pseudo-expectation values correspond to the actual expected values over a distribution of solutions which is the uniform distribution over permutations of a single solution?
We demonstrate our machinery on three problems, the knapsack problem analyzed by Grigoriev, the MOD 2 principle (which says that the complete graph K_n has no perfect matching when n is odd), and the following Turan type problem: Minimize the number of triangles in a graph G with a given edge density. For knapsack, we recover Grigoriev's lower bound exactly. For the MOD 2 principle, we tighten Grigoriev's linear degree sum of squares lower bound, making it exact. Finally, for the triangle problem, we prove a sum of squares lower bound for finding the minimum triangle density. This lower bound is completely new and gives a simple example where constant degree sum of squares methods have a constant factor error in estimating graph densities.
Sum of squares hierarchy
proof complexity
graph theory
lower bounds
Theory of computation~Proof complexity
61:1-61:20
Regular Paper
A full version of the paper is available at https://arxiv.org/abs/1711.11469.
Aaron
Potechin
Aaron Potechin
University of Chicago Department of Computer Science, 5730 S. Ellis Avenue, John Crerar Library, Chicago, IL 60637, United States
10.4230/LIPIcs.ITCS.2019.61
Sanjeev Arora, Boaz Barak, and David Steurer. Subexponential Algorithms for Unique Games and Related Problems. J. ACM, 62(5):42:1-42:25, 2015.
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Boaz Barak, Samuel Hopkins, Jonathan Kelner, Pravesh Kothari, Ankur Moitra, and Aaron Potechin. A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem. In IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 428-437, 2016.
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Boaz Barak, Jonathan A. Kelner, and David Steurer. Dictionary Learning and Tensor Decomposition via the Sum-of-Squares Method. In Proceedings of the Forty-seventh Annual ACM Symposium on Theory of Computing, STOC '15, pages 143-151, 2015.
Boaz Barak, Pravesh K. Kothari, and David Steurer. Quantum Entanglement, Sum of Squares, and the Log Rank Conjecture. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, pages 975-988, 2017.
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A. Goodman. On sets of acquaintances and strangers at any party. The American Mathematical Monthly, 66(9):778-783, 1959.
Dima Grigoriev. Complexity of Positivstellensatz proofs for the knapsack. Computational Complexity, 10(2):139-154, 2001.
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Samuel B. Hopkins, Pravesh Kothari, Aaron Henry Potechin, Prasad Raghavendra, and Tselil Schramm. On the Integrality Gap of Degree-4 Sum of Squares for Planted Clique. ACM Trans. Algorithms, 14(3):28:1-28:31, June 2018.
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Pravesh K. Kothari, Ryuhei Mori, Ryan O'Donnell, and David Witmer. Sum of Squares Lower Bounds for Refuting Any CSP. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, pages 132-145, 2017.
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Jean B. Lasserre. Global Optimization with Polynomials and the Problem of Moments. SIAM J. on Optimization, 11(3):796-817, March 2000.
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Pablo Parrilo. Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD thesis, California Institute of Technology, 2000.
Aaron Potechin and David Steurer. Exact tensor completion with sum-of-squares. In Proceedings of the 30th Conference on Learning Theory, COLT 2017, Amsterdam, The Netherlands, 7-10 July 2017, pages 1619-1673, 2017.
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Aaron Potechin
Creative Commons Attribution 3.0 Unported license
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Learning Time Dependent Choice
We explore questions dealing with the learnability of models of choice over time. We present a large class of preference models defined by a structural criterion for which we are able to obtain an exponential improvement over previously known learning bounds for more general preference models. This in particular implies that the three most important discounted utility models of intertemporal choice - exponential, hyperbolic, and quasi-hyperbolic discounting - are learnable in the PAC setting with VC dimension that grows logarithmically in the number of time periods. We also examine these models in the framework of active learning. We find that the commonly studied stream-based setting is in general difficult to analyze for preference models, but we provide a redeeming situation in which the learner can indeed improve upon the guarantees provided by PAC learning. In contrast to the stream-based setting, we show that if the learner is given full power over the data he learns from - in the form of learning via membership queries - even very naive algorithms significantly outperform the guarantees provided by higher level active learning algorithms.
Intertemporal Choice
Discounted Utility
Preference Recovery
PAC Learning
Active Learning
Theory of computation~Models of learning
62:1-62:19
Regular Paper
Zachary
Chase
Zachary Chase
Department of Mathematics, California Institute of Technology, Pasadena, USA
Siddharth
Prasad
Siddharth Prasad
Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, USA
10.4230/LIPIcs.ITCS.2019.62
M.F. Balcan, A. Daniely, R. Mehta, R. Urner, and V. V. Vazirani. Learning economic parameters from revealed preferences. In International Conference on Web and Internet Economics, pages 338-353. Springer, Cham, 2014.
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E. Beigman and R. Vohra. Learning from revealed preference. In Proceedings of the 7th ACM Conference on Electronic Commerce, pages 36-42. ACM, 2006.
G. S. Berns, D. Laibson, and G. Loewenstein. Intertemporal choice - toward an integrative framework. Trends in cognitive sciences, 11(11):482-488, 2006.
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N. Stern, S. Peters, V. Bakhshi, A. Bowen, C. Cameron, S. Catovsky, ..., and N. Edmonson. Stern Review: The economics of climate change. London: HM treasury, 2006.
M. Zadimoghaddam and A. Roth. Efficiently learning from revealed preference. In International Workshop on Internet and Network Economics, pages 114-127. Springer, Berlin, Heidelberg, 2012.
Zachary Chase and Siddharth Prasad
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Erasures vs. Errors in Local Decoding and Property Testing
We initiate the study of the role of erasures in local decoding and use our understanding to prove a separation between erasure-resilient and tolerant property testing. Local decoding in the presence of errors has been extensively studied, but has not been considered explicitly in the presence of erasures.
Motivated by applications in property testing, we begin our investigation with local list decoding in the presence of erasures. We prove an analog of a famous result of Goldreich and Levin on local list decodability of the Hadamard code. Specifically, we show that the Hadamard code is locally list decodable in the presence of a constant fraction of erasures, arbitrary close to 1, with list sizes and query complexity better than in the Goldreich-Levin theorem. We use this result to exhibit a property which is testable with a number of queries independent of the length of the input in the presence of erasures, but requires a number of queries that depends on the input length, n, for tolerant testing. We further study approximate locally list decodable codes that work against erasures and use them to strengthen our separation by constructing a property which is testable with a constant number of queries in the presence of erasures, but requires n^{Omega(1)} queries for tolerant testing.
Next, we study the general relationship between local decoding in the presence of errors and in the presence of erasures. We observe that every locally (uniquely or list) decodable code that works in the presence of errors also works in the presence of twice as many erasures (with the same parameters up to constant factors). We show that there is also an implication in the other direction for locally decodable codes (with unique decoding): specifically, that the existence of a locally decodable code that works in the presence of erasures implies the existence of a locally decodable code that works in the presence of errors and has related parameters. However, it remains open whether there is an implication in the other direction for locally list decodable codes. We relate this question to other open questions in local decoding.
Error-correcting codes
probabilistically checkable proofs (PCPs) of proximity
Hadamard code
local list decoding
tolerant testing
Theory of computation~Streaming, sublinear and near linear time algorithms
Mathematics of computing~Coding theory
63:1-63:21
Regular Paper
The first and the third author were supported by National Science Foundation under Grant No. CCF-142297.
A full version [Sofya Raskhodnikova et al., 2018] (https://eccc.weizmann.ac.il/report/2018/195) of the paper contains all the omitted proofs.
Sofya
Raskhodnikova
Sofya Raskhodnikova
Department of Computer Science, Boston University, USA
Noga
Ron-Zewi
Noga Ron-Zewi
Department of Computer Science, University of Haifa, Israel
Nithin
Varma
Nithin Varma
Department of Computer Science, Boston University, USA
https://orcid.org/0000-0002-1211-2566
10.4230/LIPIcs.ITCS.2019.63
Noga Alon, Jeff Edmonds, and Michael Luby. Linear Time Erasure Codes with Nearly Optimal Recovery. In Proceedings of FOCS 1995, pages 512-519, 1995.
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Zeev Dvir, Parikshit Gopalan, and Sergey Yekhanin. Matching Vector Codes. SIAM J. Comput., 40(4):1154-1178, 2011.
Klim Efremenko. 3-Query Locally Decodable Codes of Subexponential Length. SIAM J. on Computing, 41(6):1694-1703, 2012.
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Sivakanth Gopi, Swastik Kopparty, Rafael Mendes de Oliveira, Noga Ron-Zewi, and Shubhangi Saraf. Locally Testable and Locally Correctable Codes Approaching the Gilbert-Varshamov Bound. In Proceedings of SODA 2017, pages 2073-2091, 2017.
Aryeh Grinberg, Ronen Shaltiel, and Emanuele Viola. Indistinguishability by adaptive procedures with advice, and lower bounds on hardness amplification proofs. In FOCS, 2018. URL: https://eccc.weizmann.ac.il/report/2018/061.
https://eccc.weizmann.ac.il/report/2018/061
Alan Guo and Swastik Kopparty. List-Decoding Algorithms for Lifted Codes. IEEE Trans. Information Theory, 62(5):2719-2725, 2016.
Venkatesan Guruswami. List decoding from erasures: bounds and code constructions. IEEE Trans. Information Theory, 49(11):2826-2833, 2003.
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Venkatesan Guruswami and Salil P. Vadhan. A Lower Bound on List Size for List Decoding. IEEE Trans. Information Theory, 56(11):5681-5688, 2010.
Dan Gutfreund and Guy N. Rothblum. The Complexity of Local List Decoding. In Proceedings of RANDOM 2008, pages 455-468, 2008.
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Swastik Kopparty. List-Decoding Multiplicity Codes. Theory of Comput., 11:149-182, 2015.
Swastik Kopparty, Or Meir, Noga Ron-Zewi, and Shubhangi Saraf. High-Rate Locally Correctable and Locally Testable Codes with Sub-Polynomial Query Complexity. J. ACM, 64(2):11:1-11:42, 2017.
Swastik Kopparty and Shubhangi Saraf. Local List-Decoding and Testing of Random Linear Codes from High Error. SIAM J. Comput., 42(3):1302-1326, 2013.
Or Meir. Combinatorial PCPs with Efficient Verifiers. Comp. Complexity, 23(3):355-478, 2014.
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Michal Parnas, Dana Ron, and Ronitt Rubinfeld. Tolerant property testing and distance approximation. J. Comput. Syst. Sci., 72(6):1012-1042, 2006.
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Sofya Raskhodnikova, Noga Ron-Zewi, and Nithin Varma. Erasures versus Errors in Local Decoding and Property Testing. Electronic Colloquium on Computational Complexity (ECCC), 2018. URL: https://eccc.weizmann.ac.il/report/2018/195.
https://eccc.weizmann.ac.il/report/2018/195
Ronitt Rubinfeld and Madhu Sudan. Robust Characterizations of Polynomials with Applications to Program Testing. SIAM J. Comput., 25(2):252-271, 1996.
Madhu Sudan, Luca Trevisan, and Salil P. Vadhan. Pseudorandom Generators without the XOR Lemma. J. Comput. Syst. Sci., 62(2):236-266, 2001.
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Sergey Yekhanin. Towards 3-query locally decodable codes of subexponential length. J. of the ACM, 55(1):1:1-1:16, 2008.
Sofya Raskhodnikova, Noga Ron-Zewi, and Nithin Varma
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A New Approach to Multi-Party Peer-to-Peer Communication Complexity
We introduce new models and new information theoretic measures for the study of communication complexity in the natural peer-to-peer, multi-party, number-in-hand setting. We prove a number of properties of our new models and measures, and then, in order to exemplify their effectiveness, we use them to prove two lower bounds. The more elaborate one is a tight lower bound of Omega(kn) on the multi-party peer-to-peer randomized communication complexity of the k-player, n-bit function Disjointness, Disj_k^n. The other one is a tight lower bound of Omega(kn) on the multi-party peer-to-peer randomized communication complexity of the k-player, n-bit bitwise parity function, Par_k^n. Both lower bounds hold when n=Omega(k). The lower bound for Disj_k^n improves over the lower bound that can be inferred from the result of Braverman et al. (FOCS 2013), which was proved in the coordinator model and can yield a lower bound of Omega(kn/log k) in the peer-to-peer model.
To the best of our knowledge, our lower bounds are the first tight (non-trivial) lower bounds on communication complexity in the natural peer-to-peer multi-party setting.
In addition to the above results for communication complexity, we also prove, using the same tools, an Omega(n) lower bound on the number of random bits necessary for the (information theoretic) private computation of the function Disj_k^n.
communication complexity
multi-party communication complexity
peer-to-peer communication complexity
information complexity
private computation
Theory of computation~Communication complexity
Mathematics of computing~Information theory
Theory of computation~Cryptographic protocols
64:1-64:19
Regular Paper
Adi
Rosén
Adi Rosén
CNRS and Université Paris Diderot
Research supported in part by ANR project RDAM.
Florent
Urrutia
Florent Urrutia
Université Paris Diderot
Research supported in part by ERC QCC and by ANR project RDAM.
10.4230/LIPIcs.ITCS.2019.64
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Adi Rosén and Florent Urrutia
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
A Schur Complement Cheeger Inequality
Cheeger's inequality shows that any undirected graph G with minimum normalized Laplacian eigenvalue lambda_G has a cut with conductance at most O(sqrt{lambda_G}). Qualitatively, Cheeger's inequality says that if the mixing time of a graph is high, there is a cut that certifies this. However, this relationship is not tight, as some graphs (like cycles) do not have cuts with conductance o(sqrt{lambda_G}).
To better approximate the mixing time of a graph, we consider a more general object. Specifically, instead of bounding the mixing time with cuts, we bound it with cuts in graphs obtained by Schur complementing out vertices from the graph G. Combinatorially, these Schur complements describe random walks in G restricted to a subset of its vertices. As a result, all Schur complement cuts have conductance at least Omega(lambda_G). We show that unlike with cuts, this inequality is tight up to a constant factor. Specifically, there is a Schur complement cut with conductance at most O(lambda_G).
electrical networks
Cheeger's inequality
mixing time
conductance
Schur complements
Mathematics of computing~Spectra of graphs
65:1-65:15
Regular Paper
Aaron
Schild
Aaron Schild
University of California, Berkeley, CA, USA
Supported by NSF grant CCF-1816861.
10.4230/LIPIcs.ITCS.2019.65
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Aaron Schild
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Game Efficiency Through Linear Programming Duality
The efficiency of a game is typically quantified by the price of anarchy (PoA), defined as the worst ratio of the value of an equilibrium - solution of the game - and that of an optimal outcome. Given the tremendous impact of tools from mathematical programming in the design of algorithms and the similarity of the price of anarchy and different measures such as the approximation and competitive ratios, it is intriguing to develop a duality-based method to characterize the efficiency of games.
In the paper, we present an approach based on linear programming duality to study the efficiency of games. We show that the approach provides a general recipe to analyze the efficiency of games and also to derive concepts leading to improvements. The approach is particularly appropriate to bound the PoA. Specifically, in our approach the dual programs naturally lead to competitive PoA bounds that are (almost) optimal for several classes of games. The approach indeed captures the smoothness framework and also some current non-smooth techniques/concepts. We show the applicability to the wide variety of games and environments, from congestion games to Bayesian welfare, from full-information settings to incomplete-information ones.
Price of Anarchy
Primal-Dual
Theory of computation~Algorithmic game theory and mechanism design
66:1-66:20
Regular Paper
Research supported by the ANR project OATA no ANR-15-CE40-0015-01.
A full version of the paper is available at [Nguyen Kim Thang, 2017], https://arxiv.org/abs/1708.06499.
Nguyen
Kim Thang
Nguyen Kim Thang
IBISC, Univ Evry, University Paris Saclay, Evry, France
https://orcid.org/0000-0002-6085-9453
10.4230/LIPIcs.ITCS.2019.66
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