eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
1
1240
10.4230/LIPIcs.APPROX/RANDOM.2021
article
LIPIcs, Volume 207, APPROX/RANDOM 2021, Complete Volume
Wootters, Mary
1
https://orcid.org/0000-0002-2345-2531
Sanità, Laura
2
https://orcid.org/0000-0002-6384-1857
Stanford University, Departments of Computer Science and Electrical Engineering, CA, USA
Eindhoven University of Technology, Department of Mathematics and Computer Science, The Netherlands
LIPIcs, Volume 207, APPROX/RANDOM 2021, Complete Volume
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021/LIPIcs.APPROX-RANDOM.2021.pdf
LIPIcs, Volume 207, APPROX/RANDOM 2021, Complete Volume
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
0:i
0:x
10.4230/LIPIcs.APPROX/RANDOM.2021.0
article
Front Matter, Table of Contents, Preface, Conference Organization
Wootters, Mary
1
https://orcid.org/0000-0002-2345-2531
Sanità, Laura
2
https://orcid.org/0000-0002-6384-1857
Stanford University, Departments of Computer Science and Electrical Engineering, CA, USA
Eindhoven University of Technology, Department of Mathematics and Computer Science, The Netherlands
Front Matter, Table of Contents, Preface, Conference Organization
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.0/LIPIcs.APPROX-RANDOM.2021.0.pdf
Front Matter
Table of Contents
Preface
Conference Organization
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
1:1
1:23
10.4230/LIPIcs.APPROX/RANDOM.2021.1
article
On Approximate Envy-Freeness for Indivisible Chores and Mixed Resources
Bhaskar, Umang
1
Sricharan, A. R.
2
Vaish, Rohit
1
Tata Institute of Fundamental Research, Mumbai, India
Chennai Mathematical Institute, India
We study the fair allocation of undesirable indivisible items, or chores. While the case of desirable indivisible items (or goods) is extensively studied, with many results known for different notions of fairness, less is known about the fair division of chores. We study envy-free allocation of chores and make three contributions. First, we show that determining the existence of an envy-free allocation is NP-complete even in the simple case when agents have binary additive valuations. Second, we provide a polynomial-time algorithm for computing an allocation that satisfies envy-freeness up to one chore (EF1), correcting a claim in the existing literature. A modification of our algorithm can be used to compute an EF1 allocation for doubly monotone instances (where each agent can partition the set of items into objective goods and objective chores). Our third result applies to a mixed resources model consisting of indivisible items and a divisible, undesirable heterogeneous resource (i.e., a bad cake). We show that there always exists an allocation that satisfies envy-freeness for mixed resources (EFM) in this setting, complementing a recent result of Bei et al. [Bei et al., 2021] for indivisible goods and divisible cake.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.1/LIPIcs.APPROX-RANDOM.2021.1.pdf
Fair Division
Indivisible Chores
Approximate Envy-Freeness
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
2:1
2:18
10.4230/LIPIcs.APPROX/RANDOM.2021.2
article
Optimal Algorithms for Online b-Matching with Variable Vertex Capacities
Albers, Susanne
1
Schubert, Sebastian
1
https://orcid.org/0000-0002-3883-2297
Department of Computer Science, Technische Universität München, Germany
We study the b-matching problem, which generalizes classical online matching introduced by Karp, Vazirani and Vazirani (STOC 1990). Consider a bipartite graph G = (S ̇∪ R,E). Every vertex s ∈ S is a server with a capacity b_s, indicating the number of possible matching partners. The vertices r ∈ R are requests that arrive online and must be matched immediately to an eligible server. The goal is to maximize the cardinality of the constructed matching. In contrast to earlier work, we study the general setting where servers may have arbitrary, individual capacities. We prove that the most natural and simple online algorithms achieve optimal competitive ratios.
As for deterministic algorithms, we give a greedy algorithm RelativeBalance and analyze it by extending the primal-dual framework of Devanur, Jain and Kleinberg (SODA 2013). In the area of randomized algorithms we study the celebrated Ranking algorithm by Karp, Vazirani and Vazirani. We prove that the original Ranking strategy, simply picking a random permutation of the servers, achieves an optimal competitiveness of 1-1/e, independently of the server capacities. Hence it is not necessary to resort to a reduction, replacing every server s by b_s vertices of unit capacity and to then run Ranking on this graph with ∑_{s ∈ S} b_s vertices on the left-hand side. From a theoretical point of view our result explores the power of randomization and strictly limits the amount of required randomness. From a practical point of view it leads to more efficient allocation algorithms.
Technically, we show that the primal-dual framework of Devanur, Jain and Kleinberg cannot establish a competitiveness better than 1/2 for the original Ranking algorithm, choosing a permutation of the servers. Therefore, we formulate a new configuration LP for the b-matching problem and then conduct a primal-dual analysis. We extend this analysis approach to the vertex-weighted b-matching problem. Specifically, we show that the algorithm PerturbedGreedy by Aggarwal, Goel, Karande and Mehta (SODA 2011), again with a sole randomization over the set of servers, is (1-1/e)-competitive. Together with recent work by Huang and Zhang (STOC 2020), our results demonstrate that configuration LPs can be strictly stronger than standard LPs in the analysis of more complex matching problems.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.2/LIPIcs.APPROX-RANDOM.2021.2.pdf
Online algorithms
primal-dual analysis
configuration LP
b-matching
variable vertex capacities
unweighted matching
vertex-weighted matching
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
3:1
3:16
10.4230/LIPIcs.APPROX/RANDOM.2021.3
article
Bag-Of-Tasks Scheduling on Related Machines
Gupta, Anupam
1
Kumar, Amit
2
Singla, Sahil
3
Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
Computer Science and Engineering Department, Indian Institute of Technology, Delhi, India
Department of Computer Science, Princeton University, NJ, USA
We consider online scheduling to minimize weighted completion time on related machines, where each job consists of several tasks that can be concurrently executed. A job gets completed when all its component tasks finish. We obtain an O(K³ log² K)-competitive algorithm in the non-clairvoyant setting, where K denotes the number of distinct machine speeds. The analysis is based on dual-fitting on a precedence-constrained LP relaxation that may be of independent interest.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.3/LIPIcs.APPROX-RANDOM.2021.3.pdf
approximation algorithms
scheduling
bag-of-tasks
related machines
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
4:1
4:23
10.4230/LIPIcs.APPROX/RANDOM.2021.4
article
Hardness of Approximation for Euclidean k-Median
Bhattacharya, Anup
1
Goyal, Dishant
2
Jaiswal, Ragesh
2
School of Computer Sciences, National Institute of Science Education and Research (NISER), Khurda, India
Indian Institute of Technology Delhi, India
The Euclidean k-median problem is defined in the following manner: given a set 𝒳 of n points in d-dimensional Euclidean space ℝ^d, and an integer k, find a set C ⊂ ℝ^d of k points (called centers) such that the cost function Φ(C,𝒳) ≡ ∑_{x ∈ 𝒳} min_{c ∈ C} ‖x-c‖₂ is minimized. The Euclidean k-means problem is defined similarly by replacing the distance with squared Euclidean distance in the cost function. Various hardness of approximation results are known for the Euclidean k-means problem [Pranjal Awasthi et al., 2015; Euiwoong Lee et al., 2017; Vincent Cohen{-}Addad and {Karthik {C. S.}}, 2019]. However, no hardness of approximation result was known for the Euclidean k-median problem. In this work, assuming the unique games conjecture (UGC), we provide the hardness of approximation result for the Euclidean k-median problem in O(log k) dimensional space. This solves an open question posed explicitly in the work of Awasthi et al. [Pranjal Awasthi et al., 2015].
Furthermore, we study the hardness of approximation for the Euclidean k-means/k-median problems in the bi-criteria setting where an algorithm is allowed to choose more than k centers. That is, bi-criteria approximation algorithms are allowed to output β k centers (for constant β > 1) and the approximation ratio is computed with respect to the optimal k-means/k-median cost. We show the hardness of bi-criteria approximation result for the Euclidean k-median problem for any β < 1.015, assuming UGC. We also show a similar hardness of bi-criteria approximation result for the Euclidean k-means problem with a stronger bound of β < 1.28, again assuming UGC.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.4/LIPIcs.APPROX-RANDOM.2021.4.pdf
Hardness of approximation
bicriteria approximation
approximation algorithms
k-median
k-means
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
5:1
5:25
10.4230/LIPIcs.APPROX/RANDOM.2021.5
article
Online Directed Spanners and Steiner Forests
Grigorescu, Elena
1
https://orcid.org/0000-0001-9673-4313
Lin, Young-San
1
https://orcid.org/0000-0002-5719-6708
Quanrud, Kent
1
Department of Computer Science, Purdue University, West Lafayette, IN, USA
We present online algorithms for directed spanners and directed Steiner forests. These are well-studied network connectivity problems that fall under the unifying framework of online covering and packing linear programming formulations. This framework was developed in the seminal work of Buchbinder and Naor (Mathematics of Operations Research, 34, 2009) and is based on primal-dual techniques. Specifically, our results include the following:
- For the pairwise spanner problem, in which the pairs of vertices to be spanned arrive online, we present an efficient randomized algorithm with competitive ratio Õ(n^{4/5}) for graphs with general edge lengths, where n is the number of vertices of the given graph. For graphs with uniform edge lengths, we give an efficient randomized algorithm with competitive ratio Õ(n^{2/3+ε}), and an efficient deterministic algorithm with competitive ratio Õ(k^{1/2+ε}), where k is the number of terminal pairs. To the best of our knowledge, these are the first online algorithms for directed spanners. In the offline version, the current best approximation ratio for uniform edge lengths is Õ(n^{3/5 + ε}), due to Chlamt{á}č, Dinitz, Kortsarz, and Laekhanukit (SODA 2017, TALG 2020).
- For the directed Steiner forest problem with uniform costs, in which the pairs of vertices to be connected arrive online, we present an efficient randomized algorithm with competitive ratio Õ(n^{2/3 + ε}). The state-of-the-art online algorithm for general costs is due to Chakrabarty, Ene, Krishnaswamy, and Panigrahi (SICOMP 2018) and is Õ(k^{1/2 + ε})-competitive. In the offline version, the current best approximation ratio with uniform costs is Õ(n^{26/45 + ε}), due to Abboud and Bodwin (SODA 2018).
To obtain efficient and competitive online algorithms, we observe that a small modification of the online covering and packing framework by Buchbinder and Naor implies a polynomial-time implementation of the primal-dual approach with separation oracles, which a priori might perform exponentially many calls to the oracle. We convert the online spanner problem into an online covering problem and complete the rounding-step analysis in a problem-specific fashion.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.5/LIPIcs.APPROX-RANDOM.2021.5.pdf
online directed pairwise spanners
online directed Steiner forests
online covering/packing linear programming
primal-dual approach
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
6:1
6:15
10.4230/LIPIcs.APPROX/RANDOM.2021.6
article
Query Complexity of Global Minimum Cut
Bishnu, Arijit
1
Ghosh, Arijit
1
Mishra, Gopinath
1
Paraashar, Manaswi
1
Advanced Computing and Microelectronics Unit, Indian Statistical Institute, Kolkata, India
In this work, we resolve the query complexity of global minimum cut problem for a graph by designing a randomized algorithm for approximating the size of minimum cut in a graph, where the graph can be accessed through local queries like Degree, Neighbor, and Adjacency queries.
Given ε ∈ (0,1), the algorithm with high probability outputs an estimate t̂ satisfying the following (1-ε) t ≤ t̂ ≤ (1+ε) t, where t is the size of minimum cut in the graph. The expected number of local queries used by our algorithm is min{m+n,m/t}poly(log n,1/(ε)) where n and m are the number of vertices and edges in the graph, respectively. Eden and Rosenbaum showed that Ω(m/t) local queries are required for approximating the size of minimum cut in graphs, {but no local query based algorithm was known. Our algorithmic result coupled with the lower bound of Eden and Rosenbaum [APPROX 2018] resolve the query complexity of the problem of estimating the size of minimum cut in graphs using local queries.}
Building on the lower bound of Eden and Rosenbaum, we show that, for all t ∈ ℕ, Ω(m) local queries are required to decide if the size of the minimum cut in the graph is t or t-2. Also, we show that, for any t ∈ ℕ, Ω(m) local queries are required to find all the minimum cut edges even if it is promised that the input graph has a minimum cut of size t. Both of our lower bound results are randomized, and hold even if we can make Random Edge queries in addition to local queries.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.6/LIPIcs.APPROX-RANDOM.2021.6.pdf
Query complexity
Global mincut
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
7:1
7:14
10.4230/LIPIcs.APPROX/RANDOM.2021.7
article
A Constant-Factor Approximation for Weighted Bond Cover
Kim, Eun Jung
1
Lee, Euiwoong
2
Thilikos, Dimitrios M.
3
Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016, Paris, France
University of Michigan, Ann Arbor, MI, USA
LIRMM, Univ. Montpellier, CNRS, Montpellier, France
The Weighted ℱ-Vertex Deletion for a class ℱ of graphs asks, weighted graph G, for a minimum weight vertex set S such that G-S ∈ ℱ. The case when ℱ is minor-closed and excludes some graph as a minor has received particular attention but a constant-factor approximation remained elusive for Weighted ℱ-Vertex Deletion. Only three cases of minor-closed ℱ are known to admit constant-factor approximations, namely Vertex Cover, Feedback Vertex Set and Diamond Hitting Set. We study the problem for the class ℱ of θ_c-minor-free graphs, under the equivalent setting of the Weighted c-Bond Cover problem, and present a constant-factor approximation algorithm using the primal-dual method. For this, we leverage a structure theorem implicit in [Joret et al., SIDMA'14] which states the following: any graph G containing a θ_c-minor-model either contains a large two-terminal protrusion, or contains a constant-size θ_c-minor-model, or a collection of pairwise disjoint constant-sized connected sets that can be contracted simultaneously to yield a dense graph. In the first case, we tame the graph by replacing the protrusion with a special-purpose weighted gadget. For the second and third case, we provide a weighting scheme which guarantees a local approximation ratio. Besides making an important step in the quest of (dis)proving a constant-factor approximation for Weighted ℱ-Vertex Deletion, our result may be useful as a template for algorithms for other minor-closed families.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.7/LIPIcs.APPROX-RANDOM.2021.7.pdf
Constant-factor approximation algorithms
Primal-dual method
Bonds in graphs
Graph minors
Graph modification problems
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
8:1
8:18
10.4230/LIPIcs.APPROX/RANDOM.2021.8
article
Truly Asymptotic Lower Bounds for Online Vector Bin Packing
Balogh, János
1
Cohen, Ilan Reuven
2
Epstein, Leah
3
Levin, Asaf
4
Institute of Informatics, University of Szeged, Hungary
Faculty of Engineering, Bar-Ilan University, Ramat-Gan, Israel
Department of Mathematics, University of Haifa, Israel
Faculty of Industrial Engineering and Management, Technion, Haifa, Israel
In this work, we consider online d-dimensional vector bin packing. It is known that no algorithm can have a competitive ratio of o(d/log² d) in the absolute sense, although upper bounds for this problem have always been presented in the asymptotic sense. Since variants of bin packing are traditionally studied with respect to the asymptotic measure, and since the two measures are different, we focus on the asymptotic measure and prove new lower bounds of the asymptotic competitive ratio. The existing lower bounds prior to this work were known to be smaller than 3, even for very large d. Here, we significantly improved on the best known lower bounds of the asymptotic competitive ratio (and as a byproduct, on the absolute competitive ratio) for online vector packing of vectors with d ≥ 3 dimensions, for every dimension d. To obtain these results, we use several different constructions, one of which is an adaptive construction with a lower bound of Ω(√d). Our main result is that the lower bound of Ω(d/log² d) on the competitive ratio holds also in the asymptotic sense. This result holds also against randomized algorithms, and requires a careful adaptation of constructions for online coloring, rather than simple black-box reductions.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.8/LIPIcs.APPROX-RANDOM.2021.8.pdf
Bin packing
online algorithms
approximation algorithms
vector packing
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
9:1
9:22
10.4230/LIPIcs.APPROX/RANDOM.2021.9
article
Fine-Grained Completeness for Optimization in P
Bringmann, Karl
1
2
Cassis, Alejandro
1
2
Fischer, Nick
1
2
Künnemann, Marvin
3
Saarland University, Saarland Informatics Campus, Saarbrücken, Germany
Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
Institute for Theoretical Studies, ETH Zürich, Switzerland
We initiate the study of fine-grained completeness theorems for exact and approximate optimization in the polynomial-time regime.
Inspired by the first completeness results for decision problems in P (Gao, Impagliazzo, Kolokolova, Williams, TALG 2019) as well as the classic class MaxSNP and MaxSNP-completeness for NP optimization problems (Papadimitriou, Yannakakis, JCSS 1991), we define polynomial-time analogues MaxSP and MinSP, which contain a number of natural optimization problems in P, including Maximum Inner Product, general forms of nearest neighbor search and optimization variants of the k-XOR problem. Specifically, we define MaxSP as the class of problems definable as max_{x₁,… ,x_k} #{(y₁,… ,y_𝓁) : ϕ(x₁,… ,x_k, y₁,… ,y_𝓁)}, where ϕ is a quantifier-free first-order property over a given relational structure (with MinSP defined analogously). On m-sized structures, we can solve each such problem in time O(m^{k+𝓁-1}). Our results are:
- We determine (a sparse variant of) the Maximum/Minimum Inner Product problem as complete under deterministic fine-grained reductions: A strongly subquadratic algorithm for Maximum/Minimum Inner Product would beat the baseline running time of O(m^{k+𝓁-1}) for all problems in MaxSP/MinSP by a polynomial factor.
- This completeness transfers to approximation: Maximum/Minimum Inner Product is also complete in the sense that a strongly subquadratic c-approximation would give a (c+ε)-approximation for all MaxSP/MinSP problems in time O(m^{k+𝓁-1-δ}), where ε > 0 can be chosen arbitrarily small. Combining our completeness with (Chen, Williams, SODA 2019), we obtain the perhaps surprising consequence that refuting the OV Hypothesis is equivalent to giving a O(1)-approximation for all MinSP problems in faster-than-O(m^{k+𝓁-1}) time.
- By fine-tuning our reductions, we obtain mild algorithmic improvements for solving and approximating all problems in MaxSP and MinSP, using the fastest known algorithms for Maximum/Minimum Inner Product.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.9/LIPIcs.APPROX-RANDOM.2021.9.pdf
Fine-grained Complexity & Algorithm Design
Completeness
Hardness of Approximation in P
Dimensionality Reductions
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
10:1
10:13
10.4230/LIPIcs.APPROX/RANDOM.2021.10
article
An Estimator for Matching Size in Low Arboricity Graphs with Two Applications
Jowhari, Hossein
1
https://orcid.org/0000-0001-9032-9958
Department of Computer Science and Statistics, Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran
In this paper, we present a new degree-based estimator for the size of maximum matching in bounded arboricity graphs. When the arboricity of the graph is bounded by α, the estimator gives a α+2 factor approximation of the matching size. For planar graphs, we show the estimator does better and returns a 3.5 approximation of the matching size. Using this estimator, we get new results for approximating the matching size of planar graphs in the streaming and distributed models of computation. In particular, in the vertex-arrival streams, we get a randomized O((√n)/(ε²)log n) space algorithm for approximating the matching size of a planar graph on n vertices within (3.5+ε) factor. Similarly, we get a simultaneous protocol in the vertex-partition model for approximating the matching size within (3.5+ε) factor using O((n^{2/3})/(ε²)log n) communication from each player. In comparison with the previous estimators, the estimator in this paper does not need to know the arboricity of the input graph and improves the approximation factor in the case of planar graphs.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.10/LIPIcs.APPROX-RANDOM.2021.10.pdf
Data Stream Algorithms
Maximum Matching
Planar Graphs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
11:1
11:11
10.4230/LIPIcs.APPROX/RANDOM.2021.11
article
An Optimal Algorithm for Triangle Counting in the Stream
Jayaram, Rajesh
1
Kallaugher, John
2
Carnegie Mellon University, Pittsburgh, PA, USA
The University of Texas at Austin, Austin, TX, USA
We present a new algorithm for approximating the number of triangles in a graph G whose edges arrive as an arbitrary order stream. If m is the number of edges in G, T the number of triangles, Δ_E the maximum number of triangles which share a single edge, and Δ_V the maximum number of triangles which share a single vertex, then our algorithm requires space:
Õ(m/T⋅(Δ_E + √{Δ_V}))
Taken with the Ω((m Δ_E)/T) lower bound of Braverman, Ostrovsky, and Vilenchik (ICALP 2013), and the Ω((m √{Δ_V})/T) lower bound of Kallaugher and Price (SODA 2017), our algorithm is optimal up to log factors, resolving the complexity of a classic problem in graph streaming.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.11/LIPIcs.APPROX-RANDOM.2021.11.pdf
Triangle Counting
Streaming
Graph Algorithms
Sampling
Sketching
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
12:1
12:22
10.4230/LIPIcs.APPROX/RANDOM.2021.12
article
Matching Drivers to Riders: A Two-Stage Robust Approach
Housni, Omar El
1
Goyal, Vineet
2
Hanguir, Oussama
2
Stein, Clifford
2
School of Operations Research and Information Engineering, Cornell Tech, New York, NY, USA
Industrial Engineering and Operations Research, Columbia University, New York, NY, USA
Matching demand (riders) to supply (drivers) efficiently is a fundamental problem for ride-hailing platforms who need to match the riders (almost) as soon as the request arrives with only partial knowledge about future ride requests. A myopic approach that computes an optimal matching for current requests ignoring future uncertainty can be highly sub-optimal. In this paper, we consider a two-stage robust optimization framework for this matching problem where future demand uncertainty is modeled using a set of demand scenarios (specified explicitly or implicitly). The goal is to match the current request to drivers (in the first stage) so that the cost of first stage matching and the worst-case cost over all scenarios for the second stage matching is minimized. We show that this two-stage robust matching is NP-hard under both explicit and implicit models of uncertainty. We present constant approximation algorithms for both models of uncertainty under different settings and show they improve significantly over standard greedy approaches.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.12/LIPIcs.APPROX-RANDOM.2021.12.pdf
matching
robust optimization
approximation algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
13:1
13:23
10.4230/LIPIcs.APPROX/RANDOM.2021.13
article
Secretary Matching Meets Probing with Commitment
Borodin, Allan
1
MacRury, Calum
1
Rakheja, Akash
1
Department of Computer Science, University of Toronto, Canada
We consider the online bipartite matching problem within the context of stochastic probing with commitment. This is the one-sided online bipartite matching problem where edges adjacent to an online node must be probed to determine if they exist based on edge probabilities that become known when an online vertex arrives. If a probed edge exists, it must be used in the matching. We consider the competitiveness of online algorithms in the adversarial order model (AOM) and the secretary/random order model (ROM). More specifically, we consider an unknown bipartite stochastic graph G = (U,V,E) where U is the known set of offline vertices, V is the set of online vertices, G has edge probabilities (p_{e})_{e ∈ E}, and G has edge weights (w_{e})_{e ∈ E} or vertex weights (w_u)_{u ∈ U}. Additionally, G has a downward-closed set of probing constraints (𝒞_{v})_{v ∈ V}, where 𝒞_v indicates which sequences of edges adjacent to an online vertex v can be probed. This model generalizes the various settings of the classical bipartite matching problem (i.e. with and without probing). Our contributions include the introduction and analysis of probing within the random order model, and our generalization of probing constraints which includes budget (i.e. knapsack) constraints. Our algorithms run in polynomial time assuming access to a membership oracle for each 𝒞_v.
In the vertex weighted setting, for adversarial order arrivals, we generalize the known 1/2 competitive ratio to our setting of 𝒞_v constraints. For random order arrivals, we show that the same algorithm attains an asymptotic competitive ratio of 1-1/e, provided the edge probabilities vanish to 0 sufficiently fast. We also obtain a strict competitive ratio for non-vanishing edge probabilities when the probing constraints are sufficiently simple. For example, if each 𝒞_v corresponds to a patience constraint 𝓁_v (i.e., 𝓁_v is the maximum number of probes of edges adjacent to v), and any one of following three conditions is satisfied (each studied in previous papers), then there is a conceptually simple greedy algorithm whose competitive ratio is 1-1/e.
- When the offline vertices are unweighted.
- When the online vertex probabilities are "vertex uniform"; i.e., p_{u,v} = p_v for all (u,v) ∈ E.
- When the patience constraint 𝓁_v satisfies 𝓁_v ∈ {[1,|U|} for every online vertex; i.e., every online vertex either has unit or full patience. Finally, in the edge weighted case, we match the known optimal 1/e asymptotic competitive ratio for the classic (i.e. without probing) secretary matching problem.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.13/LIPIcs.APPROX-RANDOM.2021.13.pdf
Stochastic probing
Online algorithms
Bipartite matching
Optimization under uncertainty
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
14:1
14:18
10.4230/LIPIcs.APPROX/RANDOM.2021.14
article
Semi-Streaming Algorithms for Submodular Function Maximization Under b-Matching Constraint
Huang, Chien-Chung
1
Sellier, François
2
CNRS, DI ENS, École normale supérieure, Université PSL, France
École polytechnique, Institut Polytechnique de Paris, France
We consider the problem of maximizing a submodular function under the b-matching constraint, in the semi-streaming model. Our main results can be summarized as follows.
- When the function is linear, i.e. for the maximum weight b-matching problem, we obtain a 2+ε approximation. This improves the previous best bound of 3+ε [Roie Levin and David Wajc, 2021].
- When the function is a non-negative monotone submodular function, we obtain a 3 + 2 √2 ≈ 5.828 approximation. This matches the currently best ratio [Roie Levin and David Wajc, 2021].
- When the function is a non-negative non-monotone submodular function, we obtain a 4 + 2 √3 ≈ 7.464 approximation. This ratio is also achieved in [Roie Levin and David Wajc, 2021], but only under the simple matching constraint, while we can deal with the more general b-matching constraint.
We also consider a generalized problem, where a k-uniform hypergraph is given with an extra matroid constraint imposed on the edges, with the same goal of finding a b-matching that maximizes a submodular function. We extend our technique to this case to obtain an algorithm with an approximation of 8/3k+O(1).
Our algorithms build on the ideas of the recent works of Levin and Wajc [Roie Levin and David Wajc, 2021] and of Garg, Jordan, and Svensson [Paritosh Garg et al., 2021]. Our main technical innovation is to introduce a data structure and associate it with each vertex and the matroid, to record the extra information of the stored edges. After the streaming phase, these data structures guide the greedy algorithm to make better choices.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.14/LIPIcs.APPROX-RANDOM.2021.14.pdf
Maximum Weight Matching
Submodular Function Maximization
Streaming
Matroid
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
15:1
15:18
10.4230/LIPIcs.APPROX/RANDOM.2021.15
article
General Knapsack Problems in a Dynamic Setting
Fairstein, Yaron
1
Kulik, Ariel
1
Naor, Joseph (Seffi)
1
Raz, Danny
1
Computer Science Department, Technion, Haifa, Israel
The world is dynamic and changes over time, thus any optimization problem used to model real life problems must address this dynamic nature, taking into account the cost of changes to a solution over time. The multistage model was introduced with this goal in mind. In this model we are given a series of instances of an optimization problem, corresponding to different times, and a solution is provided for each instance. The strive for obtaining near-optimal solutions for each instance on one hand, while maintaining similar solutions for consecutive time units on the other hand, is quantified and integrated into the objective function. In this paper we consider the Generalized Multistage d-Knapsack problem, a generalization of the multistage variants of the Multiple Knapsack problem, as well as the d-Dimensional Knapsack problem. We present a PTAS for Generalized Multistage d-Knapsack.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.15/LIPIcs.APPROX-RANDOM.2021.15.pdf
Multistage
Multiple-Knapsacks
Multidimensional Knapsack
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
16:1
16:16
10.4230/LIPIcs.APPROX/RANDOM.2021.16
article
Min-Sum Clustering (With Outliers)
Banerjee, Sandip
1
Ostrovsky, Rafail
2
Rabani, Yuval
1
The Hebrew University of Jerusalem, Israel
University of California, Los Angeles, CA, USA
We give a constant factor polynomial time pseudo-approximation algorithm for min-sum clustering with or without outliers. The algorithm is allowed to exclude an arbitrarily small constant fraction of the points. For instance, we show how to compute a solution that clusters 98% of the input data points and pays no more than a constant factor times the optimal solution that clusters 99% of the input data points. More generally, we give the following bicriteria approximation: For any ε > 0, for any instance with n input points and for any positive integer n' ≤ n, we compute in polynomial time a clustering of at least (1-ε) n' points of cost at most a constant factor greater than the optimal cost of clustering n' points. The approximation guarantee grows with 1/(ε). Our results apply to instances of points in real space endowed with squared Euclidean distance, as well as to points in a metric space, where the number of clusters, and also the dimension if relevant, is arbitrary (part of the input, not an absolute constant).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.16/LIPIcs.APPROX-RANDOM.2021.16.pdf
Clustering
approximation algorithms
primal-dual
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
17:1
17:19
10.4230/LIPIcs.APPROX/RANDOM.2021.17
article
Streaming Approximation Resistance of Every Ordering CSP
Singer, Noah
1
https://orcid.org/0000-0002-0076-521X
Sudan, Madhu
2
Velusamy, Santhoshini
2
Harvard College, Cambridge, MA, USA
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA
An ordering constraint satisfaction problem (OCSP) is given by a positive integer k and a constraint predicate Π mapping permutations on {1,…,k} to {0,1}. Given an instance of OCSP(Π) on n variables and m constraints, the goal is to find an ordering of the n variables that maximizes the number of constraints that are satisfied, where a constraint specifies a sequence of k distinct variables and the constraint is satisfied by an ordering on the n variables if the ordering induced on the k variables in the constraint satisfies Π. Ordering constraint satisfaction problems capture natural problems including "Maximum acyclic subgraph (MAS)" and "Betweenness".
In this work we consider the task of approximating the maximum number of satisfiable constraints in the (single-pass) streaming setting, where an instance is presented as a stream of constraints. We show that for every Π, OCSP(Π) is approximation-resistant to o(n)-space streaming algorithms, i.e., algorithms using o(n) space cannot distinguish streams where almost every constraint is satisfiable from streams where no ordering beats the random ordering by a noticeable amount. This space bound is tight up to polylogarithmic factors. In the case of MAS our result shows that for every ε > 0, MAS is not 1/2+ε-approximable in o(n) space. The previous best inapproximability result only ruled out a 3/4-approximation in o(√ n) space.
Our results build on recent works of Chou, Golovnev, Sudan, Velingker, and Velusamy who show tight, linear-space inapproximability results for a broad class of (non-ordering) constraint satisfaction problems (CSPs) over arbitrary (finite) alphabets. Our results are obtained by building a family of appropriate CSPs (one for every q) from any given OCSP, and applying their work to this family of CSPs. To convert the resulting hardness results for CSPs back to our OCSP, we show that the hard instances from this earlier work have the following "small-set expansion" property: If the CSP instance is viewed as a hypergraph in the natural way, then for every partition of the hypergraph into small blocks most of the hyperedges are incident on vertices from distinct blocks. By exploiting this combinatorial property, in combination with the hardness results of the resulting families of CSPs, we give optimal inapproximability results for all OCSPs.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.17/LIPIcs.APPROX-RANDOM.2021.17.pdf
Streaming approximations
approximation resistance
constraint satisfaction problems
ordering constraint satisfaction problems
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
18:1
18:22
10.4230/LIPIcs.APPROX/RANDOM.2021.18
article
Upper and Lower Bounds for Complete Linkage in General Metric Spaces
Arutyunova, Anna
1
Großwendt, Anna
1
Röglin, Heiko
1
Schmidt, Melanie
2
Wargalla, Julian
2
Universität Bonn, Germany
Universität Köln, Germany
In a hierarchical clustering problem the task is to compute a series of mutually compatible clusterings of a finite metric space (P,dist). Starting with the clustering where every point forms its own cluster, one iteratively merges two clusters until only one cluster remains. Complete linkage is a well-known and popular algorithm to compute such clusterings: in every step it merges the two clusters whose union has the smallest radius (or diameter) among all currently possible merges. We prove that the radius (or diameter) of every k-clustering computed by complete linkage is at most by factor O(k) (or O(k²)) worse than an optimal k-clustering minimizing the radius (or diameter). Furthermore we give a negative answer to the question proposed by Dasgupta and Long [Sanjoy Dasgupta and Philip M. Long, 2005], who show a lower bound of Ω(log(k)) and ask if the approximation guarantee is in fact Θ(log(k)). We present instances where complete linkage performs poorly in the sense that the k-clustering computed by complete linkage is off by a factor of Ω(k) from an optimal solution for radius and diameter. We conclude that in general metric spaces complete linkage does not perform asymptotically better than single linkage, merging the two clusters with smallest inter-cluster distance, for which we prove an approximation guarantee of O(k).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.18/LIPIcs.APPROX-RANDOM.2021.18.pdf
Hierarchical Clustering
Complete Linkage
agglomerative Clustering
k-Center
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
19:1
19:18
10.4230/LIPIcs.APPROX/RANDOM.2021.19
article
On Two-Pass Streaming Algorithms for Maximum Bipartite Matching
Konrad, Christian
1
https://orcid.org/0000-0003-1802-4011
Naidu, Kheeran K.
1
https://orcid.org/0000-0002-5946-4702
Department of Computer Science, University of Bristol, UK
We study two-pass streaming algorithms for Maximum Bipartite Matching (MBM). All known two-pass streaming algorithms for MBM operate in a similar fashion: They compute a maximal matching in the first pass and find 3-augmenting paths in the second in order to augment the matching found in the first pass. Our aim is to explore the limitations of this approach and to determine whether current techniques can be used to further improve the state-of-the-art algorithms. We give the following results:
We show that every two-pass streaming algorithm that solely computes a maximal matching in the first pass and outputs a (2/3+ε)-approximation requires n^{1+Ω(1/(log log n))} space, for every ε > 0, where n is the number of vertices of the input graph. This result is obtained by extending the Ruzsa-Szemerédi graph construction of [Goel et al., SODA'12] so as to ensure that the resulting graph has a close to perfect matching, the key property needed in our construction. This result may be of independent interest.
Furthermore, we combine the two main techniques, i.e., subsampling followed by the Greedy matching algorithm [Konrad, MFCS'18] which gives a 2-√2 ≈ 0.5857-approximation, and the computation of degree-bounded semi-matchings [Esfandiari et al., ICDMW'16][Kale and Tirodkar, APPROX'17] which gives a 1/2 + 1/12 ≈ 0.5833-approximation, and obtain a meta-algorithm that yields Konrad’s and Esfandiari et al.’s algorithms as special cases. This unifies two strands of research. By optimizing parameters, we discover that Konrad’s algorithm is optimal for the implied class of algorithms and, perhaps surprisingly, that there is a second optimal algorithm. We show that the analysis of our meta-algorithm is best possible. Our results imply that further improvements, if possible, require new techniques.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.19/LIPIcs.APPROX-RANDOM.2021.19.pdf
Data streaming
matchings
lower bounds
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
20:1
20:24
10.4230/LIPIcs.APPROX/RANDOM.2021.20
article
Approximation Algorithms for Demand Strip Packing
Gálvez, Waldo
1
https://orcid.org/0000-0002-6395-3322
Grandoni, Fabrizio
2
Ameli, Afrouz Jabal
2
Khodamoradi, Kamyar
3
https://orcid.org/0000-0003-1289-6839
Technische Universität München, Germany
IDSIA, USI-SUPSI, Lugano, Switzerland
Universitä Würzburg, Germany
In the Demand Strip Packing problem (DSP), we are given a time interval and a collection of tasks, each characterized by a processing time and a demand for a given resource (such as electricity, computational power, etc.). A feasible solution consists of a schedule of the tasks within the mentioned time interval. Our goal is to minimize the peak resource consumption, i.e. the maximum total demand of tasks executed at any point in time.
It is known that DSP is NP-hard to approximate below a factor 3/2, and standard techniques for related problems imply a (polynomial-time) 2-approximation. Our main result is a (5/3+ε)-approximation algorithm for any constant ε > 0. We also achieve best-possible approximation factors for some relevant special cases.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.20/LIPIcs.APPROX-RANDOM.2021.20.pdf
Strip Packing
Two-Dimensional Packing
Approximation Algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
21:1
21:24
10.4230/LIPIcs.APPROX/RANDOM.2021.21
article
Peak Demand Minimization via Sliced Strip Packing
Deppert, Max A.
1
https://orcid.org/0000-0003-3083-7998
Jansen, Klaus
1
https://orcid.org/0000-0001-8358-6796
Khan, Arindam
2
https://orcid.org/0000-0001-7505-1687
Rau, Malin
3
https://orcid.org/0000-0002-5710-560X
Tutas, Malte
1
https://orcid.org/0000-0002-1360-4634
Universität Kiel, Germany
Department of Computer Science and Automation, Indian Institute of Science, Bengaluru, India
Universität Hamburg, Germany
We study the Nonpreemptive Peak Demand Minimization (NPDM) problem, where we are given a set of jobs, specified by their processing times and energy requirements. The goal is to schedule all jobs within a fixed time period such that the peak load (the maximum total energy requirement at any time) is minimized. This problem has recently received significant attention due to its relevance in smart-grids. Theoretically, the problem is related to the classical strip packing problem (SP). In SP, a given set of axis-aligned rectangles must be packed into a fixed-width strip, such that the height of the strip is minimized. NPDM can be modeled as strip packing with slicing and stacking constraint: each rectangle may be cut vertically into multiple slices and the slices may be packed into the strip as individual pieces. The stacking constraint forbids solutions where two slices of the same rectangle are intersected by the same vertical line. Nonpreemption enforces the slices to be placed in contiguous horizontal locations (but may be placed at different vertical locations).
We obtain a (5/3+ε)-approximation algorithm for the problem. We also provide an asymptotic efficient polynomial-time approximation scheme (AEPTAS) which generates a schedule for almost all jobs with energy consumption (1+ε) OPT. The remaining jobs fit into a thin container of height 1. The previous best result for NPDM was a 2.7 approximation based on FFDH [Ranjan et al., 2015]. One of our key ideas is providing several new lower bounds on the optimal solution of a geometric packing, which could be useful in other related problems. These lower bounds help us to obtain approximative solutions based on Steinberg’s algorithm in many cases. In addition, we show how to split schedules generated by the AEPTAS into few segments and to rearrange the corresponding jobs to insert the thin container mentioned above.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.21/LIPIcs.APPROX-RANDOM.2021.21.pdf
scheduling
peak demand minimization
approximation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
22:1
22:23
10.4230/LIPIcs.APPROX/RANDOM.2021.22
article
Tight Approximation Algorithms For Geometric Bin Packing with Skewed Items
Khan, Arindam
1
Sharma, Eklavya
1
Department of Computer Science and Automation, Indian Institute of Science, Bengaluru, India
In the Two-dimensional Bin Packing (2BP) problem, we are given a set of rectangles of height and width at most one and our goal is to find an axis-aligned nonoverlapping packing of these rectangles into the minimum number of unit square bins. The problem admits no APTAS and the current best approximation ratio is 1.406 by Bansal and Khan [SODA'14]. A well-studied variant of the problem is Guillotine Two-dimensional Bin Packing (G2BP), where all rectangles must be packed in such a way that every rectangle in the packing can be obtained by recursively applying a sequence of end-to-end axis-parallel cuts, also called guillotine cuts. Bansal, Lodi, and Sviridenko [FOCS'05] obtained an APTAS for this problem. Let λ be the smallest constant such that for every set I of items, the number of bins in the optimal solution to G2BP for I is upper bounded by λ opt(I) + c, where opt(I) is the number of bins in the optimal solution to 2BP for I and c is a constant. It is known that 4/3 ≤ λ ≤ 1.692. Bansal and Khan [SODA'14] conjectured that λ = 4/3. The conjecture, if true, will imply a (4/3+ε)-approximation algorithm for 2BP. According to convention, for a given constant δ > 0, a rectangle is large if both its height and width are at least δ, and otherwise it is called skewed. We make progress towards the conjecture by showing λ = 4/3 for skewed instance, i.e., when all input rectangles are skewed. Even for this case, the previous best upper bound on λ was roughly 1.692. We also give an APTAS for 2BP for skewed instance, though general 2BP does not admit an APTAS.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.22/LIPIcs.APPROX-RANDOM.2021.22.pdf
Geometric bin packing
guillotine separability
approximation algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
23:1
23:22
10.4230/LIPIcs.APPROX/RANDOM.2021.23
article
Approximating Two-Stage Stochastic Supplier Problems
Brubach, Brian
1
Grammel, Nathaniel
2
Harris, David G.
2
Srinivasan, Aravind
2
Tsepenekas, Leonidas
2
Vullikanti, Anil
3
Wellesley College, MA, USA
University of Maryland at College Park, MD, USA
University of Virginia, Charlottesville, VA, USA
The main focus of this paper is radius-based (supplier) clustering in the two-stage stochastic setting with recourse, where the inherent stochasticity of the model comes in the form of a budget constraint. We also explore a number of variants where additional constraints are imposed on the first-stage decisions, specifically matroid and multi-knapsack constraints.
Our eventual goal is to provide results for supplier problems in the most general distributional setting, where there is only black-box access to the underlying distribution. To that end, we follow a two-step approach. First, we develop algorithms for a restricted version of each problem, in which all possible scenarios are explicitly provided; second, we employ a novel scenario-discarding variant of the standard Sample Average Approximation (SAA) method, in which we crucially exploit properties of the restricted-case algorithms. We finally note that the scenario-discarding modification to the SAA method is necessary in order to optimize over the radius.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.23/LIPIcs.APPROX-RANDOM.2021.23.pdf
Approximation Algorithms
Stochastic Optimization
Two-Stage Recourse Model
Clustering Problems
Knapsack Supplier
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
24:1
24:21
10.4230/LIPIcs.APPROX/RANDOM.2021.24
article
Fast Approximation Algorithms for Bounded Degree and Crossing Spanning Tree Problems
Chekuri, Chandra
1
Quanrud, Kent
2
Torres, Manuel R.
1
University of Illinois at Urbana-Champaign, IL, USA
Purdue University, West Lafayette, IN, USA
We develop fast approximation algorithms for the minimum-cost version of the Bounded-Degree MST problem (BD-MST) and its generalization the Crossing Spanning Tree problem (Crossing-ST). We solve the underlying LP to within a (1+ε) approximation factor in near-linear time via the multiplicative weight update (MWU) technique. This yields, in particular, a near-linear time algorithm that outputs an estimate B such that B ≤ B^* ≤ ⌈(1+ε)B⌉+1 where B^* is the minimum-degree of a spanning tree of a given graph. To round the fractional solution, in our main technical contribution, we describe a fast near-linear time implementation of swap-rounding in the spanning tree polytope of a graph. The fractional solution can also be used to sparsify the input graph that can in turn be used to speed up existing combinatorial algorithms. Together, these ideas lead to significantly faster approximation algorithms than known before for the two problems of interest. In addition, a fast algorithm for swap rounding in the graphic matroid is a generic tool that has other applications, including to TSP and submodular function maximization.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.24/LIPIcs.APPROX-RANDOM.2021.24.pdf
bounded degree spanning tree
crossing spanning tree
swap rounding
fast approximation algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
25:1
25:23
10.4230/LIPIcs.APPROX/RANDOM.2021.25
article
Hitting Weighted Even Cycles in Planar Graphs
Göke, Alexander
1
Koenemann, Jochen
2
Mnich, Matthias
1
https://orcid.org/0000-0002-4721-5354
Sun, Hao
2
Hamburg University of Technology, Institute for Algorithms and Complexity, Germany
University of Waterloo, Canada
A classical branch of graph algorithms is graph transversals, where one seeks a minimum-weight subset of nodes in a node-weighted graph G which intersects all copies of subgraphs F from a fixed family F. Many such graph transversal problems have been shown to admit polynomial-time approximation schemes (PTAS) for planar input graphs G, using a variety of techniques like the shifting technique (Baker, J. ACM 1994), bidimensionality (Fomin et al., SODA 2011), or connectivity domination (Cohen-Addad et al., STOC 2016). These techniques do not seem to apply to graph transversals with parity constraints, which have recently received significant attention, but for which no PTASs are known.
In the even-cycle transversal (ECT) problem, the goal is to find a minimum-weight hitting set for the set of even cycles in an undirected graph. For ECT, Fiorini et al. (IPCO 2010) showed that the integrality gap of the standard covering LP relaxation is Θ(log n), and that adding sparsity inequalities reduces the integrality gap to 10.
Our main result is a primal-dual algorithm that yields a 47/7 ≈ 6.71-approximation for ECT on node-weighted planar graphs, and an integrality gap of the same value for the standard LP relaxation on node-weighted planar graphs.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.25/LIPIcs.APPROX-RANDOM.2021.25.pdf
Even cycles
planar graphs
integrality gap
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
26:1
26:16
10.4230/LIPIcs.APPROX/RANDOM.2021.26
article
Revenue Maximization in Transportation Networks
Bhawalkar, Kshipra
1
Kollias, Kostas
1
Purohit, Manish
1
Google Research, Mountain View, CA, USA
We study the joint optimization problem of pricing trips in a transportation network and serving the induced demands by routing a fleet of available service vehicles to maximize revenue. Our framework encompasses applications that include traditional transportation networks (e.g., airplanes, buses) and their more modern counterparts (e.g., ride-sharing systems). We describe a simple combinatorial model, in which each edge in the network is endowed with a curve that gives the demand for traveling between its endpoints at any given price. We are supplied with a number of vehicles and a time budget to serve the demands induced by the prices that we set, seeking to maximize revenue. We first focus on a (preliminary) special case of our model with unit distances and unit time horizon. We show that this version of the problem can be solved optimally in polynomial time. Switching to the general case of our model, we first present a two-stage approach that separately optimizes for prices and routes, achieving a logarithmic approximation to revenue in the process. Next, using the insights gathered in the first two results, we present a constant factor approximation algorithm that jointly optimizes for prices and routes for the supply vehicles. Finally, we discuss how our algorithms can handle capacitated vehicles, impatient demands, and selfish (wage-maximizing) drivers.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.26/LIPIcs.APPROX-RANDOM.2021.26.pdf
Pricing
networks
approximation algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
27:1
27:14
10.4230/LIPIcs.APPROX/RANDOM.2021.27
article
Connected k-Partition of k-Connected Graphs and c-Claw-Free Graphs
Borndörfer, Ralf
1
https://orcid.org/0000-0001-7223-9174
Casel, Katrin
2
https://orcid.org/0000-0001-6146-8684
Issac, Davis
2
https://orcid.org/0000-0001-5559-7471
Niklanovits, Aikaterini
2
https://orcid.org/0000-0002-4911-4493
Schwartz, Stephan
1
https://orcid.org/0000-0003-2901-5065
Zeif, Ziena
2
https://orcid.org/0000-0003-0378-1458
Zuse Institute Berlin, Germany
Hasso Plattner Institute, University of Potsdam, Germany
A connected partition is a partition of the vertices of a graph into sets that induce connected subgraphs. Such partitions naturally occur in many application areas such as road networks, and image processing. In these settings, it is often desirable to partition into a fixed number of parts of roughly of the same size or weight. The resulting computational problem is called Balanced Connected Partition (BCP). The two classical objectives for BCP are to maximize the weight of the smallest, or minimize the weight of the largest component. We study BCP on c-claw-free graphs, the class of graphs that do not have K_{1,c} as an induced subgraph, and present efficient (c-1)-approximation algorithms for both objectives. In particular, for 3-claw-free graphs, also simply known as claw-free graphs, we obtain a 2-approximation. Due to the claw-freeness of line graphs, this also implies a 2-approximation for the edge-partition version of BCP in general graphs.
A harder connected partition problem arises from demanding a connected partition into k parts that have (possibly) heterogeneous target weights w₁,…,w_k. In the 1970s Győri and Lovász showed that if G is k-connected and the target weights sum to the total size of G, such a partition exists. However, to this day no polynomial algorithm to compute such partitions exists for k > 4. Towards finding such a partition T₁,…, T_k in k-connected graphs for general k, we show how to efficiently compute connected partitions that at least approximately meet the target weights, subject to the mild assumption that each w_i is greater than the weight of the heaviest vertex. In particular, we give a 3-approximation for both the lower and the upper bounded version i.e. we guarantee that each T_i has weight at least (w_i)/3 or that each T_i has weight most 3w_i, respectively. Also, we present a both-side bounded version that produces a connected partition where each T_i has size at least (w_i)/3 and at most max({r,3}) w_i, where r ≥ 1 is the ratio between the largest and smallest value in w₁, … , w_k. In particular for the balanced version, i.e. w₁ = w₂ = , … , = w_k, this gives a partition with 1/3w_i ≤ w(T_i) ≤ 3w_i.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.27/LIPIcs.APPROX-RANDOM.2021.27.pdf
connected partition
Győri-Lovász
balanced partition
approximation algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
28:1
28:23
10.4230/LIPIcs.APPROX/RANDOM.2021.28
article
Better Pseudodistributions and Derandomization for Space-Bounded Computation
Hoza, William M.
1
https://orcid.org/0000-0001-5162-9181
Simons Institute for the Theory of Computing, Berkeley, CA, USA
Three decades ago, Nisan constructed an explicit pseudorandom generator (PRG) that fools width-n length-n read-once branching programs (ROBPs) with error ε and seed length O(log² n + log n ⋅ log(1/ε)) [Nisan, 1992]. Nisan’s generator remains the best explicit PRG known for this important model of computation. However, a recent line of work starting with Braverman, Cohen, and Garg [Braverman et al., 2020; Chattopadhyay and Liao, 2020; Cohen et al., 2021; Pyne and Vadhan, 2021] has shown how to construct weighted pseudorandom generators (WPRGs, aka pseudorandom pseudodistribution generators) with better seed lengths than Nisan’s generator when the error parameter ε is small.
In this work, we present an explicit WPRG for width-n length-n ROBPs with seed length O(log² n + log(1/ε)). Our seed length eliminates log log factors from prior constructions, and our generator completes this line of research in the sense that further improvements would require beating Nisan’s generator in the standard constant-error regime. Our technique is a variation of a recently-discovered reduction that converts moderate-error PRGs into low-error WPRGs [Cohen et al., 2021; Pyne and Vadhan, 2021]. Our version of the reduction uses averaging samplers.
We also point out that as a consequence of the recent work on WPRGs, any randomized space-S decision algorithm can be simulated deterministically in space O (S^{3/2} / √{log S}). This is a slight improvement over Saks and Zhou’s celebrated O(S^{3/2}) bound [Saks and Zhou, 1999]. For this application, our improved WPRG is not necessary.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.28/LIPIcs.APPROX-RANDOM.2021.28.pdf
Weighted pseudorandom generator
pseudorandom pseudodistribution
read-once branching program
derandomization
space complexity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
29:1
29:19
10.4230/LIPIcs.APPROX/RANDOM.2021.29
article
On the Hardness of Average-Case k-SUM
Brakerski, Zvika
1
Stephens-Davidowitz, Noah
2
Vaikuntanathan, Vinod
3
Weizmann Institute of Science, Rehovot, Israel
Cornell University, Ithaca, NY, USA
Massachusetts Institute of Technology, Cambridge, MA, USA
In this work, we show the first worst-case to average-case reduction for the classical k-SUM problem. A k-SUM instance is a collection of m integers, and the goal of the k-SUM problem is to find a subset of k integers that sums to 0. In the average-case version, the m elements are chosen uniformly at random from some interval [-u,u].
We consider the total setting where m is sufficiently large (with respect to u and k), so that we are guaranteed (with high probability) that solutions must exist. In particular, m = u^{Ω(1/k)} suffices for totality. Much of the appeal of k-SUM, in particular connections to problems in computational geometry, extends to the total setting.
The best known algorithm in the average-case total setting is due to Wagner (following the approach of Blum-Kalai-Wasserman), and achieves a running time of u^{Θ(1/log k)} when m = u^{Θ(1/log k)}. This beats the known (conditional) lower bounds for worst-case k-SUM, raising the natural question of whether it can be improved even further. However, in this work, we show a matching average-case lower bound, by showing a reduction from worst-case lattice problems, thus introducing a new family of techniques into the field of fine-grained complexity. In particular, we show that any algorithm solving average-case k-SUM on m elements in time u^{o(1/log k)} will give a super-polynomial improvement in the complexity of algorithms for lattice problems.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.29/LIPIcs.APPROX-RANDOM.2021.29.pdf
k-SUM
fine-grained complexity
average-case hardness
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
30:1
30:23
10.4230/LIPIcs.APPROX/RANDOM.2021.30
article
Improved Hitting Set for Orbit of ROABPs
Bhargava, Vishwas
1
Ghosh, Sumanta
2
Department of Computer Science, Rutgers University, Piscataway, NJ, USA
Department of Computer Science, IIT Bombay, India
The orbit of an n-variate polynomial f(x) over a field 𝔽 is the set {f(Ax+b) ∣ A ∈ GL(n, 𝔽) and b ∈ 𝔽ⁿ}, and the orbit of a polynomial class is the union of orbits of all the polynomials in it. In this paper, we give improved constructions of hitting-sets for the orbit of read-once oblivious algebraic branching programs (ROABPs) and a related model. Over fields with characteristic zero or greater than d, we construct a hitting set of size (ndw)^{O(w²log n⋅ min{w², dlog w})} for the orbit of ROABPs in unknown variable order where d is the individual degree and w is the width of ROABPs. We also give a hitting set of size (ndw)^{O(min{w²,dlog w})} for the orbit of polynomials computed by w-width ROABPs in any variable order. Our hitting sets improve upon the results of Saha and Thankey [Chandan Saha and Bhargav Thankey, 2021] who gave an (ndw)^{O(dlog w)} size hitting set for the orbit of commutative ROABPs (a subclass of any-order ROABPs) and (nw)^{O(w⁶log n)} size hitting set for the orbit of multilinear ROABPs. Designing better hitting sets in large individual degree regime, for instance d > n, was asked as an open problem by [Chandan Saha and Bhargav Thankey, 2021] and this work solves it in small width setting.
We prove some new rank concentration results by establishing low-cone concentration for the polynomials over vector spaces, and they strengthen some previously known low-support based rank concentrations shown in [Michael A. Forbes et al., 2013]. These new low-cone concentration results are crucial in our hitting set construction, and may be of independent interest. To the best of our knowledge, this is the first time when low-cone rank concentration has been used for designing hitting sets.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.30/LIPIcs.APPROX-RANDOM.2021.30.pdf
Hitting Set
Low Cone Concentration
Orbits
PIT
ROABP
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
31:1
31:25
10.4230/LIPIcs.APPROX/RANDOM.2021.31
article
A New Notion of Commutativity for the Algorithmic Lovász Local Lemma
Harris, David G.
1
Iliopoulos, Fotis
2
Kolmogorov, Vladimir
3
University of Maryland, College Park, MD, USA
Institute for Advanced Study, Princeton, NJ, USA
Institute of Science and Technology, Klosterneuburg, Austria
The Lovász Local Lemma (LLL) is a powerful tool in probabilistic combinatorics which can be used to establish the existence of objects that satisfy certain properties. The breakthrough paper of Moser & Tardos and follow-up works revealed that the LLL has intimate connections with a class of stochastic local search algorithms for finding such desirable objects. In particular, it can be seen as a sufficient condition for this type of algorithms to converge fast.
Besides conditions for convergence, many other natural questions can be asked about algorithms; for instance, "are they parallelizable?", "how many solutions can they output?", "what is the expected "weight" of a solution?". These questions and more have been answered for a class of LLL-inspired algorithms called commutative. In this paper we introduce a new, very natural and more general notion of commutativity (essentially matrix commutativity) which allows us to show a number of new refined properties of LLL-inspired local search algorithms with significantly simpler proofs.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.31/LIPIcs.APPROX-RANDOM.2021.31.pdf
Lovász Local Lemma
Resampling
Moser-Tardos algorithm
latin transversal
commutativity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
32:1
32:21
10.4230/LIPIcs.APPROX/RANDOM.2021.32
article
From Coupling to Spectral Independence and Blackbox Comparison with the Down-Up Walk
Liu, Kuikui
1
University of Washington, Seattle, WA, USA
We show that the existence of a "good" coupling w.r.t. Hamming distance for any local Markov chain on a discrete product space implies rapid mixing of the Glauber dynamics in a blackbox fashion. More specifically, we only require the expected distance between successive iterates under the coupling to be summable, as opposed to being one-step contractive in the worst case. Combined with recent local-to-global arguments [Chen et al., 2021], we establish asymptotically optimal lower bounds on the standard and modified log-Sobolev constants for the Glauber dynamics for sampling from spin systems on bounded-degree graphs when a curvature condition [Ollivier, 2009] is satisfied. To achieve this, we use Stein’s method for Markov chains [Bresler and Nagaraj, 2019; Reinert and Ross, 2019] to show that a "good" coupling for a local Markov chain yields strong bounds on the spectral independence of the distribution in the sense of [Anari et al., 2020].
Our primary application is to sampling proper list-colorings on bounded-degree graphs. In particular, combining the coupling for the flip dynamics given by [Vigoda, 2000; Chen et al., 2019] with our techniques, we show optimal O(nlog n) mixing for the Glauber dynamics for sampling proper list-colorings on any bounded-degree graph with maximum degree Δ whenever the size of the color lists are at least ({11/6 - ε}) Δ, where ε ≈ 10^{-5} is small constant. While O(n²) mixing was already known before, our approach additionally yields Chernoff-type concentration bounds for Hamming Lipschitz functions in this regime, which was not known before. Our approach is markedly different from prior works establishing spectral independence for spin systems using spatial mixing [Anari et al., 2020; Z. {Chen} et al., 2020; Chen et al., 2021; Feng et al., 2021], which crucially is still open in this regime for proper list-colorings.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.32/LIPIcs.APPROX-RANDOM.2021.32.pdf
Markov chains
Approximate counting
Spectral independence
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
33:1
33:16
10.4230/LIPIcs.APPROX/RANDOM.2021.33
article
Singularity of Random Integer Matrices with Large Entries
Karingula, Sankeerth Rao
1
https://orcid.org/0000-0003-2212-4322
Lovett, Shachar
1
https://orcid.org/0000-0003-4552-1443
Department of Computer Science, University of California San Diego, CA, USA
We study the singularity probability of random integer matrices. Concretely, the probability that a random n × n matrix, with integer entries chosen uniformly from {-m,…,m}, is singular. This problem has been well studied in two regimes: large n and constant m; or large m and constant n. In this paper, we extend previous techniques to handle the regime where both n,m are large. We show that the probability that such a matrix is singular is m^{-cn} for some absolute constant c > 0. We also provide some connections of our result to coding theory.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.33/LIPIcs.APPROX-RANDOM.2021.33.pdf
Coding Theory
Random matrix theory
Singularity probability MDS codes
Error correction codes
Littlewood Offord
Fourier Analysis
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
34:1
34:23
10.4230/LIPIcs.APPROX/RANDOM.2021.34
article
Interplay Between Graph Isomorphism and Earth Mover’s Distance in the Query and Communication Worlds
Chakraborty, Sourav
1
Ghosh, Arijit
1
Mishra, Gopinath
1
Sen, Sayantan
1
Indian Statistical Institute, Kolkata, India
The graph isomorphism distance between two graphs G_u and G_k is the fraction of entries in the adjacency matrix that has to be changed to make G_u isomorphic to G_k. We study the problem of estimating, up to a constant additive factor, the graph isomorphism distance between two graphs in the query model. In other words, if G_k is a known graph and G_u is an unknown graph whose adjacency matrix has to be accessed by querying the entries, what is the query complexity for testing whether the graph isomorphism distance between G_u and G_k is less than γ₁ or more than γ₂, where γ₁ and γ₂ are two constants with 0 ≤ γ₁ < γ₂ ≤ 1. It is also called the tolerant property testing of graph isomorphism in the dense graph model. The non-tolerant version (where γ₁ is 0) has been studied by Fischer and Matsliah (SICOMP'08).
In this paper, we prove a (interesting) connection between tolerant graph isomorphism testing and tolerant testing of the well studied Earth Mover’s Distance (EMD). We prove that deciding tolerant graph isomorphism is equivalent to deciding tolerant EMD testing between multi-sets in the query setting. Moreover, the reductions between tolerant graph isomorphism and tolerant EMD testing (in query setting) can also be extended directly to work in the two party Alice-Bob communication model (where Alice and Bob have one graph each and they want to solve tolerant graph isomorphism problem by communicating bits), and possibly in other sublinear models as well.
Testing tolerant EMD between two probability distributions is equivalent to testing EMD between two multi-sets, where the multiplicity of each element is taken appropriately, and we sample elements from the unknown multi-set with replacement. In this paper, our (main) contribution is to introduce the problem of {(tolerant) EMD testing between multi-sets (over Hamming cube) when we get samples from the unknown multi-set without replacement} and to show that this variant of tolerant testing of EMD is as hard as tolerant testing of graph isomorphism between two graphs. {Thus, while testing of equivalence between distributions is at the heart of the non-tolerant testing of graph isomorphism, we are showing that the estimation of the EMD over a Hamming cube (when we are allowed to sample without replacement) is at the heart of tolerant graph isomorphism.} We believe that the introduction of the problem of testing EMD between multi-sets (when we get samples without replacement) opens an entirely new direction in the world of testing properties of distributions.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.34/LIPIcs.APPROX-RANDOM.2021.34.pdf
Graph Isomorphism
Earth Mover Distance
Query Complexity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
35:1
35:22
10.4230/LIPIcs.APPROX/RANDOM.2021.35
article
The Product of Gaussian Matrices Is Close to Gaussian
Li, Yi
1
https://orcid.org/0000-0002-6420-653X
Woodruff, David P.
2
Division of Mathematical Sciences, Nanyang Technological University, Singapore, Singapore
Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA
We study the distribution of the matrix product G₁ G₂ ⋯ G_r of r independent Gaussian matrices of various sizes, where G_i is d_{i-1} × d_i, and we denote p = d₀, q = d_r, and require d₁ = d_{r-1}. Here the entries in each G_i are standard normal random variables with mean 0 and variance 1. Such products arise in the study of wireless communication, dynamical systems, and quantum transport, among other places. We show that, provided each d_i, i = 1, …, r, satisfies d_i ≥ C p ⋅ q, where C ≥ C₀ for a constant C₀ > 0 depending on r, then the matrix product G₁ G₂ ⋯ G_r has variation distance at most δ to a p × q matrix G of i.i.d. standard normal random variables with mean 0 and variance ∏_{i = 1}^{r-1} d_i. Here δ → 0 as C → ∞. Moreover, we show a converse for constant r that if d_i < C' max{p,q}^{1/2}min{p,q}^{3/2} for some i, then this total variation distance is at least δ', for an absolute constant δ' > 0 depending on C' and r. This converse is best possible when p = Θ(q).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.35/LIPIcs.APPROX-RANDOM.2021.35.pdf
random matrix theory
total variation distance
matrix product
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
36:1
36:13
10.4230/LIPIcs.APPROX/RANDOM.2021.36
article
Fast Mixing via Polymers for Random Graphs with Unbounded Degree
Galanis, Andreas
1
Goldberg, Leslie Ann
1
Stewart, James
1
Department of Computer Science, University of Oxford, UK
The polymer model framework is a classical tool from statistical mechanics that has recently been used to obtain approximation algorithms for spin systems on classes of bounded-degree graphs; examples include the ferromagnetic Potts model on expanders and on the grid. One of the key ingredients in the analysis of polymer models is controlling the growth rate of the number of polymers, which has been typically achieved so far by invoking the bounded-degree assumption. Nevertheless, this assumption is often restrictive and obstructs the applicability of the method to more general graphs. For example, sparse random graphs typically have bounded average degree and good expansion properties, but they include vertices with unbounded degree, and therefore are excluded from the current polymer-model framework.
We develop a less restrictive framework for polymer models that relaxes the standard bounded-degree assumption, by reworking the relevant polymer models from the edge perspective. The edge perspective allows us to bound the growth rate of the number of polymers in terms of the total degree of polymers, which in turn can be related more easily to the expansion properties of the underlying graph. To apply our methods, we consider random graphs with unbounded degrees from a fixed degree sequence (with minimum degree at least 3) and obtain approximation algorithms for the ferromagnetic Potts model, which is a standard benchmark for polymer models. Our techniques also extend to more general spin systems.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.36/LIPIcs.APPROX-RANDOM.2021.36.pdf
Markov chains
approximate counting
Potts model
expander graphs
random graphs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
37:1
37:18
10.4230/LIPIcs.APPROX/RANDOM.2021.37
article
Deterministic Approximate Counting of Polynomial Threshold Functions via a Derandomized Regularity Lemma
Servedio, Rocco A.
1
Tan, Li-Yang
2
Columbia University, New York, NY, USA
Stanford University, CA, USA
We study the problem of deterministically approximating the number of satisfying assignments of a polynomial threshold function (PTF) over Boolean space. We present and analyze a scheme for transforming such algorithms for PTFs over Gaussian space into algorithms for the more challenging and more standard setting of Boolean space. Applying this transformation to existing algorithms for Gaussian space leads to new algorithms for Boolean space that improve on prior state-of-the-art results due to Meka and Zuckerman [Meka and Zuckerman, 2013] and Kane [Kane, 2012]. Our approach is based on a bias-preserving derandomization of Meka and Zuckerman’s regularity lemma for polynomials [Meka and Zuckerman, 2013] using the [Rocco A. Servedio and Li-Yang Tan, 2018] pseudorandom generator for PTFs.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.37/LIPIcs.APPROX-RANDOM.2021.37.pdf
Derandomization
Polynomial threshold functions
deterministic approximate counting
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
38:1
38:17
10.4230/LIPIcs.APPROX/RANDOM.2021.38
article
Improved Product-Based High-Dimensional Expanders
Golowich, Louis
1
Department of Computer Science, Harvard University, Cambridge, MA, USA
High-dimensional expanders generalize the notion of expander graphs to higher-dimensional simplicial complexes. In contrast to expander graphs, only a handful of high-dimensional expander constructions have been proposed, and no elementary combinatorial construction with near-optimal expansion is known. In this paper, we introduce an improved combinatorial high-dimensional expander construction, by modifying a previous construction of Liu, Mohanty, and Yang (ITCS 2020), which is based on a high-dimensional variant of a tensor product. Our construction achieves a spectral gap of Ω(1/(k²)) for random walks on the k-dimensional faces, which is only quadratically worse than the optimal bound of Θ(1/k). Previous combinatorial constructions, including that of Liu, Mohanty, and Yang, only achieved a spectral gap that is exponentially small in k. We also present reasoning that suggests our construction is optimal among similar product-based constructions.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.38/LIPIcs.APPROX-RANDOM.2021.38.pdf
High-Dimensional Expander
Expander Graph
Random Walk
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
39:1
39:16
10.4230/LIPIcs.APPROX/RANDOM.2021.39
article
Improved Bounds for Coloring Locally Sparse Hypergraphs
Iliopoulos, Fotis
1
2
Institute for Advanced Study, Princeton, NJ, USA
Princeton University, NJ, USA
We show that, for every k ≥ 2, every k-uniform hypergaph of degree Δ and girth at least 5 is efficiently (1+o(1))(k-1) (Δ / ln Δ)^{1/(k-1)}-list colorable. As an application we obtain the currently best deterministic algorithm for list-coloring random hypergraphs of bounded average degree.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.39/LIPIcs.APPROX-RANDOM.2021.39.pdf
hypergaph coloring
semi-random method
locally sparse
random hypergraphs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
40:1
40:21
10.4230/LIPIcs.APPROX/RANDOM.2021.40
article
Smoothed Analysis of the Condition Number Under Low-Rank Perturbations
Shah, Rikhav
1
Silwal, Sandeep
2
University of California at Berkeley, CA, USA
Massachusetts Institute of Technology, Cambridge, MA, USA
Let M be an arbitrary n by n matrix of rank n-k. We study the condition number of M plus a low-rank perturbation UV^T where U, V are n by k random Gaussian matrices. Under some necessary assumptions, it is shown that M+UV^T is unlikely to have a large condition number. The main advantages of this kind of perturbation over the well-studied dense Gaussian perturbation, where every entry is independently perturbed, is the O(nk) cost to store U,V and the O(nk) increase in time complexity for performing the matrix-vector multiplication (M+UV^T)x. This improves the Ω(n²) space and time complexity increase required by a dense perturbation, which is especially burdensome if M is originally sparse. Our results also extend to the case where U and V have rank larger than k and to symmetric and complex settings. We also give an application to linear systems solving and perform some numerical experiments. Lastly, barriers in applying low-rank noise to other problems studied in the smoothed analysis framework are discussed.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.40/LIPIcs.APPROX-RANDOM.2021.40.pdf
Smoothed analysis
condition number
low rank noise
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
41:1
41:16
10.4230/LIPIcs.APPROX/RANDOM.2021.41
article
Matroid Intersection: A Pseudo-Deterministic Parallel Reduction from Search to Weighted-Decision
Ghosh, Sumanta
1
Gurjar, Rohit
1
Indian Institute of Technology Bombay, India
We study the matroid intersection problem from the parallel complexity perspective. Given two matroids over the same ground set, the problem asks to decide whether they have a common base and its search version asks to find a common base, if one exists. Another widely studied variant is the weighted decision version where with the two matroids, we are given small weights on the ground set elements and a target weight W, and the question is to decide whether there is a common base of weight at least W. From the perspective of parallel complexity, the relation between the search and the decision versions is not well understood. We make a significant progress on this question by giving a pseudo-deterministic parallel (NC) algorithm for the search version that uses an oracle access to the weighted decision.
The notion of pseudo-deterministic NC was recently introduced by Goldwasser and Grossman [Shafi Goldwasser and Ofer Grossman, 2017], which is a relaxation of NC. A pseudo-deterministic NC algorithm for a search problem is a randomized NC algorithm that, for a given input, outputs a fixed solution with high probability. In case the given matroids are linearly representable, our result implies a pseudo-deterministic NC algorithm (without the weighted decision oracle). This resolves an open question posed by Anari and Vazirani [Nima Anari and Vijay V. Vazirani, 2020].
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.41/LIPIcs.APPROX-RANDOM.2021.41.pdf
Linear Matroid
Matroid Intersection
Parallel Complexity
Pseudo-deterministic NC
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
42:1
42:20
10.4230/LIPIcs.APPROX/RANDOM.2021.42
article
On the Probabilistic Degree of an n-Variate Boolean Function
Srinivasan, Srikanth
1
Venkitesh, S.
2
Department of Computer Science, Aarhus University, Aarhus, Denmark
Department of Mathematics, IIT Bombay, Mumbai, India
Nisan and Szegedy (CC 1994) showed that any Boolean function f:{0,1}ⁿ → {0,1} that depends on all its input variables, when represented as a real-valued multivariate polynomial P(x₁,…,x_n), has degree at least log n - O(log log n). This was improved to a tight (log n - O(1)) bound by Chiarelli, Hatami and Saks (Combinatorica 2020). Similar statements are also known for other Boolean function complexity measures such as Sensitivity (Simon (FCT 1983)), Quantum query complexity, and Approximate degree (Ambainis and de Wolf (CC 2014)).
In this paper, we address this question for Probabilistic degree. The function f has probabilistic degree at most d if there is a random real-valued polynomial of degree at most d that agrees with f at each input with high probability. Our understanding of this complexity measure is significantly weaker than those above: for instance, we do not even know the probabilistic degree of the OR function, the best-known bounds put it between (log n)^{1/2-o(1)} and O(log n) (Beigel, Reingold, Spielman (STOC 1991); Tarui (TCS 1993); Harsha, Srinivasan (RSA 2019)).
Here we can give a near-optimal understanding of the probabilistic degree of n-variate functions f, modulo our lack of understanding of the probabilistic degree of OR. We show that if the probabilistic degree of OR is (log n)^c, then the minimum possible probabilistic degree of such an f is at least (log n)^{c/(c+1)-o(1)}, and we show this is tight up to (log n)^{o(1)} factors.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.42/LIPIcs.APPROX-RANDOM.2021.42.pdf
truly n-variate
Boolean function
probabilistic polynomial
probabilistic degree
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
43:1
43:15
10.4230/LIPIcs.APPROX/RANDOM.2021.43
article
The Swendsen-Wang Dynamics on Trees
Blanca, Antonio
1
Chen, Zongchen
2
Štefankovič, Daniel
3
Vigoda, Eric
2
Pennsylvania State University, University Park, PA, USA
Georgia Institute of Technology, Atlanta, GA, USA
University of Rochester, NY, USA
The Swendsen-Wang algorithm is a sophisticated, widely-used Markov chain for sampling from the Gibbs distribution for the ferromagnetic Ising and Potts models. This chain has proved difficult to analyze, due in part to the global nature of its updates. We present optimal bounds on the convergence rate of the Swendsen-Wang algorithm for the complete d-ary tree. Our bounds extend to the non-uniqueness region and apply to all boundary conditions. We show that the spatial mixing conditions known as Variance Mixing and Entropy Mixing, introduced in the study of local Markov chains by Martinelli et al. (2003), imply Ω(1) spectral gap and O(log n) mixing time, respectively, for the Swendsen-Wang dynamics on the d-ary tree. We also show that these bounds are asymptotically optimal. As a consequence, we establish Θ(log n) mixing for the Swendsen-Wang dynamics for all boundary conditions throughout the tree uniqueness region; in fact, our bounds hold beyond the uniqueness threshold for the Ising model, and for the q-state Potts model when q is small with respect to d. Our proofs feature a novel spectral view of the Variance Mixing condition inspired by several recent rapid mixing results on high-dimensional expanders and utilize recent work on block factorization of entropy under spatial mixing conditions.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.43/LIPIcs.APPROX-RANDOM.2021.43.pdf
Markov Chains
mixing times
Ising and Potts models
Swendsen-Wang dynamics
trees
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
44:1
44:22
10.4230/LIPIcs.APPROX/RANDOM.2021.44
article
Distance Estimation Between Unknown Matrices Using Sublinear Projections on Hamming Cube
Bishnu, Arijit
1
Ghosh, Arijit
1
Mishra, Gopinath
1
Indian Statistical Institute, Kolkata, India
Using geometric techniques like projection and dimensionality reduction, we show that there exists a randomized sub-linear time algorithm that can estimate the Hamming distance between two matrices. Consider two matrices A and B of size n × n whose dimensions are known to the algorithm but the entries are not. The entries of the matrix are real numbers. The access to any matrix is through an oracle that computes the projection of a row (or a column) of the matrix on a vector in {0,1}ⁿ. We call this query oracle to be an Inner Product oracle (shortened as IP). We show that our algorithm returns a (1± ε) approximation to {D}_M (A,B) with high probability by making O(n/(√{{D)_M (A,B)}}poly(log n, 1/(ε))) oracle queries, where {D}_M (A,B) denotes the Hamming distance (the number of corresponding entries in which A and B differ) between two matrices A and B of size n × n. We also show a matching lower bound on the number of such IP queries needed. Though our main result is on estimating {D}_M (A,B) using IP, we also compare our results with other query models.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.44/LIPIcs.APPROX-RANDOM.2021.44.pdf
Distance estimation
Property testing
Dimensionality reduction
Sub-linear algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
45:1
45:16
10.4230/LIPIcs.APPROX/RANDOM.2021.45
article
Decision Tree Heuristics Can Fail, Even in the Smoothed Setting
Blanc, Guy
1
Lange, Jane
2
Qiao, Mingda
1
Tan, Li-Yang
1
Stanford University, CA, USA
Massachusetts Institute of Technology, Cambridge, MA, USA
Greedy decision tree learning heuristics are mainstays of machine learning practice, but theoretical justification for their empirical success remains elusive. In fact, it has long been known that there are simple target functions for which they fail badly (Kearns and Mansour, STOC 1996).
Recent work of Brutzkus, Daniely, and Malach (COLT 2020) considered the smoothed analysis model as a possible avenue towards resolving this disconnect. Within the smoothed setting and for targets f that are k-juntas, they showed that these heuristics successfully learn f with depth-k decision tree hypotheses. They conjectured that the same guarantee holds more generally for targets that are depth-k decision trees.
We provide a counterexample to this conjecture: we construct targets that are depth-k decision trees and show that even in the smoothed setting, these heuristics build trees of depth 2^{Ω(k)} before achieving high accuracy. We also show that the guarantees of Brutzkus et al. cannot extend to the agnostic setting: there are targets that are very close to k-juntas, for which these heuristics build trees of depth 2^{Ω(k)} before achieving high accuracy.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.45/LIPIcs.APPROX-RANDOM.2021.45.pdf
decision trees
learning theory
smoothed analysis
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
46:1
46:23
10.4230/LIPIcs.APPROX/RANDOM.2021.46
article
On the Structure of Learnability Beyond P/Poly
Rajgopal, Ninad
1
https://orcid.org/0000-0001-6945-2345
Santhanam, Rahul
2
University of Warwick, Coventry, UK
University of Oxford, UK
Motivated by the goal of showing stronger structural results about the complexity of learning, we study the learnability of strong concept classes beyond P/poly, such as PSPACE/poly and EXP/poly. We show the following:
1) (Unconditional Lower Bounds for Learning) Building on [Adam R. Klivans et al., 2013], we prove unconditionally that BPE/poly cannot be weakly learned in polynomial time over the uniform distribution, even with membership and equivalence queries.
2) (Robustness of Learning) For the concept classes EXP/poly and PSPACE/poly, we show unconditionally that worst-case and average-case learning are equivalent, that PAC-learnability and learnability over the uniform distribution are equivalent, and that membership queries do not help in either case.
3) (Reducing Succinct Search to Decision for Learning) For the decision problems R_{Kt} and R_{KS} capturing the complexity of learning EXP/poly and PSPACE/poly respectively, we show a succinct search to decision reduction: for each of these problems, the problem is in BPP iff there is a probabilistic polynomial-time algorithm computing circuits encoding proofs for positive instances of the problem. This is shown via a more general result giving succinct search to decision results for PSPACE, EXP and NEXP, which might be of independent interest.
4) (Implausibility of Oblivious Strongly Black-Box Reductions showing NP-hardness of learning NP/poly) We define a natural notion of hardness of learning with respect to oblivious strongly black-box reductions. We show that learning PSPACE/poly is PSPACE-hard with respect to oblivious strongly black-box reductions. On the other hand, if learning NP/poly is NP-hard with respect to oblivious strongly black-box reductions, the Polynomial Hierarchy collapses.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.46/LIPIcs.APPROX-RANDOM.2021.46.pdf
Hardness of Learning
Oracle Circuit Classes
Succinct Search
Black-Box Reductions
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
47:1
47:15
10.4230/LIPIcs.APPROX/RANDOM.2021.47
article
The Critical Mean-Field Chayes-Machta Dynamics
Blanca, Antonio
1
Sinclair, Alistair
2
Zhang, Xusheng
1
Pennsylvania State University, University Park, PA, USA
University of California at Berkeley, CA, USA
The random-cluster model is a unifying framework for studying random graphs, spin systems and electrical networks that plays a fundamental role in designing efficient Markov Chain Monte Carlo (MCMC) sampling algorithms for the classical ferromagnetic Ising and Potts models. In this paper, we study a natural non-local Markov chain known as the Chayes-Machta dynamics for the mean-field case of the random-cluster model, where the underlying graph is the complete graph on n vertices. The random-cluster model is parametrized by an edge probability p and a cluster weight q. Our focus is on the critical regime: p = p_c(q) and q ∈ (1,2), where p_c(q) is the threshold corresponding to the order-disorder phase transition of the model. We show that the mixing time of the Chayes-Machta dynamics is O(log n ⋅ log log n) in this parameter regime, which reveals that the dynamics does not undergo an exponential slowdown at criticality, a surprising fact that had been predicted (but not proved) by statistical physicists. This also provides a nearly optimal bound (up to the log log n factor) for the mixing time of the mean-field Chayes-Machta dynamics in the only regime of parameters where no non-trivial bound was previously known. Our proof consists of a multi-phased coupling argument that combines several key ingredients, including a new local limit theorem, a precise bound on the maximum of symmetric random walks with varying step sizes, and tailored estimates for critical random graphs. In addition, we derive an improved comparison inequality between the mixing time of the Chayes-Machta dynamics and that of the local Glauber dynamics on general graphs; this results in better mixing time bounds for the local dynamics in the mean-field setting.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.47/LIPIcs.APPROX-RANDOM.2021.47.pdf
Markov Chains
Mixing Times
Random-cluster Model
Ising and Potts Models
Mean-field
Chayes-Machta Dynamics
Random Graphs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
48:1
48:17
10.4230/LIPIcs.APPROX/RANDOM.2021.48
article
On the Robust Communication Complexity of Bipartite Matching
Assadi, Sepehr
1
Behnezhad, Soheil
2
Rutgers University, Piscataway, NJ, USA
University of Maryland, College Park, MD, USA
We study the robust - à la Chakrabarti, Cormode, and McGregor [STOC'08] - communication complexity of the maximum bipartite matching problem. The edges of an adversarially chosen n-vertex bipartite graph G are partitioned randomly between Alice and Bob. Alice has to send a single message to Bob, using which Bob has to output an approximate maximum matching of G. We are particularly interested in understanding the best approximation ratio possible by protocols that use a near-optimal message size of n ⋅ polylog(n).
The communication complexity of bipartite matching in this setting under an adversarial partitioning is well-understood. In their beautiful paper, Goel, Kapralov, and Khanna [SODA'12] gave a rac{2} {3}-approximate protocol with O(n) communication and showed that this approximation is tight unless we allow more than a near-linear communication. The complexity of the robust version, i.e., with a random partitioning of the edges, however remains wide open. The best known protocol, implied by a very recent random-order streaming algorithm of the authors [ICALP'21], uses O(n log n) communication to obtain a (rac{2} {3} + ε₀)-approximation for a constant ε₀ ∼ 10^{-14}. The best known lower bound, on the other hand, leaves open the possibility of all the way up to even a (1-ε)-approximation using near-linear communication for constant ε > 0.
In this work, we give a new protocol with a significantly better approximation. Particularly, our protocol achieves a 0.716 expected approximation using O(n) communication. This protocol is based on a new notion of distribution-dependent sparsifiers which give a natural way of sparsifying graphs sampled from a known distribution. We then show how to lift the assumption on knowing the graph’s distribution via minimax theorems. We believe this is a particularly powerful method of designing communication protocols and might find further applications.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.48/LIPIcs.APPROX-RANDOM.2021.48.pdf
Maximum Matching
Communication Complexity
Random-Order Streaming
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
49:1
49:21
10.4230/LIPIcs.APPROX/RANDOM.2021.49
article
L1 Regression with Lewis Weights Subsampling
Parulekar, Aditya
1
Parulekar, Advait
2
Price, Eric
1
Department of Computer Science, University of Texas at Austin, TX, USA
Department of Electrical and Computer Engineering, University of Texas at Austin, TX, USA
We consider the problem of finding an approximate solution to 𝓁₁ regression while only observing a small number of labels. Given an n × d unlabeled data matrix X, we must choose a small set of m ≪ n rows to observe the labels of, then output an estimate β̂ whose error on the original problem is within a 1 + ε factor of optimal. We show that sampling from X according to its Lewis weights and outputting the empirical minimizer succeeds with probability 1-δ for m > O(1/(ε²) d log d/(ε δ)). This is analogous to the performance of sampling according to leverage scores for 𝓁₂ regression, but with exponentially better dependence on δ. We also give a corresponding lower bound of Ω(d/(ε²) + (d + 1/(ε²)) log 1/(δ)).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.49/LIPIcs.APPROX-RANDOM.2021.49.pdf
Active regression
Lewis weights
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
50:1
50:26
10.4230/LIPIcs.APPROX/RANDOM.2021.50
article
Hitting Sets for Orbits of Circuit Classes and Polynomial Families
Saha, Chandan
1
Thankey, Bhargav
1
Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India
The orbit of an n-variate polynomial f(𝐱) over a field 𝔽 is the set {f(A𝐱+𝐛) : A ∈ GL(n,𝔽) and 𝐛 ∈ 𝔽ⁿ}. In this paper, we initiate the study of explicit hitting sets for the orbits of polynomials computable by several natural and well-studied circuit classes and polynomial families. In particular, we give quasi-polynomial time hitting sets for the orbits of:
1) Low-individual-degree polynomials computable by commutative ROABPs. This implies quasi-polynomial time hitting sets for the orbits of the elementary symmetric polynomials.
2) Multilinear polynomials computable by constant-width ROABPs. This implies a quasi-polynomial time hitting set for the orbits of the family {IMM_{3,d}}_{d ∈ ℕ}, which is complete for arithmetic formulas.
3) Polynomials computable by constant-depth, constant-occur formulas. This implies quasi-polynomial time hitting sets for the orbits of multilinear depth-4 circuits with constant top fan-in, and also polynomial-time hitting sets for the orbits of the power symmetric and the sum-product polynomials.
4) Polynomials computable by occur-once formulas.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.50/LIPIcs.APPROX-RANDOM.2021.50.pdf
Hitting Sets
Orbits
ROABPs
Rank Concentration
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
51:1
51:15
10.4230/LIPIcs.APPROX/RANDOM.2021.51
article
Sampling Multiple Edges Efficiently
Eden, Talya
1
https://orcid.org/0000-0001-8470-9508
Mossel, Saleet
1
Rubinfeld, Ronitt
1
CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
We present a sublinear time algorithm that allows one to sample multiple edges from a distribution that is pointwise ε-close to the uniform distribution, in an amortized-efficient fashion. We consider the adjacency list query model, where access to a graph G is given via degree and neighbor queries.
The problem of sampling a single edge in this model has been raised by Eden and Rosenbaum (SOSA 18). Let n and m denote the number of vertices and edges of G, respectively. Eden and Rosenbaum provided upper and lower bounds of Θ^*(n/√ m) for sampling a single edge in general graphs (where O^*(⋅) suppresses poly(1/ε) and poly(log n) dependencies). We ask whether the query complexity lower bound for sampling a single edge can be circumvented when multiple samples are required. That is, can we get an improved amortized per-sample cost if we allow a preprocessing phase? We answer in the affirmative.
We present an algorithm that, if one knows the number of required samples q in advance, has an overall cost that is sublinear in q, namely, O^*(√ q ⋅(n/√ m)), which is strictly preferable to O^*(q⋅ (n/√ m)) cost resulting from q invocations of the algorithm by Eden and Rosenbaum.
Subsequent to a preliminary version of this work, Tětek and Thorup (arXiv, preprint) proved that this bound is essentially optimal.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.51/LIPIcs.APPROX-RANDOM.2021.51.pdf
Sampling edges
graph algorithm
sublinear algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
52:1
52:14
10.4230/LIPIcs.APPROX/RANDOM.2021.52
article
Lower Bounds for XOR of Forrelations
Girish, Uma
1
Raz, Ran
1
Zhan, Wei
1
Department of Computer Science, Princeton University, NJ, USA
The Forrelation problem, first introduced by Aaronson [Scott Aaronson, 2010] and Aaronson and Ambainis [Scott Aaronson and Andris Ambainis, 2015], is a well studied computational problem in the context of separating quantum and classical computational models. Variants of this problem were used to give tight separations between quantum and classical query complexity [Scott Aaronson and Andris Ambainis, 2015]; the first separation between poly-logarithmic quantum query complexity and bounded-depth circuits of super-polynomial size, a result that also implied an oracle separation of the classes BQP and PH [Ran Raz and Avishay Tal, 2019]; and improved separations between quantum and classical communication complexity [Uma Girish et al., 2021]. In all these separations, the lower bound for the classical model only holds when the advantage of the protocol (over a random guess) is more than ≈ 1/√N, that is, the success probability is larger than ≈ 1/2 + 1/√N. This is unavoidable as ≈ 1/√N is the correlation between two coordinates of an input that is sampled from the Forrelation distribution, and hence there are simple classical protocols that achieve advantage ≈ 1/√N, in all these models.
To achieve separations when the classical protocol has smaller advantage, we study in this work the xor of k independent copies of (a variant of) the Forrelation function (where k≪ N). We prove a very general result that shows that any family of Boolean functions that is closed under restrictions, whose Fourier mass at level 2k is bounded by α^k (that is, the sum of the absolute values of all Fourier coefficients at level 2k is bounded by α^k), cannot compute the xor of k independent copies of the Forrelation function with advantage better than O((α^k)/(N^{k/2})). This is a strengthening of a result of [Eshan Chattopadhyay et al., 2019], that gave a similar statement for k = 1, using the technique of [Ran Raz and Avishay Tal, 2019]. We give several applications of our result. In particular, we obtain the following separations:
Quantum versus Classical Communication Complexity. We give the first example of a partial Boolean function that can be computed by a simultaneous-message quantum protocol with communication complexity polylog(N) (where Alice and Bob also share polylog(N) EPR pairs), and such that, any classical randomized protocol of communication complexity at most õ(N^{1/4}), with any number of rounds, has quasipolynomially small advantage over a random guess. Previously, only separations where the classical protocol has polynomially small advantage were known between these models [Dmitry Gavinsky, 2016; Uma Girish et al., 2021].
Quantum Query Complexity versus Bounded Depth Circuits. We give the first example of a partial Boolean function that has a quantum query algorithm with query complexity polylog(N), and such that, any constant-depth circuit of quasipolynomial size has quasipolynomially small advantage over a random guess. Previously, only separations where the constant-depth circuit has polynomially small advantage were known [Ran Raz and Avishay Tal, 2019].
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.52/LIPIcs.APPROX-RANDOM.2021.52.pdf
Forrelation
Quasipolynomial
Separation
Quantum versus Classical
Xor
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
53:1
53:20
10.4230/LIPIcs.APPROX/RANDOM.2021.53
article
Fourier Growth of Structured 𝔽₂-Polynomials and Applications
Błasiok, Jarosław
1
Ivanov, Peter
2
Jin, Yaonan
1
Lee, Chin Ho
1
Servedio, Rocco A.
1
Viola, Emanuele
2
Columbia University, New York, NY, USA
Northeastern University, Boston, MA, USA
We analyze the Fourier growth, i.e. the L₁ Fourier weight at level k (denoted L_{1,k}), of various well-studied classes of "structured" m F₂-polynomials. This study is motivated by applications in pseudorandomness, in particular recent results and conjectures due to [Chattopadhyay et al., 2019; Chattopadhyay et al., 2019; Eshan Chattopadhyay et al., 2020] which show that upper bounds on Fourier growth (even at level k = 2) give unconditional pseudorandom generators.
Our main structural results on Fourier growth are as follows:
- We show that any symmetric degree-d m F₂-polynomial p has L_{1,k}(p) ≤ Pr [p = 1] ⋅ O(d)^k. This quadratically strengthens an earlier bound that was implicit in [Omer Reingold et al., 2013].
- We show that any read-Δ degree-d m F₂-polynomial p has L_{1,k}(p) ≤ Pr [p = 1] ⋅ (k Δ d)^{O(k)}.
- We establish a composition theorem which gives L_{1,k} bounds on disjoint compositions of functions that are closed under restrictions and admit L_{1,k} bounds.
Finally, we apply the above structural results to obtain new unconditional pseudorandom generators and new correlation bounds for various classes of m F₂-polynomials.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.53/LIPIcs.APPROX-RANDOM.2021.53.pdf
Fourier analysis
Pseudorandomness
Fourier growth
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
54:1
54:22
10.4230/LIPIcs.APPROX/RANDOM.2021.54
article
Candidate Tree Codes via Pascal Determinant Cubes
Ben Yaacov, Inbar
1
Cohen, Gil
1
Narayanan, Anand Kumar
2
The Blavatnik School of Computer Science, Tel-Aviv University, Israel
CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Tree codes are combinatorial structures introduced by Schulman [Schulman, 1993] as key ingredients in interactive coding schemes. Asymptotically-good tree codes are long known to exist, yet their explicit construction remains a notoriously hard open problem. Even proposing a plausible construction, without the burden of proof, is difficult and the defining tree code property requires structure that remains elusive. To the best of our knowledge, only one candidate appears in the literature, due to Moore and Schulman [Moore and Schulman, 2014].
We put forth a new candidate for an explicit asymptotically-good tree code. Our construction is an extension of the vanishing rate tree code by Cohen-Haeupler-Schulman [Cohen et al., 2018], and its correctness relies on a conjecture that we introduce on certain Pascal determinants indexed by the points of the Boolean hypercube. Furthermore, using the vanishing distance tree code by Gelles et al. [Gelles et al., 2016] enables us to present a construction that relies on an even weaker assumption. We furnish evidence supporting our conjecture through numerical computation, combinatorial arguments from planar path graphs and based on well-studied heuristics from arithmetic geometry.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.54/LIPIcs.APPROX-RANDOM.2021.54.pdf
Tree codes
Sparse polynomials
Explicit constructions
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
55:1
55:19
10.4230/LIPIcs.APPROX/RANDOM.2021.55
article
Towards a Decomposition-Optimal Algorithm for Counting and Sampling Arbitrary Motifs in Sublinear Time
Biswas, Amartya Shankha
1
Eden, Talya
1
2
https://orcid.org/0000-0001-8470-9508
Rubinfeld, Ronitt
1
CSAIL, Massachusetts Institute of Technology, Cambridge MA, USA
sites.google.com/view/edentalya/home
We consider the problem of sampling and approximately counting an arbitrary given motif H in a graph G, where access to G is given via queries: degree, neighbor, and pair, as well as uniform edge sample queries. Previous algorithms for these tasks were based on a decomposition of H into a collection of odd cycles and stars, denoted D^*(H) = {O_{k₁},...,O_{k_q}, S_{p₁},...,S_{p_𝓁}}. These algorithms were shown to be optimal for the case where H is a clique or an odd-length cycle, but no other lower bounds were known.
We present a new algorithm for sampling arbitrary motifs which, up to poly(log n) factors, is always at least as good, and for most graphs G is strictly better. The main ingredient leading to this improvement is an improved uniform algorithm for sampling stars, which might be of independent interest, as it allows to sample vertices according to the p-th moment of the degree distribution.
Finally, we prove that this algorithm is decomposition-optimal for decompositions that contain at least one odd cycle. These are the first lower bounds for motifs H with a nontrivial decomposition, i.e., motifs that have more than a single component in their decomposition.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.55/LIPIcs.APPROX-RANDOM.2021.55.pdf
Sublinear time algorithms
Graph algorithms
Sampling subgraphs
Approximate counting
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
56:1
56:21
10.4230/LIPIcs.APPROX/RANDOM.2021.56
article
Ideal-Theoretic Explanation of Capacity-Achieving Decoding
Bhandari, Siddharth
1
Harsha, Prahladh
1
https://orcid.org/0000-0002-2739-5642
Kumar, Mrinal
2
Sudan, Madhu
3
Tata Institute of Fundamental Research, Mumbai, India
Department of Computer Science and Engineering, IIT Bombay, India
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA
In this work, we present an abstract framework for some algebraic error-correcting codes with the aim of capturing codes that are list-decodable to capacity, along with their decoding algorithm. In the polynomial ideal framework, a code is specified by some ideals in a polynomial ring, messages are polynomials and their encoding is the residue modulo the ideals. We present an alternate way of viewing this class of codes in terms of linear operators, and show that this alternate view makes their algorithmic list-decodability amenable to analysis.
Our framework leads to a new class of codes that we call affine Folded Reed-Solomon codes (which are themselves a special case of the broader class we explore). These codes are common generalizations of the well-studied Folded Reed-Solomon codes and Univariate Multiplicity codes, while also capturing the less-studied Additive Folded Reed-Solomon codes as well as a large family of codes that were not previously known/studied.
More significantly our framework also captures the algorithmic list-decodability of the constituent codes. Specifically, we present a unified view of the decoding algorithm for ideal-theoretic codes and show that the decodability reduces to the analysis of the distance of some related codes. We show that good bounds on this distance lead to capacity-achieving performance of the underlying code, providing a unifying explanation of known capacity-achieving results. In the specific case of affine Folded Reed-Solomon codes, our framework shows that they are list-decodable up to capacity (for appropriate setting of the parameters), thereby unifying the previous results for Folded Reed-Solomon, Multiplicity and Additive Folded Reed-Solomon codes.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.56/LIPIcs.APPROX-RANDOM.2021.56.pdf
List Decodability
List Decoding Capacity
Polynomial Ideal Codes
Multiplicity Codes
Folded Reed-Solomon Codes
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
57:1
57:18
10.4230/LIPIcs.APPROX/RANDOM.2021.57
article
Visible Rank and Codes with Locality
Alrabiah, Omar
1
Guruswami, Venkatesan
2
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, USA
Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
We propose a framework to study the effect of local recovery requirements of codeword symbols on the dimension of linear codes, based on a combinatorial proxy that we call visible rank. The locality constraints of a linear code are stipulated by a matrix H of ⋆’s and 0’s (which we call a "stencil"), whose rows correspond to the local parity checks (with the ⋆’s indicating the support of the check). The visible rank of H is the largest r for which there is a r × r submatrix in H with a unique generalized diagonal of ⋆’s. The visible rank yields a field-independent combinatorial lower bound on the rank of H and thus the co-dimension of the code.
We point out connections of the visible rank to other notions in the literature such as unique restricted graph matchings, matroids, spanoids, and min-rank. In particular, we prove a rank-nullity type theorem relating visible rank to the rank of an associated construct called symmetric spanoid, which was introduced by Dvir, Gopi, Gu, and Wigderson [Zeev Dvir et al., 2020]. Using this connection and a construction of appropriate stencils, we answer a question posed in [Zeev Dvir et al., 2020] and demonstrate that symmetric spanoid rank cannot improve the currently best known Õ(n^{(q-2)/(q-1)}) upper bound on the dimension of q-query locally correctable codes (LCCs) of length n. This also pins down the efficacy of visible rank as a proxy for the dimension of LCCs.
We also study the t-Disjoint Repair Group Property (t-DRGP) of codes where each codeword symbol must belong to t disjoint check equations. It is known that linear codes with 2-DRGP must have co-dimension Ω(√n) (which is matched by a simple product code construction). We show that there are stencils corresponding to 2-DRGP with visible rank as small as O(log n). However, we show the second tensor of any 2-DRGP stencil has visible rank Ω(n), thus recovering the Ω(√n) lower bound for 2-DRGP. For q-LCC, however, the k'th tensor power for k ⩽ n^{o(1)} is unable to improve the Õ(n^{(q-2)/(q-1)}) upper bound on the dimension of q-LCCs by a polynomial factor.Inspired by this and as a notion of intrinsic interest, we define the notion of visible capacity of a stencil as the limiting visible rank of high tensor powers, analogous to Shannon capacity, and pose the question whether there can be large gaps between visible capacity and algebraic rank.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.57/LIPIcs.APPROX-RANDOM.2021.57.pdf
Visible Rank
Stencils
Locality
DRGP Codes
Locally Correctable Codes
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
58:1
58:21
10.4230/LIPIcs.APPROX/RANDOM.2021.58
article
Pseudorandom Generators for Read-Once Monotone Branching Programs
Doron, Dean
1
Meka, Raghu
2
Reingold, Omer
1
Tal, Avishay
3
Vadhan, Salil
4
Department of Computer Science, Stanford University, CA, USA
Department of Computer Science, University of California at Los Angeles, CA, USA
Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, CA, USA
John A. Paulson School of Engineering & Applied Sciences, Harvard University, Cambridge, MA, USA
Motivated by the derandomization of space-bounded computation, there has been a long line of work on constructing pseudorandom generators (PRGs) against various forms of read-once branching programs (ROBPs), with a goal of improving the O(log² n) seed length of Nisan’s classic construction [Noam Nisan, 1992] to the optimal O(log n).
In this work, we construct an explicit PRG with seed length Õ(log n) for constant-width ROBPs that are monotone, meaning that the states at each time step can be ordered so that edges with the same labels never cross each other. Equivalently, for each fixed input, the transition functions are a monotone function of the state. This result is complementary to a line of work that gave PRGs with seed length O(log n) for (ordered) permutation ROBPs of constant width [Braverman et al., 2014; Koucký et al., 2011; De, 2011; Thomas Steinke, 2012], since the monotonicity constraint can be seen as the "opposite" of the permutation constraint.
Our PRG also works for monotone ROBPs that can read the input bits in any order, which are strictly more powerful than read-once AC⁰. Our PRG achieves better parameters (in terms of the dependence on the depth of the circuit) than the best previous pseudorandom generator for read-once AC⁰, due to Doron, Hatami, and Hoza [Doron et al., 2019].
Our pseudorandom generator construction follows Ajtai and Wigderson’s approach of iterated pseudorandom restrictions [Ajtai and Wigderson, 1989; Gopalan et al., 2012]. We give a randomness-efficient width-reduction process which proves that the branching program simplifies to an O(log n)-junta after only O(log log n) independent applications of the Forbes-Kelley pseudorandom restrictions [Michael A. Forbes and Zander Kelley, 2018].
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.58/LIPIcs.APPROX-RANDOM.2021.58.pdf
Branching programs
pseudorandom generators
constant depth circuits
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
59:1
59:17
10.4230/LIPIcs.APPROX/RANDOM.2021.59
article
On the Power of Choice for k-Colorability of Random Graphs
Dani, Varsha
1
2
Gupta, Diksha
3
Hayes, Thomas P.
4
5
Ronin Institute, Montclair, NJ, USA
Dept. of Computer Science, Rochester Institute of Technology, NY, USA
School of Computing, National University of Singapore, Singapore
Dept. of Computer Science, University of New Mexico, Albuquerque, NM, USA
http:cs.unm.edu/ hayest
In an r-choice Achlioptas process, random edges are generated r at a time, and an online strategy is used to select one of them for inclusion in a graph. We investigate the problem of whether such a selection strategy can shift the k-colorability transition; that is, the number of edges at which the graph goes from being k-colorable to non-k-colorable.
We show that, for k ≥ 9, two choices suffice to delay the k-colorability threshold, and that for every k ≥ 2, six choices suffice.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.59/LIPIcs.APPROX-RANDOM.2021.59.pdf
Random graphs
Achlioptas Processes
Phase Transition
Graph Colorability
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
60:1
60:19
10.4230/LIPIcs.APPROX/RANDOM.2021.60
article
Memory-Sample Lower Bounds for Learning Parity with Noise
Garg, Sumegha
1
Kothari, Pravesh K.
2
Liu, Pengda
3
Raz, Ran
4
Department of Computer Science, Harvard University, Cambridge, MA, USA
Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA
Department of Computer Science, Stanford University, CA, USA
Department of Computer Science, Princeton University, NJ, USA
In this work, we show, for the well-studied problem of learning parity under noise, where a learner tries to learn x = (x₁,…,x_n) ∈ {0,1}ⁿ from a stream of random linear equations over 𝔽₂ that are correct with probability 1/2+ε and flipped with probability 1/2-ε (0 < ε < 1/2), that any learning algorithm requires either a memory of size Ω(n²/ε) or an exponential number of samples.
In fact, we study memory-sample lower bounds for a large class of learning problems, as characterized by [Garg et al., 2018], when the samples are noisy. A matrix M: A × X → {-1,1} corresponds to the following learning problem with error parameter ε: an unknown element x ∈ X is chosen uniformly at random. A learner tries to learn x from a stream of samples, (a₁, b₁), (a₂, b₂) …, where for every i, a_i ∈ A is chosen uniformly at random and b_i = M(a_i,x) with probability 1/2+ε and b_i = -M(a_i,x) with probability 1/2-ε (0 < ε < 1/2). Assume that k,𝓁, r are such that any submatrix of M of at least 2^{-k} ⋅ |A| rows and at least 2^{-𝓁} ⋅ |X| columns, has a bias of at most 2^{-r}. We show that any learning algorithm for the learning problem corresponding to M, with error parameter ε, requires either a memory of size at least Ω((k⋅𝓁)/ε), or at least 2^{Ω(r)} samples. The result holds even if the learner has an exponentially small success probability (of 2^{-Ω(r)}). In particular, this shows that for a large class of learning problems, same as those in [Garg et al., 2018], any learning algorithm requires either a memory of size at least Ω(((log|X|)⋅(log|A|))/ε) or an exponential number of noisy samples.
Our proof is based on adapting the arguments in [Ran Raz, 2017; Garg et al., 2018] to the noisy case.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.60/LIPIcs.APPROX-RANDOM.2021.60.pdf
memory-sample tradeoffs
learning parity under noise
space lower bound
branching program
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
61:1
61:23
10.4230/LIPIcs.APPROX/RANDOM.2021.61
article
Testing Hamiltonicity (And Other Problems) in Minor-Free Graphs
Levi, Reut
1
https://orcid.org/0000-0003-3167-1766
Shoshan, Nadav
1
Efi Arazi School of Computer Science, The Interdisciplinary Center Herzliya, Israel
In this paper we provide sub-linear algorithms for several fundamental problems in the setting in which the input graph excludes a fixed minor, i.e., is a minor-free graph. In particular, we provide the following algorithms for minor-free unbounded degree graphs.
1) A tester for Hamiltonicity with two-sided error with poly(1/ε)-query complexity, where ε is the proximity parameter.
2) A local algorithm, as defined by Rubinfeld et al. (ICS 2011), for constructing a spanning subgraph with almost minimum weight, specifically, at most a factor (1+ε) of the optimum, with poly(1/ε)-query complexity. Both our algorithms use partition oracles, a tool introduced by Hassidim et al. (FOCS 2009), which are oracles that provide access to a partition of the graph such that the number of cut-edges is small and each part of the partition is small. The polynomial dependence in 1/ε of our algorithms is achieved by combining the recent poly(d/ε)-query partition oracle of Kumar-Seshadhri-Stolman (ECCC 2021) for minor-free graphs with degree bounded by d.
For bounded degree minor-free graphs we introduce the notion of covering partition oracles which is a relaxed version of partition oracles and design a poly(d/ε)-time covering partition oracle for this family of graphs. Using our covering partition oracle we provide the same results as above (except that the tester for Hamiltonicity has one-sided error) for minor-free bounded degree graphs, as well as showing that any property which is monotone and additive (e.g. bipartiteness) can be tested in minor-free graphs by making poly(d/ε)-queries.
The benefit of using the covering partition oracle rather than the partition oracle in our algorithms is its simplicity and an improved polynomial dependence in 1/ε in the obtained query complexity.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.61/LIPIcs.APPROX-RANDOM.2021.61.pdf
Property Testing
Hamiltonian path
minor free graphs
sparse spanning sub-graphs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
207
62:1
62:19
10.4230/LIPIcs.APPROX/RANDOM.2021.62
article
Parallel Repetition for the GHZ Game: A Simpler Proof
Girish, Uma
1
Holmgren, Justin
2
Mittal, Kunal
1
Raz, Ran
1
Zhan, Wei
1
Department of Computer Science, Princeton University, NJ, USA
NTT Research, Sunnyvale, CA, USA
We give a new proof of the fact that the parallel repetition of the (3-player) GHZ game reduces the value of the game to zero polynomially quickly. That is, we show that the value of the n-fold GHZ game is at most n^{-Ω(1)}. This was first established by Holmgren and Raz [Holmgren and Raz, 2020]. We present a new proof of this theorem that we believe to be simpler and more direct. Unlike most previous works on parallel repetition, our proof makes no use of information theory, and relies on the use of Fourier analysis.
The GHZ game [Greenberger et al., 1989] has played a foundational role in the understanding of quantum information theory, due in part to the fact that quantum strategies can win the GHZ game with probability 1. It is possible that improved parallel repetition bounds may find applications in this setting.
Recently, Dinur, Harsha, Venkat, and Yuen [Dinur et al., 2017] highlighted the GHZ game as a simple three-player game, which is in some sense maximally far from the class of multi-player games whose behavior under parallel repetition is well understood. Dinur et al. conjectured that parallel repetition decreases the value of the GHZ game exponentially quickly, and speculated that progress on proving this would shed light on parallel repetition for general multi-player (multi-prover) games.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.62/LIPIcs.APPROX-RANDOM.2021.62.pdf
Parallel Repetition
GHZ
Polynomial
Multi-player