Volume
LIPIcs, Volume 116
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)
Publication Details
- published at: 2018-08-13
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-085-9
- DBLP: db/conf/approx/approx2018
Access Numbers
- Detailed Access Statistics available here
-
Total Document Accesses (updated on a weekly basis):
0PDF Downloads
Documents
No documents found matching your filter selection.
Document
Complete Volume
LIPIcs, Volume 116, APPROX/RANDOM'18, Complete Volume
Abstract
LIPIcs, Volume 116, APPROX/RANDOM'18, Complete Volume
Cite as
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@Proceedings{blais_et_al:LIPIcs.APPROX-RANDOM.2018,
title = {{LIPIcs, Volume 116, APPROX/RANDOM'18, Complete Volume}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018},
URN = {urn:nbn:de:0030-drops-97254},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018},
annote = {Keywords: Mathematics of computing, Theory of computation}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{blais_et_al:LIPIcs.APPROX-RANDOM.2018.0,
author = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {0:i--0:xvi},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.0},
URN = {urn:nbn:de:0030-drops-94043},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Polylogarithmic Approximation Algorithms for Weighted-F-Deletion Problems
Abstract
Let F be a family of graphs. A canonical vertex deletion problem corresponding to F is defined as follows: given an n-vertex undirected graph G and a weight function w: V(G) - >R^+, find a minimum weight subset S subseteq V(G) such that G-S belongs to F. This is known as Weighted F Vertex Deletion problem. In this paper we devise a recursive scheme to obtain O(log^{O(1)} n)-approximation algorithms for such problems, building upon the classical technique of finding balanced separators in a graph. Roughly speaking, our scheme applies to those problems, where an optimum solution S together with a well-structured set X, form a balanced separator of the input graph. In this paper, we obtain the first O(log^{O(1)} n)-approximation algorithms for the following vertex deletion problems.
- Let {F} be a finite set of graphs containing a planar graph, and F=G(F) be the family of graphs such that every graph H in G(F) excludes all graphs in F as minors. The vertex deletion problem corresponding to F=G(F) is the Weighted Planar F-Minor-Free Deletion (WPF-MFD) problem. We give randomized and deterministic approximation algorithms for WPF-MFD with ratios O(log^{1.5} n) and O(log^2 n), respectively. Previously, only a randomized constant factor approximation algorithm for the unweighted version of the problem was known [FOCS 2012].
- We give an O(log^2 n)-factor approximation algorithm for Weighted Chordal Vertex Deletion (WCVD), the vertex deletion problem to the family of chordal graphs. On the way to this algorithm, we also obtain a constant factor approximation algorithm for Multicut on chordal graphs.
- We give an O(log^3 n)-factor approximation algorithm for Weighted Distance Hereditary Vertex Deletion (WDHVD), also known as Weighted Rankwidth-1 Vertex Deletion (WR-1VD). This is the vertex deletion problem to the family of distance hereditary graphs, or equivalently, the family of graphs of rankwidth one.
We believe that our recursive scheme can be applied to obtain O(log^{O(1)} n)-approximation algorithms for many other problems as well.
Cite as
Akanksha Agrawal, Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, and Meirav Zehavi. Polylogarithmic Approximation Algorithms for Weighted-F-Deletion Problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 1:1-1:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{agrawal_et_al:LIPIcs.APPROX-RANDOM.2018.1,
author = {Agrawal, Akanksha and Lokshtanov, Daniel and Misra, Pranabendu and Saurabh, Saket and Zehavi, Meirav},
title = {{Polylogarithmic Approximation Algorithms for Weighted-F-Deletion Problems}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {1:1--1:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.1},
URN = {urn:nbn:de:0030-drops-94058},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.1},
annote = {Keywords: Approximation Algorithms, Planar- F-Deletion, Separator}
}
Document
Improved Approximation Bounds for the Minimum Constraint Removal Problem
Abstract
In the minimum constraint removal problem, we are given a set of geometric objects as obstacles in the plane, and we want to find the minimum number of obstacles that must be removed to reach a target point t from the source point s by an obstacle-free path. The problem is known to be intractable, and (perhaps surprisingly) no sub-linear approximations are known even for simple obstacles such as rectangles and disks. The main result of our paper is a new approximation technique that gives O(sqrt{n})-approximation for rectangles, disks as well as rectilinear polygons. The technique also gives O(sqrt{n})-approximation for the minimum color path problem in graphs. We also present some inapproximability results for the geometric constraint removal problem.
Cite as
Sayan Bandyapadhyay, Neeraj Kumar, Subhash Suri, and Kasturi Varadarajan. Improved Approximation Bounds for the Minimum Constraint Removal Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 2:1-2:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{bandyapadhyay_et_al:LIPIcs.APPROX-RANDOM.2018.2,
author = {Bandyapadhyay, Sayan and Kumar, Neeraj and Suri, Subhash and Varadarajan, Kasturi},
title = {{Improved Approximation Bounds for the Minimum Constraint Removal Problem}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {2:1--2:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.2},
URN = {urn:nbn:de:0030-drops-94066},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.2},
annote = {Keywords: Minimum Constraint Removal, Minimum Color Path, Barrier Resilience, Obstacle Removal, Obstacle Free Path, Approximation}
}
Document
A Tight 4/3 Approximation for Capacitated Vehicle Routing in Trees
Abstract
Given a set of clients with demands, the Capacitated Vehicle Routing problem is to find a set of tours that collectively cover all client demand, such that the capacity of each vehicle is not exceeded and such that the sum of the tour lengths is minimized. In this paper, we provide a 4/3-approximation algorithm for Capacitated Vehicle Routing on trees, improving over the previous best-known approximation ratio of (sqrt{41}-1)/4 by Asano et al.[Asano et al., 2001], while using the same lower bound. Asano et al. show that there exist instances whose optimal cost is 4/3 times this lower bound. Notably, our 4/3 approximation ratio is therefore tight for this lower bound, achieving the best-possible performance.
Cite as
Amariah Becker. A Tight 4/3 Approximation for Capacitated Vehicle Routing in Trees. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{becker:LIPIcs.APPROX-RANDOM.2018.3,
author = {Becker, Amariah},
title = {{A Tight 4/3 Approximation for Capacitated Vehicle Routing in Trees}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {3:1--3:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.3},
URN = {urn:nbn:de:0030-drops-94075},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.3},
annote = {Keywords: Approximation algorithms, Graph algorithms, Capacitated vehicle routing}
}
Document
Low Rank Approximation in the Presence of Outliers
Abstract
We consider the problem of principal component analysis (PCA) in the presence of outliers. Given a matrix A (d x n) and parameters k, m, the goal is to remove a set of at most m columns of A (outliers), so as to minimize the rank-k approximation error of the remaining matrix (inliers). While much of the work on this problem has focused on recovery of the rank-k subspace under assumptions on the inliers and outliers, we focus on the approximation problem. Our main result shows that sampling-based methods developed in the outlier-free case give non-trivial guarantees even in the presence of outliers. Using this insight, we develop a simple algorithm that has bi-criteria guarantees. Further, unlike similar formulations for clustering, we show that bi-criteria guarantees are unavoidable for the problem, under appropriate complexity assumptions.
Cite as
Aditya Bhaskara and Srivatsan Kumar. Low Rank Approximation in the Presence of Outliers. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 4:1-4:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{bhaskara_et_al:LIPIcs.APPROX-RANDOM.2018.4,
author = {Bhaskara, Aditya and Kumar, Srivatsan},
title = {{Low Rank Approximation in the Presence of Outliers}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {4:1--4:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.4},
URN = {urn:nbn:de:0030-drops-94087},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.4},
annote = {Keywords: Low rank approximation, PCA, Robustness to outliers}
}
Document
Greedy Bipartite Matching in Random Type Poisson Arrival Model
Abstract
We introduce a new random input model for bipartite matching which we call the Random Type Poisson Arrival Model. Just like in the known i.i.d. model (introduced by Feldman et al. [Feldman et al., 2009]), online nodes have types in our model. In contrast to the adversarial types studied in the known i.i.d. model, following the random graphs studied in Mastin and Jaillet [A. Mastin, 2013], in our model each type graph is generated randomly by including each offline node in the neighborhood of an online node with probability c/n independently. In our model, nodes of the same type appear consecutively in the input and the number of times each type node appears is distributed according to the Poisson distribution with parameter 1. We analyze the performance of the simple greedy algorithm under this input model. The performance is controlled by the parameter c and we are able to exactly characterize the competitive ratio for the regimes c = o(1) and c = omega(1). We also provide a precise bound on the expected size of the matching in the remaining regime of constant c. We compare our results to the previous work of Mastin and Jaillet who analyzed the simple greedy algorithm in the G_{n,n,p} model where each online node type occurs exactly once. We essentially show that the approach of Mastin and Jaillet can be extended to work for the Random Type Poisson Arrival Model, although several nontrivial technical challenges need to be overcome. Intuitively, one can view the Random Type Poisson Arrival Model as the G_{n,n,p} model with less randomness; that is, instead of each online node having a new type, each online node has a chance of repeating the previous type.
Cite as
Allan Borodin, Christodoulos Karavasilis, and Denis Pankratov. Greedy Bipartite Matching in Random Type Poisson Arrival Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 5:1-5:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{borodin_et_al:LIPIcs.APPROX-RANDOM.2018.5,
author = {Borodin, Allan and Karavasilis, Christodoulos and Pankratov, Denis},
title = {{Greedy Bipartite Matching in Random Type Poisson Arrival Model}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {5:1--5:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.5},
URN = {urn:nbn:de:0030-drops-94098},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.5},
annote = {Keywords: bipartite matching, stochastic input models, online algorithms, greedy algorithms}
}
Document
Semi-Direct Sum Theorem and Nearest Neighbor under l_infty
Abstract
We introduce semi-direct sum theorem as a framework for proving asymmetric communication lower bounds for the functions of the form V_{i=1}^n f(x,y_i). Utilizing tools developed in proving direct sum theorem for information complexity, we show that if the function is of the form V_{i=1}^n f(x,y_i) where Alice is given x and Bob is given y_i's, it suffices to prove a lower bound for a single f(x,y_i). This opens a new avenue of attack other than the conventional combinatorial technique (i.e. "richness lemma" from [Miltersen et al., 1995]) for proving randomized lower bounds for asymmetric communication for functions of such form.
As the main technical result and an application of semi-direct sum framework, we prove an information lower bound on c-approximate Nearest Neighbor (ANN) under l_infty which implies that the algorithm of [Indyk, 2001] for c-approximate Nearest Neighbor under l_infty is optimal even under randomization for both decision tree and cell probe data structure model (under certain parameter assumption for the latter). In particular, this shows that randomization cannot improve [Indyk, 2001] under decision tree model. Previously only a deterministic lower bound was known by [Andoni et al., 2008] and randomized lower bound for cell probe model by [Kapralov and Panigrahy, 2012]. We suspect further applications of our framework in exhibiting randomized asymmetric communication lower bounds for big data applications.
Cite as
Mark Braverman and Young Kun Ko. Semi-Direct Sum Theorem and Nearest Neighbor under l_infty. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{braverman_et_al:LIPIcs.APPROX-RANDOM.2018.6,
author = {Braverman, Mark and Ko, Young Kun},
title = {{Semi-Direct Sum Theorem and Nearest Neighbor under l\underlineinfty}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {6:1--6:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.6},
URN = {urn:nbn:de:0030-drops-94101},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.6},
annote = {Keywords: Asymmetric Communication Lower Bound, Data Structure Lower Bound, Nearest Neighbor Search}
}
Document
Nearly Optimal Distinct Elements and Heavy Hitters on Sliding Windows
Abstract
We study the distinct elements and l_p-heavy hitters problems in the sliding window model, where only the most recent n elements in the data stream form the underlying set. We first introduce the composable histogram, a simple twist on the exponential (Datar et al., SODA 2002) and smooth histograms (Braverman and Ostrovsky, FOCS 2007) that may be of independent interest. We then show that the composable histogram{} along with a careful combination of existing techniques to track either the identity or frequency of a few specific items suffices to obtain algorithms for both distinct elements and l_p-heavy hitters that are nearly optimal in both n and epsilon.
Applying our new composable histogram framework, we provide an algorithm that outputs a (1+epsilon)-approximation to the number of distinct elements in the sliding window model and uses O{1/(epsilon^2) log n log (1/epsilon)log log n+ (1/epsilon) log^2 n} bits of space. For l_p-heavy hitters, we provide an algorithm using space O{(1/epsilon^p) log^2 n (log^2 log n+log 1/epsilon)} for 0<p <=2, improving upon the best-known algorithm for l_2-heavy hitters (Braverman et al., COCOON 2014), which has space complexity O{1/epsilon^4 log^3 n}. We also show complementing nearly optimal lower bounds of Omega ((1/epsilon) log^2 n+(1/epsilon^2) log n) for distinct elements and Omega ((1/epsilon^p) log^2 n) for l_p-heavy hitters, both tight up to O{log log n} and O{log 1/epsilon} factors.
Cite as
Vladimir Braverman, Elena Grigorescu, Harry Lang, David P. Woodruff, and Samson Zhou. Nearly Optimal Distinct Elements and Heavy Hitters on Sliding Windows. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 7:1-7:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{braverman_et_al:LIPIcs.APPROX-RANDOM.2018.7,
author = {Braverman, Vladimir and Grigorescu, Elena and Lang, Harry and Woodruff, David P. and Zhou, Samson},
title = {{Nearly Optimal Distinct Elements and Heavy Hitters on Sliding Windows}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {7:1--7:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.7},
URN = {urn:nbn:de:0030-drops-94118},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.7},
annote = {Keywords: Streaming algorithms, sliding windows, heavy hitters, distinct elements}
}
Document
Survivable Network Design for Group Connectivity in Low-Treewidth Graphs
Abstract
In the Group Steiner Tree problem (GST), we are given a (edge or vertex)-weighted graph G=(V,E) on n vertices, together with a root vertex r and a collection of groups {S_i}_{i in [h]}: S_i subseteq V(G). The goal is to find a minimum-cost subgraph H that connects the root to every group. We consider a fault-tolerant variant of GST, which we call Restricted (Rooted) Group SNDP. In this setting, each group S_i has a demand k_i in [k], k in N, and we wish to find a minimum-cost subgraph H subseteq G such that, for each group S_i, there is a vertex in the group that is connected to the root via k_i (vertex or edge) disjoint paths.
While GST admits O(log^2 n log h) approximation, its higher connectivity variants are known to be Label-Cover hard, and for the vertex-weighted version, the hardness holds even when k=2 (it is widely believed that there is no subpolynomial approximation for the Label-Cover problem [Bellare et al., STOC 1993]). More precisely, the problem admits no 2^{log^{1-epsilon}n}-approximation unless NP subseteq DTIME(n^{polylog(n)}). Previously, positive results were known only for the edge-weighted version when k=2 [Gupta et al., SODA 2010; Khandekar et al., Theor. Comput. Sci., 2012] and for a relaxed variant where k_i disjoint paths from r may end at different vertices in a group [Chalermsook et al., SODA 2015], for which the authors gave a bicriteria approximation. For k >= 3, there is no non-trivial approximation algorithm known for edge-weighted Restricted Group SNDP, except for the special case of the relaxed variant on trees (folklore).
Our main result is an O(log n log h) approximation algorithm for Restricted Group SNDP that runs in time n^{f(k, w)}, where w is the treewidth of the input graph. Our algorithm works for both edge and vertex weighted variants, and the approximation ratio nearly matches the lower bound when k and w are constants. The key to achieving this result is a non-trivial extension of a framework introduced in [Chalermsook et al., SODA 2017]. This framework first embeds all feasible solutions to the problem into a dynamic program (DP) table. However, finding the optimal solution in the DP table remains intractable. We formulate a linear program relaxation for the DP and obtain an approximate solution via randomized rounding. This framework also allows us to systematically construct DP tables for high-connectivity problems. As a result, we present new exact algorithms for several variants of survivable network design problems in low-treewidth graphs.
Cite as
Parinya Chalermsook, Syamantak Das, Guy Even, Bundit Laekhanukit, and Daniel Vaz. Survivable Network Design for Group Connectivity in Low-Treewidth Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 8:1-8:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{chalermsook_et_al:LIPIcs.APPROX-RANDOM.2018.8,
author = {Chalermsook, Parinya and Das, Syamantak and Even, Guy and Laekhanukit, Bundit and Vaz, Daniel},
title = {{Survivable Network Design for Group Connectivity in Low-Treewidth Graphs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {8:1--8:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.8},
URN = {urn:nbn:de:0030-drops-94127},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.8},
annote = {Keywords: Approximation Algorithms, Hardness of Approximation, Survivable Network Design, Group Steiner Tree}
}
Document
Perturbation Resilient Clustering for k-Center and Related Problems via LP Relaxations
Abstract
We consider clustering in the perturbation resilience model that has been studied since the work of Bilu and Linial [Yonatan Bilu and Nathan Linial, 2010] and Awasthi, Blum and Sheffet [Awasthi et al., 2012]. A clustering instance I is said to be alpha-perturbation resilient if the optimal solution does not change when the pairwise distances are modified by a factor of alpha and the perturbed distances satisfy the metric property - this is the metric perturbation resilience property introduced in [Angelidakis et al., 2017] and a weaker requirement than prior models. We make two high-level contributions.
- We show that the natural LP relaxation of k-center and asymmetric k-center is integral for 2-perturbation resilient instances. We belive that demonstrating the goodness of standard LP relaxations complements existing results [Maria{-}Florina Balcan et al., 2016; Angelidakis et al., 2017] that are based on new algorithms designed for the perturbation model.
- We define a simple new model of perturbation resilience for clustering with outliers. Using this model we show that the unified MST and dynamic programming based algorithm proposed in [Angelidakis et al., 2017] exactly solves the clustering with outliers problem for several common center based objectives (like k-center, k-means, k-median) when the instances is 2-perturbation resilient. We further show that a natural LP relxation is integral for 2-perturbation resilient instances of k-center with outliers.
Cite as
Chandra Chekuri and Shalmoli Gupta. Perturbation Resilient Clustering for k-Center and Related Problems via LP Relaxations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{chekuri_et_al:LIPIcs.APPROX-RANDOM.2018.9,
author = {Chekuri, Chandra and Gupta, Shalmoli},
title = {{Perturbation Resilient Clustering for k-Center and Related Problems via LP Relaxations}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {9:1--9:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.9},
URN = {urn:nbn:de:0030-drops-94136},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.9},
annote = {Keywords: Clustering, Perturbation Resilience, LP Integrality, Outliers, Beyond Worst Case Analysis}
}
Document
Sherali-Adams Integrality Gaps Matching the Log-Density Threshold
Abstract
The log-density method is a powerful algorithmic framework which in recent years has given rise to the best-known approximations for a variety of problems, including Densest-k-Subgraph and Small Set Bipartite Vertex Expansion. These approximations have been conjectured to be optimal based on various instantiations of a general conjecture: that it is hard to distinguish a fully random combinatorial structure from one which contains a similar planted sub-structure with the same "log-density".
We bolster this conjecture by showing that in a random hypergraph with edge probability n^{-alpha}, Omega(log n) rounds of Sherali-Adams cannot rule out the existence of a k-subhypergraph with edge density k^{-alpha-o(1)}, for any k and alpha. This holds even when the bound on the objective function is lifted. This gives strong integrality gaps which exactly match the gap in the above distinguishing problems, as well as the best-known approximations, for Densest k-Subgraph, Smallest p-Edge Subgraph, their hypergraph extensions, and Small Set Bipartite Vertex Expansion (or equivalently, Minimum p-Union). Previously, such integrality gaps were known only for Densest k-Subgraph for one specific parameter setting.
Cite as
Eden Chlamtác and Pasin Manurangsi. Sherali-Adams Integrality Gaps Matching the Log-Density Threshold. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 10:1-10:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{chlamtac_et_al:LIPIcs.APPROX-RANDOM.2018.10,
author = {Chlamt\'{a}c, Eden and Manurangsi, Pasin},
title = {{Sherali-Adams Integrality Gaps Matching the Log-Density Threshold}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {10:1--10:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.10},
URN = {urn:nbn:de:0030-drops-94142},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.10},
annote = {Keywords: Approximation algorithms, integrality gaps, lift-and-project, log-density, Densest k-Subgraph}
}
Document
Lower Bounds for Approximating Graph Parameters via Communication Complexity
Abstract
In a celebrated work, Blais, Brody, and Matulef [Blais et al., 2012] developed a technique for proving property testing lower bounds via reductions from communication complexity. Their work focused on testing properties of functions, and yielded new lower bounds as well as simplified analyses of known lower bounds. Here, we take a further step in generalizing the methodology of [Blais et al., 2012] to analyze the query complexity of graph parameter estimation problems. In particular, our technique decouples the lower bound arguments from the representation of the graph, allowing it to work with any query type.
We illustrate our technique by providing new simpler proofs of previously known tight lower bounds for the query complexity of several graph problems: estimating the number of edges in a graph, sampling edges from an almost-uniform distribution, estimating the number of triangles (and more generally, r-cliques) in a graph, and estimating the moments of the degree distribution of a graph. We also prove new lower bounds for estimating the edge connectivity of a graph and estimating the number of instances of any fixed subgraph in a graph. We show that the lower bounds for estimating the number of triangles and edge connectivity also hold in a strictly stronger computational model that allows access to uniformly random edge samples.
Cite as
Talya Eden and Will Rosenbaum. Lower Bounds for Approximating Graph Parameters via Communication Complexity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 11:1-11:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{eden_et_al:LIPIcs.APPROX-RANDOM.2018.11,
author = {Eden, Talya and Rosenbaum, Will},
title = {{Lower Bounds for Approximating Graph Parameters via Communication Complexity}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {11:1--11:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.11},
URN = {urn:nbn:de:0030-drops-94156},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.11},
annote = {Keywords: sublinear graph parameter estimation, lower bounds, communication complexity}
}
Document
Communication Complexity of Correlated Equilibrium with Small Support
Abstract
We define a two-player N x N game called the 2-cycle game, that has a unique pure Nash equilibrium which is also the only correlated equilibrium of the game. In this game, every 1/poly(N)-approximate correlated equilibrium is concentrated on the pure Nash equilibrium. We show that the randomized communication complexity of finding any 1/poly(N)-approximate correlated equilibrium of the game is Omega(N). For small approximation values, our lower bound answers an open question of Babichenko and Rubinstein (STOC 2017).
Cite as
Anat Ganor and Karthik C. S.. Communication Complexity of Correlated Equilibrium with Small Support. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{ganor_et_al:LIPIcs.APPROX-RANDOM.2018.12,
author = {Ganor, Anat and C. S., Karthik},
title = {{Communication Complexity of Correlated Equilibrium with Small Support}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {12:1--12:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.12},
URN = {urn:nbn:de:0030-drops-94163},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.12},
annote = {Keywords: Correlated equilibrium, Nash equilibrium, Communication complexity}
}
Document
On Minrank and the Lovász Theta Function
Abstract
Two classical upper bounds on the Shannon capacity of graphs are the theta-function due to Lovász and the minrank parameter due to Haemers. We provide several explicit constructions of n-vertex graphs with a constant theta-function and minrank at least n^delta for a constant delta>0 (over various prime order fields). This implies a limitation on the theta-function-based algorithmic approach to approximating the minrank parameter of graphs. The proofs involve linear spaces of multivariate polynomials and the method of higher incidence matrices.
Cite as
Ishay Haviv. On Minrank and the Lovász Theta Function. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{haviv:LIPIcs.APPROX-RANDOM.2018.13,
author = {Haviv, Ishay},
title = {{On Minrank and the Lov\'{a}sz Theta Function}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {13:1--13:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.13},
URN = {urn:nbn:de:0030-drops-94170},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.13},
annote = {Keywords: Minrank, Theta Function, Shannon capacity, Multivariate polynomials, Higher incidence matrices}
}
Document
Online Makespan Minimization: The Power of Restart
Abstract
We consider the online makespan minimization problem on identical machines. Chen and Vestjens (ORL 1997) show that the largest processing time first (LPT) algorithm is 1.5-competitive. For the special case of two machines, Noga and Seiden (TCS 2001) introduce the SLEEPY algorithm that achieves a competitive ratio of (5 - sqrt{5})/2 ~~ 1.382, matching the lower bound by Chen and Vestjens (ORL 1997). Furthermore, Noga and Seiden note that in many applications one can kill a job and restart it later, and they leave an open problem whether algorithms with restart can obtain better competitive ratios.
We resolve this long-standing open problem on the positive end. Our algorithm has a natural rule for killing a processing job: a newly-arrived job replaces the smallest processing job if 1) the new job is larger than other pending jobs, 2) the new job is much larger than the processing one, and 3) the processed portion is small relative to the size of the new job. With appropriate choice of parameters, we show that our algorithm improves the 1.5 competitive ratio for the general case, and the 1.382 competitive ratio for the two-machine case.
Cite as
Zhiyi Huang, Ning Kang, Zhihao Gavin Tang, Xiaowei Wu, and Yuhao Zhang. Online Makespan Minimization: The Power of Restart. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 14:1-14:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{huang_et_al:LIPIcs.APPROX-RANDOM.2018.14,
author = {Huang, Zhiyi and Kang, Ning and Tang, Zhihao Gavin and Wu, Xiaowei and Zhang, Yuhao},
title = {{Online Makespan Minimization: The Power of Restart}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {14:1--14:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.14},
URN = {urn:nbn:de:0030-drops-94182},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.14},
annote = {Keywords: Online Scheduling, Makespan Minimization, Identical Machines}
}
Document
On Sketching the q to p Norms
Abstract
We initiate the study of data dimensionality reduction, or sketching, for the q -> p norms. Given an n x d matrix A, the q -> p norm, denoted |A |_{q -> p} = sup_{x in R^d \ 0} |Ax |_p / |x |_q, is a natural generalization of several matrix and vector norms studied in the data stream and sketching models, with applications to datamining, hardness of approximation, and oblivious routing. We say a distribution S on random matrices L in R^{nd} - > R^k is a (k,alpha)-sketching family if from L(A), one can approximate |A |_{q -> p} up to a factor alpha with constant probability. We provide upper and lower bounds on the sketching dimension k for every p, q in [1, infty], and in a number of cases our bounds are tight. While we mostly focus on constant alpha, we also consider large approximation factors alpha, as well as other variants of the problem such as when A has low rank.
Cite as
Aditya Krishnan, Sidhanth Mohanty, and David P. Woodruff. On Sketching the q to p Norms. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 15:1-15:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{krishnan_et_al:LIPIcs.APPROX-RANDOM.2018.15,
author = {Krishnan, Aditya and Mohanty, Sidhanth and Woodruff, David P.},
title = {{On Sketching the q to p Norms}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {15:1--15:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.15},
URN = {urn:nbn:de:0030-drops-94192},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.15},
annote = {Keywords: Dimensionality Reduction, Norms, Sketching, Streaming}
}
Document
Flow-time Optimization for Concurrent Open-Shop and Precedence Constrained Scheduling Models
Abstract
Scheduling a set of jobs over a collection of machines is a fundamental problem that needs to be solved millions of times a day in various computing platforms: in operating systems, in large data clusters, and in data centers. Along with makespan, flow-time, which measures the length of time a job spends in a system before it completes, is arguably the most important metric to measure the performance of a scheduling algorithm. In recent years, there has been a remarkable progress in understanding flow-time based objective functions in diverse settings such as unrelated machines scheduling, broadcast scheduling, multi-dimensional scheduling, to name a few.
Yet, our understanding of the flow-time objective is limited mostly to the scenarios where jobs have no dependencies. On the other hand, in almost all real world applications, think of MapReduce settings for example, jobs have dependencies that need to be respected while making scheduling decisions. In this paper, we take first steps towards understanding this complex problem. In particular, we consider two classical scheduling problems that capture dependencies across jobs: 1) concurrent open-shop scheduling (COSSP) and 2) precedence constrained scheduling. Our main motivation to study these problems specifically comes from their relevance to two scheduling problems that have gained importance in the context of data centers: co-flow scheduling and DAG scheduling. We design almost optimal approximation algorithms for COSSP and PCSP, and show hardness results.
Cite as
Janardhan Kulkarni and Shi Li. Flow-time Optimization for Concurrent Open-Shop and Precedence Constrained Scheduling Models. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 16:1-16:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{kulkarni_et_al:LIPIcs.APPROX-RANDOM.2018.16,
author = {Kulkarni, Janardhan and Li, Shi},
title = {{Flow-time Optimization for Concurrent Open-Shop and Precedence Constrained Scheduling Models}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {16:1--16:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.16},
URN = {urn:nbn:de:0030-drops-94205},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.16},
annote = {Keywords: Approximation, Weighted Flow Time, Concurrent Open Shop, Precedence Constraints}
}
Document
Sublinear-Time Quadratic Minimization via Spectral Decomposition of Matrices
Abstract
We design a sublinear-time approximation algorithm for quadratic function minimization problems with a better error bound than the previous algorithm by Hayashi and Yoshida (NIPS'16). Our approximation algorithm can be modified to handle the case where the minimization is done over a sphere. The analysis of our algorithms is obtained by combining results from graph limit theory, along with a novel spectral decomposition of matrices. Specifically, we prove that a matrix A can be decomposed into a structured part and a pseudorandom part, where the structured part is a block matrix with a polylogarithmic number of blocks, such that in each block all the entries are the same, and the pseudorandom part has a small spectral norm, achieving better error bound than the existing decomposition theorem of Frieze and Kannan (FOCS'96). As an additional application of the decomposition theorem, we give a sublinear-time approximation algorithm for computing the top singular values of a matrix.
Cite as
Amit Levi and Yuichi Yoshida. Sublinear-Time Quadratic Minimization via Spectral Decomposition of Matrices. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 17:1-17:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{levi_et_al:LIPIcs.APPROX-RANDOM.2018.17,
author = {Levi, Amit and Yoshida, Yuichi},
title = {{Sublinear-Time Quadratic Minimization via Spectral Decomposition of Matrices}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {17:1--17:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.17},
URN = {urn:nbn:de:0030-drops-94210},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.17},
annote = {Keywords: Qudratic function minimization, Approximation Algorithms, Matrix spectral decomposition, Graph limits}
}
Document
Deterministic Heavy Hitters with Sublinear Query Time
Abstract
We study the classic problem of finding l_1 heavy hitters in the streaming model. In the general turnstile model, we give the first deterministic sublinear-time sketching algorithm which takes a linear sketch of length O(epsilon^{-2} log n * log^*(epsilon^{-1})), which is only a factor of log^*(epsilon^{-1}) more than the best existing polynomial-time sketching algorithm (Nelson et al., RANDOM '12). Our approach is based on an iterative procedure, where most unrecovered heavy hitters are identified in each iteration. Although this technique has been extensively employed in the related problem of sparse recovery, this is the first time, to the best of our knowledge, that it has been used in the context of heavy hitters. Along the way we also obtain a sublinear time algorithm for the closely related problem of the l_1/l_1 compressed sensing, matching the space usage of previous (super-)linear time algorithms. In the strict turnstile model, we show that the runtime can be improved and the sketching matrix can be made strongly explicit with O(epsilon^{-2}log^3 n/log^3(1/epsilon)) rows.
Cite as
Yi Li and Vasileios Nakos. Deterministic Heavy Hitters with Sublinear Query Time. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 18:1-18:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{li_et_al:LIPIcs.APPROX-RANDOM.2018.18,
author = {Li, Yi and Nakos, Vasileios},
title = {{Deterministic Heavy Hitters with Sublinear Query Time}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {18:1--18:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.18},
URN = {urn:nbn:de:0030-drops-94221},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.18},
annote = {Keywords: heavy hitters, turnstile model, sketching algorithm, strongly explicit}
}
Document
On Low-Risk Heavy Hitters and Sparse Recovery Schemes
Abstract
We study the heavy hitters and related sparse recovery problems in the low failure probability regime. This regime is not well-understood, and the main previous work on this is by Gilbert et al. (ICALP'13). We recognize an error in their analysis, improve their results, and contribute new sparse recovery algorithms, as well as provide upper and lower bounds for the heavy hitters problem with low failure probability. Our results are summarized as follows:
1) (Heavy Hitters) We study three natural variants for finding heavy hitters in the strict turnstile model, where the variant depends on the quality of the desired output. For the weakest variant, we give a randomized algorithm improving the failure probability analysis of the ubiquitous Count-Min data structure. We also give a new lower bound for deterministic schemes, resolving a question about this variant posed in Question 4 in the IITK Workshop on Algorithms for Data Streams (2006). Under the strongest and well-studied l_{infty}/ l_2 variant, we show that the classical Count-Sketch data structure is optimal for very low failure probabilities, which was previously unknown.
2) (Sparse Recovery Algorithms) For non-adaptive sparse-recovery, we give sublinear-time algorithms with low-failure probability, which improve upon Gilbert et al. (ICALP'13). In the adaptive case, we improve the failure probability from a constant by Indyk et al. (FOCS '11) to e^{-k^{0.99}}, where k is the sparsity parameter.
3) (Optimal Average-Case Sparse Recovery Bounds) We give matching upper and lower bounds in all parameters, including the failure probability, for the measurement complexity of the l_2/l_2 sparse recovery problem in the spiked-covariance model, completely settling its complexity in this model.
Cite as
Yi Li, Vasileios Nakos, and David P. Woodruff. On Low-Risk Heavy Hitters and Sparse Recovery Schemes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 19:1-19:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{li_et_al:LIPIcs.APPROX-RANDOM.2018.19,
author = {Li, Yi and Nakos, Vasileios and Woodruff, David P.},
title = {{On Low-Risk Heavy Hitters and Sparse Recovery Schemes}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {19:1--19:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.19},
URN = {urn:nbn:de:0030-drops-94237},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.19},
annote = {Keywords: heavy hitters, sparse recovery, turnstile model, spike covariance model, lower bounds}
}
Document
Mildly Exponential Time Approximation Algorithms for Vertex Cover, Balanced Separator and Uniform Sparsest Cut
Abstract
In this work, we study the trade-off between the running time of approximation algorithms and their approximation guarantees. By leveraging a structure of the "hard" instances of the Arora-Rao-Vazirani lemma [Sanjeev Arora et al., 2009; James R. Lee, 2005], we show that the Sum-of-Squares hierarchy can be adapted to provide "fast", but still exponential time, approximation algorithms for several problems in the regime where they are believed to be NP-hard. Specifically, our framework yields the following algorithms; here n denote the number of vertices of the graph and r can be any positive real number greater than 1 (possibly depending on n).
- A (2 - 1/(O(r)))-approximation algorithm for Vertex Cover that runs in exp (n/(2^{r^2)})n^{O(1)} time.
- An O(r)-approximation algorithms for Uniform Sparsest Cut and Balanced Separator that runs in exp (n/(2^{r^2)})n^{O(1)} time.
Our algorithm for Vertex Cover improves upon Bansal et al.'s algorithm [Nikhil Bansal et al., 2017] which achieves (2 - 1/(O(r)))-approximation in time exp (n/(r^r))n^{O(1)}. For Uniform Sparsest Cut and Balanced Separator, our algorithms improve upon O(r)-approximation exp (n/(2^r))n^{O(1)}-time algorithms that follow from a work of Charikar et al. [Moses Charikar et al., 2010].
Cite as
Pasin Manurangsi and Luca Trevisan. Mildly Exponential Time Approximation Algorithms for Vertex Cover, Balanced Separator and Uniform Sparsest Cut. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{manurangsi_et_al:LIPIcs.APPROX-RANDOM.2018.20,
author = {Manurangsi, Pasin and Trevisan, Luca},
title = {{Mildly Exponential Time Approximation Algorithms for Vertex Cover, Balanced Separator and Uniform Sparsest Cut}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {20:1--20:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.20},
URN = {urn:nbn:de:0030-drops-94241},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.20},
annote = {Keywords: Approximation algorithms, Exponential-time algorithms, Vertex Cover, Sparsest Cut, Balanced Separator}
}
Document
Deterministic O(1)-Approximation Algorithms to 1-Center Clustering with Outliers
Abstract
The 1-center clustering with outliers problem asks about identifying a prototypical robust statistic that approximates the location of a cluster of points. Given some constant 0 < alpha < 1 and n points such that alpha n of them are in some (unknown) ball of radius r, the goal is to compute a ball of radius O(r) that also contains alpha n points. This problem can be formulated with the points in a normed vector space such as R^d or in a general metric space.
The problem has a simple randomized solution: a randomly selected point is a correct solution with constant probability, and its correctness can be verified in linear time. However, the deterministic complexity of this problem was not known. In this paper, for any L^p vector space, we show an O(nd)-time solution with a ball of radius O(r) for a fixed alpha > 1/2, and for any normed vector space, we show an O(nd)-time solution with a ball of radius O(r) when alpha > 1/2 as well as an O(nd log^{(k)}(n))-time solution with a ball of radius O(r) for all alpha > 0, k in N, where log^{(k)}(n) represents the kth iterated logarithm, assuming distance computation and vector space operations take O(d) time. For an arbitrary metric space, we show for any C in N an O(n^{1+1/C})-time solution that finds a ball of radius 2Cr, assuming distance computation between any pair of points takes O(1)-time, and show that for any alpha, C, an O(n^{1+1/C})-time solution that finds a ball of radius ((2C-3)(1-alpha)-1)r cannot exist.
Cite as
Shyam Narayanan. Deterministic O(1)-Approximation Algorithms to 1-Center Clustering with Outliers. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 21:1-21:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{narayanan:LIPIcs.APPROX-RANDOM.2018.21,
author = {Narayanan, Shyam},
title = {{Deterministic O(1)-Approximation Algorithms to 1-Center Clustering with Outliers}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {21:1--21:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.21},
URN = {urn:nbn:de:0030-drops-94253},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.21},
annote = {Keywords: Deterministic, Approximation Algorithm, Cluster, Statistic}
}
Document
Robust Online Speed Scaling With Deadline Uncertainty
Abstract
A speed scaling problem is considered, where time is divided into slots, and jobs with payoff v arrive at the beginning of the slot with associated deadlines d. Each job takes one slot to be processed, and multiple jobs can be processed by the server in each slot with energy cost g(k) for processing k jobs in one slot. The payoff is accrued by the algorithm only if the job is processed by its deadline. We consider a robust version of this speed scaling problem, where a job on its arrival reveals its payoff v, however, the deadline is hidden to the online algorithm, which could potentially be chosen adversarially and known to the optimal offline algorithm. The objective is to derive a robust (to deadlines) and optimal online algorithm that achieves the best competitive ratio. We propose an algorithm (called min-LCR) and show that it is an optimal online algorithm for any convex energy cost function g(.). We do so without actually evaluating the optimal competitive ratio, and give a general proof that works for any convex g, which is rather novel. For the popular choice of energy cost function g(k) = k^alpha, alpha >= 2, we give concrete bounds on the competitive ratio of the algorithm, which ranges between 2.618 and 3 depending on the value of alpha. The best known online algorithm for the same problem, but where deadlines are revealed to the online algorithm has competitive ratio of 2 and a lower bound of sqrt{2}. Thus, importantly, lack of deadline knowledge does not make the problem degenerate, and the effect of deadline information on the optimal competitive ratio is limited.
Cite as
Goonwanth Reddy and Rahul Vaze. Robust Online Speed Scaling With Deadline Uncertainty. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{reddy_et_al:LIPIcs.APPROX-RANDOM.2018.22,
author = {Reddy, Goonwanth and Vaze, Rahul},
title = {{Robust Online Speed Scaling With Deadline Uncertainty}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {22:1--22:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.22},
URN = {urn:nbn:de:0030-drops-94269},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.22},
annote = {Keywords: Online Algorithms, Speed Scaling, Greedy Algorithms, Scheduling}
}
Document
Multi-Agent Submodular Optimization
Abstract
Recent years have seen many algorithmic advances in the area of submodular optimization: (SO) min/max~f(S): S in F, where F is a given family of feasible sets over a ground set V and f:2^V - > R is submodular. This progress has been coupled with a wealth of new applications for these models. Our focus is on a more general class of multi-agent submodular optimization (MASO) min/max Sum_{i=1}^{k} f_i(S_i): S_1 u+ S_2 u+ ... u+ S_k in F. Here we use u+ to denote disjoint union and hence this model is attractive where resources are being allocated across k agents, each with its own submodular cost function f_i(). This was introduced in the minimization setting by Goel et al. In this paper we explore the extent to which the approximability of the multi-agent problems are linked to their single-agent versions, referred to informally as the multi-agent gap.
We present different reductions that transform a multi-agent problem into a single-agent one. For minimization, we show that (MASO) has an O(alpha * min{k, log^2 (n)})-approximation whenever (SO) admits an alpha-approximation over the convex formulation. In addition, we discuss the class of "bounded blocker" families where there is a provably tight O(log n) multi-agent gap between (MASO) and (SO). For maximization, we show that monotone (resp. nonmonotone) (MASO) admits an alpha (1-1/e) (resp. alpha * 0.385) approximation whenever monotone (resp. nonmonotone) (SO) admits an alpha-approximation over the multilinear formulation; and the 1-1/e multi-agent gap for monotone objectives is tight. We also discuss several families (such as spanning trees, matroids, and p-systems) that have an (optimal) multi-agent gap of 1. These results substantially expand the family of tractable models for submodular maximization.
Cite as
Richard Santiago and F. Bruce Shepherd. Multi-Agent Submodular Optimization. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 23:1-23:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{santiago_et_al:LIPIcs.APPROX-RANDOM.2018.23,
author = {Santiago, Richard and Shepherd, F. Bruce},
title = {{Multi-Agent Submodular Optimization}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {23:1--23:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.23},
URN = {urn:nbn:de:0030-drops-94276},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.23},
annote = {Keywords: submodular optimization, multi-agent, approximation algorithms}
}
Document
Generalized Assignment of Time-Sensitive Item Groups
Abstract
We study the generalized assignment problem with time-sensitive item groups (chi-AGAP). It has central applications in advertisement placement on the Internet, and in virtual network embedding in Cloud data centers. We are given a set of items, partitioned into n groups, and a set of T identical bins (or, time-slots). Each group 1 <= j <= n has a time-window chi_j = [r_j, d_j]subseteq [T] in which it can be packed. Each item i in group j has a size s_i>0 and a non-negative utility u_{it} when packed into bin t in chi_j. A bin can accommodate at most one item from each group and the total size of the items in a bin cannot exceed its capacity. The goal is to find a feasible packing of a subset of the items in the bins such that the total utility from groups that are completely packed is maximized. Our main result is an Omega(1)-approximation algorithm for chi-AGAP. Our approximation technique relies on a non-trivial rounding of a configuration LP, which can be adapted to other common scenarios of resource allocation in Cloud data centers.
Cite as
Kanthi Sarpatwar, Baruch Schieber, and Hadas Shachnai. Generalized Assignment of Time-Sensitive Item Groups. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 24:1-24:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{sarpatwar_et_al:LIPIcs.APPROX-RANDOM.2018.24,
author = {Sarpatwar, Kanthi and Schieber, Baruch and Shachnai, Hadas},
title = {{Generalized Assignment of Time-Sensitive Item Groups}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {24:1--24:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.24},
URN = {urn:nbn:de:0030-drops-94287},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.24},
annote = {Keywords: Approximation Algorithms, Packing and Covering problems, Generalized Assignment problem}
}
Document
On Geodesically Convex Formulations for the Brascamp-Lieb Constant
Abstract
We consider two non-convex formulations for computing the optimal constant in the Brascamp-Lieb inequality corresponding to a given datum and show that they are geodesically log-concave on the manifold of positive definite matrices endowed with the Riemannian metric corresponding to the Hessian of the log-determinant function. The first formulation is present in the work of Lieb [Lieb, 1990] and the second is new and inspired by the work of Bennett et al. [Bennett et al., 2008]. Recent work of Garg et al. [Ankit Garg et al., 2017] also implies a geodesically log-concave formulation of the Brascamp-Lieb constant through a reduction to the operator scaling problem. However, the dimension of the arising optimization problem in their reduction depends exponentially on the number of bits needed to describe the Brascamp-Lieb datum. The formulations presented here have dimensions that are polynomial in the bit complexity of the input datum.
Cite as
Suvrit Sra, Nisheeth K. Vishnoi, and Ozan Yildiz. On Geodesically Convex Formulations for the Brascamp-Lieb Constant. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 25:1-25:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{sra_et_al:LIPIcs.APPROX-RANDOM.2018.25,
author = {Sra, Suvrit and Vishnoi, Nisheeth K. and Yildiz, Ozan},
title = {{On Geodesically Convex Formulations for the Brascamp-Lieb Constant}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {25:1--25:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.25},
URN = {urn:nbn:de:0030-drops-94296},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.25},
annote = {Keywords: Geodesic convexity, positive definite cone, geodesics, Brascamp-Lieb constant}
}
Document
Tensor Rank is Hard to Approximate
Abstract
We prove that approximating the rank of a 3-tensor to within a factor of 1 + 1/1852 - delta, for any delta > 0, is NP-hard over any field. We do this via reduction from bounded occurrence 2-SAT.
Cite as
Joseph Swernofsky. Tensor Rank is Hard to Approximate. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 26:1-26:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{swernofsky:LIPIcs.APPROX-RANDOM.2018.26,
author = {Swernofsky, Joseph},
title = {{Tensor Rank is Hard to Approximate}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {26:1--26:9},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.26},
URN = {urn:nbn:de:0030-drops-94309},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.26},
annote = {Keywords: tensor rank, high rank tensor, slice elimination, approximation algorithm, hardness of approximation}
}
Document
An O(1)-Approximation Algorithm for Dynamic Weighted Vertex Cover with Soft Capacity
Abstract
This study considers the soft capacitated vertex cover problem in a dynamic setting. This problem generalizes the dynamic model of the vertex cover problem, which has been intensively studied in recent years. Given a dynamically changing vertex-weighted graph G=(V,E), which allows edge insertions and edge deletions, the goal is to design a data structure that maintains an approximate minimum vertex cover while satisfying the capacity constraint of each vertex. That is, when picking a copy of a vertex v in the cover, the number of v's incident edges covered by the copy is up to a given capacity of v. We extend Bhattacharya et al.'s work [SODA'15 and ICALP'15] to obtain a deterministic primal-dual algorithm for maintaining a constant-factor approximate minimum capacitated vertex cover with O(log n / epsilon) amortized update time, where n is the number of vertices in the graph. The algorithm can be extended to (1) a more general model in which each edge is associated with a non-uniform and unsplittable demand, and (2) the more general capacitated set cover problem.
Cite as
Hao-Ting Wei, Wing-Kai Hon, Paul Horn, Chung-Shou Liao, and Kunihiko Sadakane. An O(1)-Approximation Algorithm for Dynamic Weighted Vertex Cover with Soft Capacity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{wei_et_al:LIPIcs.APPROX-RANDOM.2018.27,
author = {Wei, Hao-Ting and Hon, Wing-Kai and Horn, Paul and Liao, Chung-Shou and Sadakane, Kunihiko},
title = {{An O(1)-Approximation Algorithm for Dynamic Weighted Vertex Cover with Soft Capacity}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {27:1--27:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.27},
URN = {urn:nbn:de:0030-drops-94312},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.27},
annote = {Keywords: approximation algorithm, dynamic algorithm, primal-dual, vertex cover}
}
Document
Fixed-Parameter Approximation Schemes for Weighted Flowtime
Abstract
Given a set of n jobs with integral release dates, processing times and weights, it is a natural and important scheduling problem to compute a schedule that minimizes the sum of the weighted flow times of the jobs. There are strong lower bounds for the possible approximation ratios. In the non-preemptive case, even on a single machine the best known result is a O(sqrt{n})-approximation which is best possible. In the preemptive case on m identical machines there is a O(log min{n/m,P})-approximation (where P denotes the maximum job size) which is also best possible.
We study the problem in the parametrized setting where our parameter k is an upper bound on the maximum (integral) processing time and weight of a job, a standard parameter for scheduling problems. We present a (1+epsilon)-approximation algorithm for the preemptive and the non-preemptive case of minimizing weighted flow time on m machines with a running time of f(k,epsilon,m)* n^{O(1)}, i.e., our combined parameters are k,epsilon, and m. Key to our results is to distinguish time intervals according to whether in the optimal solution the pending jobs have large or small total weight. Depending on this we employ dynamic programming, linear programming, greedy routines, or combinations of the latter to compute the schedule for each respective interval.
Cite as
Andreas Wiese. Fixed-Parameter Approximation Schemes for Weighted Flowtime. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 28:1-28:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{wiese:LIPIcs.APPROX-RANDOM.2018.28,
author = {Wiese, Andreas},
title = {{Fixed-Parameter Approximation Schemes for Weighted Flowtime}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {28:1--28:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.28},
URN = {urn:nbn:de:0030-drops-94326},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.28},
annote = {Keywords: Scheduling, fixed-parameter algorithms, approximation algorithms, approximation schemes}
}
Document
List-Decoding Homomorphism Codes with Arbitrary Codomains
Abstract
The codewords of the homomorphism code aHom(G,H) are the affine homomorphisms between two finite groups, G and H, generalizing Hadamard codes. Following the work of Goldreich-Levin (1989), Grigorescu et al. (2006), Dinur et al. (2008), and Guo and Sudan (2014), we further expand the range of groups for which local list-decoding is possible up to mindist, the minimum distance of the code. In particular, for the first time, we do not require either G or H to be solvable. Specifically, we demonstrate a poly(1/epsilon) bound on the list size, i. e., on the number of codewords within distance (mindist-epsilon) from any received word, when G is either abelian or an alternating group, and H is an arbitrary (finite or infinite) group. We conjecture that a similar bound holds for all finite simple groups as domains; the alternating groups serve as the first test case.
The abelian vs. arbitrary result permits us to adapt previous techniques to obtain efficient local list-decoding for this case. We also obtain efficient local list-decoding for the permutation representations of alternating groups (the codomain is a symmetric group) under the restriction that the domain G=A_n is paired with codomain H=S_m satisfying m < 2^{n-1}/sqrt{n}.
The limitations on the codomain in the latter case arise from severe technical difficulties stemming from the need to solve the homomorphism extension (HomExt) problem in certain cases; these are addressed in a separate paper (Wuu 2018).
We introduce an intermediate "semi-algorithmic" model we call Certificate List-Decoding that bypasses the HomExt bottleneck and works in the alternating vs. arbitrary setting. A certificate list-decoder produces partial homomorphisms that uniquely extend to the homomorphisms in the list. A homomorphism extender applied to a list of certificates yields the desired list.
Cite as
László Babai, Timothy J. F. Black, and Angela Wuu. List-Decoding Homomorphism Codes with Arbitrary Codomains. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 29:1-29:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{babai_et_al:LIPIcs.APPROX-RANDOM.2018.29,
author = {Babai, L\'{a}szl\'{o} and Black, Timothy J. F. and Wuu, Angela},
title = {{List-Decoding Homomorphism Codes with Arbitrary Codomains}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {29:1--29:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.29},
URN = {urn:nbn:de:0030-drops-94338},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.29},
annote = {Keywords: Error-correcting codes, Local algorithms, Local list-decoding, Finite groups, Homomorphism codes}
}
Document
Optimal Deterministic Extractors for Generalized Santha-Vazirani Sources
Abstract
Let F be a finite alphabet and D be a finite set of distributions over F. A Generalized Santha-Vazirani (GSV) source of type (F, D), introduced by Beigi, Etesami and Gohari (ICALP 2015, SICOMP 2017), is a random sequence (F_1, ..., F_n) in F^n, where F_i is a sample from some distribution d in D whose choice may depend on F_1, ..., F_{i-1}.
We show that all GSV source types (F, D) fall into one of three categories: (1) non-extractable; (2) extractable with error n^{-Theta(1)}; (3) extractable with error 2^{-Omega(n)}.
We provide essentially randomness-optimal extraction algorithms for extractable sources. Our algorithm for category (2) sources extracts one bit with error epsilon from n = poly(1/epsilon) samples in time linear in n. Our algorithm for category (3) sources extracts m bits with error epsilon from n = O(m + log 1/epsilon) samples in time min{O(m2^m * n),n^{O(|F|)}}.
We also give algorithms for classifying a GSV source type (F, D): Membership in category (1) can be decided in NP, while membership in category (3) is polynomial-time decidable.
Cite as
Salman Beigi, Andrej Bogdanov, Omid Etesami, and Siyao Guo. Optimal Deterministic Extractors for Generalized Santha-Vazirani Sources. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 30:1-30:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{beigi_et_al:LIPIcs.APPROX-RANDOM.2018.30,
author = {Beigi, Salman and Bogdanov, Andrej and Etesami, Omid and Guo, Siyao},
title = {{Optimal Deterministic Extractors for Generalized Santha-Vazirani Sources}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {30:1--30:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.30},
URN = {urn:nbn:de:0030-drops-94349},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.30},
annote = {Keywords: feasibility of randomness extraction, extractor lower bounds, martingales}
}
Document
Adaptive Lower Bound for Testing Monotonicity on the Line
Abstract
In the property testing model, the task is to distinguish objects possessing some property from the objects that are far from it. One of such properties is monotonicity, when the objects are functions from one poset to another. This is an active area of research. In this paper we study query complexity of epsilon-testing monotonicity of a function f : [n]->[r]. All our lower bounds are for adaptive two-sided testers.
- We prove a nearly tight lower bound for this problem in terms of r. The bound is Omega((log r)/(log log r)) when epsilon = 1/2. No previous satisfactory lower bound in terms of r was known.
- We completely characterise query complexity of this problem in terms of n for smaller values of epsilon. The complexity is Theta(epsilon^{-1} log (epsilon n)). Apart from giving the lower bound, this improves on the best known upper bound.
Finally, we give an alternative proof of the Omega(epsilon^{-1}d log n - epsilon^{-1}log epsilon^{-1}) lower bound for testing monotonicity on the hypergrid [n]^d due to Chakrabarty and Seshadhri (RANDOM'13).
Cite as
Aleksandrs Belovs. Adaptive Lower Bound for Testing Monotonicity on the Line. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 31:1-31:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{belovs:LIPIcs.APPROX-RANDOM.2018.31,
author = {Belovs, Aleksandrs},
title = {{Adaptive Lower Bound for Testing Monotonicity on the Line}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {31:1--31:10},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.31},
URN = {urn:nbn:de:0030-drops-94350},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.31},
annote = {Keywords: property testing, monotonicity on the line, monotonicity on the hypergrid}
}
Document
Swendsen-Wang Dynamics for General Graphs in the Tree Uniqueness Region
Abstract
The Swendsen-Wang dynamics is a popular algorithm for sampling from the Gibbs distribution for the ferromagnetic Ising model on a graph G=(V,E). The dynamics is a "global" Markov chain which is conjectured to converge to equilibrium in O(|V|^{1/4}) steps for any graph G at any (inverse) temperature beta. It was recently proved by Guo and Jerrum (2017) that the Swendsen-Wang dynamics has polynomial mixing time on any graph at all temperatures, yet there are few results providing o(|V|) upper bounds on its convergence time.
We prove fast convergence of the Swendsen-Wang dynamics on general graphs in the tree uniqueness region of the ferromagnetic Ising model. In particular, when beta < beta_c(d) where beta_c(d) denotes the uniqueness/non-uniqueness threshold on infinite d-regular trees, we prove that the relaxation time (i.e., the inverse spectral gap) of the Swendsen-Wang dynamics is Theta(1) on any graph of maximum degree d >= 3. Our proof utilizes a version of the Swendsen-Wang dynamics which only updates isolated vertices. We establish that this variant of the Swendsen-Wang dynamics has mixing time O(log{|V|}) and relaxation time Theta(1) on any graph of maximum degree d for all beta < beta_c(d). We believe that this Markov chain may be of independent interest, as it is a monotone Swendsen-Wang type chain. As part of our proofs, we provide modest extensions of the technology of Mossel and Sly (2013) for analyzing mixing times and of the censoring result of Peres and Winkler (2013). Both of these results are for the Glauber dynamics, and we extend them here to general monotone Markov chains. This class of dynamics includes for example the heat-bath block dynamics, for which we obtain new tight mixing time bounds.
Cite as
Antonio Blanca, Zongchen Chen, and Eric Vigoda. Swendsen-Wang Dynamics for General Graphs in the Tree Uniqueness Region. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{blanca_et_al:LIPIcs.APPROX-RANDOM.2018.32,
author = {Blanca, Antonio and Chen, Zongchen and Vigoda, Eric},
title = {{Swendsen-Wang Dynamics for General Graphs in the Tree Uniqueness Region}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {32:1--32:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.32},
URN = {urn:nbn:de:0030-drops-94365},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.32},
annote = {Keywords: Swendsen-Wang dynamics, mixing time, relaxation time, spatial mixing, censoring}
}
Document
Sampling in Uniqueness from the Potts and Random-Cluster Models on Random Regular Graphs
Abstract
We consider the problem of sampling from the Potts model on random regular graphs. It is conjectured that sampling is possible when the temperature of the model is in the so-called uniqueness regime of the regular tree, but positive algorithmic results have been for the most part elusive. In this paper, for all integers q >= 3 and Delta >= 3, we develop algorithms that produce samples within error o(1) from the q-state Potts model on random Delta-regular graphs, whenever the temperature is in uniqueness, for both the ferromagnetic and antiferromagnetic cases.
The algorithm for the antiferromagnetic Potts model is based on iteratively adding the edges of the graph and resampling a bichromatic class that contains the endpoints of the newly added edge. Key to the algorithm is how to perform the resampling step efficiently since bichromatic classes can potentially induce linear-sized components. To this end, we exploit the tree uniqueness to show that the average growth of bichromatic components is typically small, which allows us to use correlation decay algorithms for the resampling step. While the precise uniqueness threshold on the tree is not known for general values of q and Delta in the antiferromagnetic case, our algorithm works throughout uniqueness regardless of its value.
In the case of the ferromagnetic Potts model, we are able to simplify the algorithm significantly by utilising the random-cluster representation of the model. In particular, we demonstrate that a percolation-type algorithm succeeds in sampling from the random-cluster model with parameters p,q on random Delta-regular graphs for all values of q >= 1 and p<p_c(q,Delta), where p_c(q,Delta) corresponds to a uniqueness threshold for the model on the Delta-regular tree. When restricted to integer values of q, this yields a simplified algorithm for the ferromagnetic Potts model on random Delta-regular graphs.
Cite as
Antonio Blanca, Andreas Galanis, Leslie Ann Goldberg, Daniel Stefankovic, Eric Vigoda, and Kuan Yang. Sampling in Uniqueness from the Potts and Random-Cluster Models on Random Regular Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{blanca_et_al:LIPIcs.APPROX-RANDOM.2018.33,
author = {Blanca, Antonio and Galanis, Andreas and Goldberg, Leslie Ann and Stefankovic, Daniel and Vigoda, Eric and Yang, Kuan},
title = {{Sampling in Uniqueness from the Potts and Random-Cluster Models on Random Regular Graphs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {33:1--33:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.33},
URN = {urn:nbn:de:0030-drops-94371},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.33},
annote = {Keywords: sampling, Potts model, random regular graphs, phase transitions}
}
Document
Polar Codes with Exponentially Small Error at Finite Block Length
Abstract
We show that the entire class of polar codes (up to a natural necessary condition) converge to capacity at block lengths polynomial in the gap to capacity, while simultaneously achieving failure probabilities that are exponentially small in the block length (i.e., decoding fails with probability exp(-N^{Omega(1)}) for codes of length N). Previously this combination was known only for one specific family within the class of polar codes, whereas we establish this whenever the polar code exhibits a condition necessary for any polarization.
Our results adapt and strengthen a local analysis of polar codes due to the authors with Nakkiran and Rudra [Proc. STOC 2018]. Their analysis related the time-local behavior of a martingale to its global convergence, and this allowed them to prove that the broad class of polar codes converge to capacity at polynomial block lengths. Their analysis easily adapts to show exponentially small failure probabilities, provided the associated martingale, the "Arikan martingale", exhibits a corresponding strong local effect. The main contribution of this work is a much stronger local analysis of the Arikan martingale. This leads to the general result claimed above.
In addition to our general result, we also show, for the first time, polar codes that achieve failure probability exp(-N^{beta}) for any beta < 1 while converging to capacity at block length polynomial in the gap to capacity. Finally we also show that the "local" approach can be combined with any analysis of failure probability of an arbitrary polar code to get essentially the same failure probability while achieving block length polynomial in the gap to capacity.
Cite as
Jaroslaw Blasiok, Venkatesan Guruswami, and Madhu Sudan. Polar Codes with Exponentially Small Error at Finite Block Length. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 34:1-34:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{blasiok_et_al:LIPIcs.APPROX-RANDOM.2018.34,
author = {Blasiok, Jaroslaw and Guruswami, Venkatesan and Sudan, Madhu},
title = {{Polar Codes with Exponentially Small Error at Finite Block Length}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {34:1--34:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.34},
URN = {urn:nbn:de:0030-drops-94382},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.34},
annote = {Keywords: Polar codes, error exponent, rate of polarization}
}
Document
Approximate Degree and the Complexity of Depth Three Circuits
Abstract
Threshold weight, margin complexity, and Majority-of-Threshold circuit size are basic complexity measures of Boolean functions that arise in learning theory, communication complexity, and circuit complexity. Each of these measures might exhibit a chasm at depth three: namely, all polynomial size Boolean circuits of depth two have polynomial complexity under the measure, but there may exist Boolean circuits of depth three that have essentially maximal complexity exp(Theta(n)). However, existing techniques are far from showing this: for all three measures, the best lower bound for depth three circuits is exp(Omega(n^{2/5})). Moreover, prior methods exclusively study block-composed functions. Such methods appear intrinsically unable to prove lower bounds better than exp(Omega(sqrt{n})) even for depth four circuits, and have yet to prove lower bounds better than exp(Omega(sqrt{n})) for circuits of any constant depth.
We take a step toward showing that all of these complexity measures indeed exhibit a chasm at depth three. Specifically, for any arbitrarily small constant delta > 0, we exhibit a depth three circuit of polynomial size (in fact, an O(log n)-decision list) of complexity exp(Omega(n^{1/2-delta})) under each of these measures.
Our methods go beyond the block-composed functions studied in prior work, and hence may not be subject to the same barriers. Accordingly, we suggest natural candidate functions that may exhibit stronger bounds.
Cite as
Mark Bun and Justin Thaler. Approximate Degree and the Complexity of Depth Three Circuits. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 35:1-35:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{bun_et_al:LIPIcs.APPROX-RANDOM.2018.35,
author = {Bun, Mark and Thaler, Justin},
title = {{Approximate Degree and the Complexity of Depth Three Circuits}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {35:1--35:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.35},
URN = {urn:nbn:de:0030-drops-94390},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.35},
annote = {Keywords: approximate degree, communication complexity, learning theory, polynomial approximation, threshold circuits}
}
Document
Speeding up Switch Markov Chains for Sampling Bipartite Graphs with Given Degree Sequence
Abstract
We consider the well-studied problem of uniformly sampling (bipartite) graphs with a given degree sequence, or equivalently, the uniform sampling of binary matrices with fixed row and column sums. In particular, we focus on Markov Chain Monte Carlo (MCMC) approaches, which proceed by making small changes that preserve the degree sequence to a given graph. Such Markov chains converge to the uniform distribution, but the challenge is to show that they do so quickly, i.e., that they are rapidly mixing.
The standard example of this Markov chain approach for sampling bipartite graphs is the switch algorithm, that proceeds by locally switching two edges while preserving the degree sequence. The Curveball algorithm is a variation on this approach in which essentially multiple switches (trades) are performed simultaneously, with the goal of speeding up switch-based algorithms. Even though the Curveball algorithm is expected to mix faster than switch-based algorithms for many degree sequences, nothing is currently known about its mixing time. On the other hand, the switch algorithm has been proven to be rapidly mixing for several classes of degree sequences.
In this work we present the first results regarding the mixing time of the Curveball algorithm. We give a theoretical comparison between the switch and Curveball algorithms in terms of their underlying Markov chains. As our main result, we show that the Curveball chain is rapidly mixing whenever a switch-based chain is rapidly mixing. We do this using a novel state space graph decomposition of the switch chain into Johnson graphs. This decomposition is of independent interest.
Cite as
Corrie Jacobien Carstens and Pieter Kleer. Speeding up Switch Markov Chains for Sampling Bipartite Graphs with Given Degree Sequence. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 36:1-36:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{carstens_et_al:LIPIcs.APPROX-RANDOM.2018.36,
author = {Carstens, Corrie Jacobien and Kleer, Pieter},
title = {{Speeding up Switch Markov Chains for Sampling Bipartite Graphs with Given Degree Sequence}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {36:1--36:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.36},
URN = {urn:nbn:de:0030-drops-94403},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.36},
annote = {Keywords: Binary matrix, graph sampling, Curveball, switch, Markov chain decomposition, Johnson graph}
}
Document
Randomness Extraction in AC0 and with Small Locality
Abstract
Randomness extractors, which extract high quality (almost-uniform) random bits from biased random sources, are important objects both in theory and in practice. While there have been significant progress in obtaining near optimal constructions of randomness extractors in various settings, the computational complexity of randomness extractors is still much less studied. In particular, it is not clear whether randomness extractors with good parameters can be computed in several interesting complexity classes that are much weaker than P.
In this paper we study randomness extractors in the following two models of computation: (1) constant-depth circuits (AC^0), and (2) the local computation model. Previous work in these models, such as [Viola, 2005], [Goldreich et al., 2015] and [Bogdanov and Guo, 2013], only achieve constructions with weak parameters. In this work we give explicit constructions of randomness extractors with much better parameters. Our results on AC^0 extractors refute a conjecture in [Goldreich et al., 2015] and answer several open problems there. We also provide a lower bound on the error of extractors in AC^0, which together with the entropy lower bound in [Viola, 2005; Goldreich et al., 2015] almost completely characterizes extractors in this class. Our results on local extractors also significantly improve the seed length in [Bogdanov and Guo, 2013]. As an application, we use our AC^0 extractors to study pseudorandom generators in AC^0, and show that we can construct both cryptographic pseudorandom generators (under reasonable computational assumptions) and unconditional pseudorandom generators for space bounded computation with very good parameters.
Our constructions combine several previous techniques in randomness extractors, as well as introduce new techniques to reduce or preserve the complexity of extractors, which may be of independent interest. These include (1) a general way to reduce the error of strong seeded extractors while preserving the AC^0 property and small locality, and (2) a seeded randomness condenser with small locality.
Cite as
Kuan Cheng and Xin Li. Randomness Extraction in AC0 and with Small Locality. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 37:1-37:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{cheng_et_al:LIPIcs.APPROX-RANDOM.2018.37,
author = {Cheng, Kuan and Li, Xin},
title = {{Randomness Extraction in AC0 and with Small Locality}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {37:1--37:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.37},
URN = {urn:nbn:de:0030-drops-94414},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.37},
annote = {Keywords: Randomness Extraction, AC0, Locality, Pseudorandom Generator}
}
Document
Boolean Function Analysis on High-Dimensional Expanders
Abstract
We initiate the study of Boolean function analysis on high-dimensional expanders. We describe an analog of the Fourier expansion and of the Fourier levels on simplicial complexes, and generalize the FKN theorem to high-dimensional expanders.
Our results demonstrate that a high-dimensional expanding complex X can sometimes serve as a sparse model for the Boolean slice or hypercube, and quite possibly additional results from Boolean function analysis can be carried over to this sparse model. Therefore, this model can be viewed as a derandomization of the Boolean slice, containing |X(k)|=O(n) points in comparison to binom{n}{k+1} points in the (k+1)-slice (which consists of all n-bit strings with exactly k+1 ones).
Cite as
Yotam Dikstein, Irit Dinur, Yuval Filmus, and Prahladh Harsha. Boolean Function Analysis on High-Dimensional Expanders. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 38:1-38:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{dikstein_et_al:LIPIcs.APPROX-RANDOM.2018.38,
author = {Dikstein, Yotam and Dinur, Irit and Filmus, Yuval and Harsha, Prahladh},
title = {{Boolean Function Analysis on High-Dimensional Expanders}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {38:1--38:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.38},
URN = {urn:nbn:de:0030-drops-94421},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.38},
annote = {Keywords: high dimensional expanders, Boolean function analysis, sparse model}
}
Document
Percolation of Lipschitz Surface and Tight Bounds on the Spread of Information Among Mobile Agents
Abstract
We consider the problem of spread of information among mobile agents on the torus. The agents are initially distributed as a Poisson point process on the torus, and move as independent simple random walks. Two agents can share information whenever they are at the same vertex of the torus. We study the so-called flooding time: the amount of time it takes for information to be known by all agents. We establish a tight upper bound on the flooding time, and introduce a technique which we believe can be applicable to analyze other processes involving mobile agents.
Cite as
Peter Gracar and Alexandre Stauffer. Percolation of Lipschitz Surface and Tight Bounds on the Spread of Information Among Mobile Agents. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 39:1-39:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{gracar_et_al:LIPIcs.APPROX-RANDOM.2018.39,
author = {Gracar, Peter and Stauffer, Alexandre},
title = {{Percolation of Lipschitz Surface and Tight Bounds on the Spread of Information Among Mobile Agents}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {39:1--39:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.39},
URN = {urn:nbn:de:0030-drops-94439},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.39},
annote = {Keywords: Lipschitz surface, spread of information, flooding time, moving agents}
}
Document
Flipping out with Many Flips: Hardness of Testing k-Monotonicity
Abstract
A function f:{0,1}^n - > {0,1} is said to be k-monotone if it flips between 0 and 1 at most k times on every ascending chain. Such functions represent a natural generalization of (1-)monotone functions, and have been recently studied in circuit complexity, PAC learning, and cryptography. Our work is part of a renewed focus in understanding testability of properties characterized by freeness of arbitrary order patterns as a generalization of monotonicity. Recently, Canonne et al. (ITCS 2017) initiate the study of k-monotone functions in the area of property testing, and Newman et al. (SODA 2017) study testability of families characterized by freeness from order patterns on real-valued functions over the line [n] domain.
We study k-monotone functions in the more relaxed parametrized property testing model, introduced by Parnas et al. (JCSS, 72(6), 2006). In this process we resolve a problem left open in previous work. Specifically, our results include the following.
1) Testing 2-monotonicity on the hypercube non-adaptively with one-sided error requires an exponential in sqrt{n} number of queries. This behavior shows a stark contrast with testing (1-)monotonicity, which only needs O~(sqrt{n}) queries (Khot et al. (FOCS 2015)). Furthermore, even the apparently easier task of distinguishing 2-monotone functions from functions that are far from being n^{.01}-monotone also requires an exponential number of queries.
2) On the hypercube [n]^d domain, there exists a testing algorithm that makes a constant number of queries and distinguishes functions that are k-monotone from functions that are far from being O(kd^2) -monotone. Such a dependency is likely necessary, given the lower bound above for the hypercube.
Cite as
Elena Grigorescu, Akash Kumar, and Karl Wimmer. Flipping out with Many Flips: Hardness of Testing k-Monotonicity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 40:1-40:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{grigorescu_et_al:LIPIcs.APPROX-RANDOM.2018.40,
author = {Grigorescu, Elena and Kumar, Akash and Wimmer, Karl},
title = {{Flipping out with Many Flips: Hardness of Testing k-Monotonicity}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {40:1--40:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.40},
URN = {urn:nbn:de:0030-drops-94448},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.40},
annote = {Keywords: Property Testing, Boolean Functions, k-Monotonicity, Lower Bounds}
}
Document
How Long Can Optimal Locally Repairable Codes Be?
Abstract
A locally repairable code (LRC) with locality r allows for the recovery of any erased codeword symbol using only r other codeword symbols. A Singleton-type bound dictates the best possible trade-off between the dimension and distance of LRCs - an LRC attaining this trade-off is deemed optimal. Such optimal LRCs have been constructed over alphabets growing linearly in the block length. Unlike the classical Singleton bound, however, it was not known if such a linear growth in the alphabet size is necessary, or for that matter even if the alphabet needs to grow at all with the block length. Indeed, for small code distances 3,4, arbitrarily long optimal LRCs were known over fixed alphabets.
Here, we prove that for distances d >=slant 5, the code length n of an optimal LRC over an alphabet of size q must be at most roughly O(d q^3). For the case d=5, our upper bound is O(q^2). We complement these bounds by showing the existence of optimal LRCs of length Omega_{d,r}(q^{1+1/floor[(d-3)/2]}) when d <=slant r+2. Our bounds match when d=5, pinning down n=Theta(q^2) as the asymptotically largest length of an optimal LRC for this case.
Cite as
Venkatesan Guruswami, Chaoping Xing, and Chen Yuan. How Long Can Optimal Locally Repairable Codes Be?. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 41:1-41:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{guruswami_et_al:LIPIcs.APPROX-RANDOM.2018.41,
author = {Guruswami, Venkatesan and Xing, Chaoping and Yuan, Chen},
title = {{How Long Can Optimal Locally Repairable Codes Be?}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {41:1--41:11},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.41},
URN = {urn:nbn:de:0030-drops-94458},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.41},
annote = {Keywords: Locally Repairable Code, Singleton Bound}
}
Document
On Minrank and Forbidden Subgraphs
Abstract
The minrank over a field F of a graph G on the vertex set {1,2,...,n} is the minimum possible rank of a matrix M in F^{n x n} such that M_{i,i} != 0 for every i, and M_{i,j}=0 for every distinct non-adjacent vertices i and j in G. For an integer n, a graph H, and a field F, let g(n,H,F) denote the maximum possible minrank over F of an n-vertex graph whose complement contains no copy of H. In this paper we study this quantity for various graphs H and fields F. For finite fields, we prove by a probabilistic argument a general lower bound on g(n,H,F), which yields a nearly tight bound of Omega(sqrt{n}/log n) for the triangle H=K_3. For the real field, we prove by an explicit construction that for every non-bipartite graph H, g(n,H,R) >= n^delta for some delta = delta(H)>0. As a by-product of this construction, we disprove a conjecture of Codenotti, Pudlák, and Resta. The results are motivated by questions in information theory, circuit complexity, and geometry.
Cite as
Ishay Haviv. On Minrank and Forbidden Subgraphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 42:1-42:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{haviv:LIPIcs.APPROX-RANDOM.2018.42,
author = {Haviv, Ishay},
title = {{On Minrank and Forbidden Subgraphs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {42:1--42:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.42},
URN = {urn:nbn:de:0030-drops-94461},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.42},
annote = {Keywords: Minrank, Forbidden subgraphs, Shannon capacity, Circuit Complexity}
}
Document
Preserving Randomness for Adaptive Algorithms
Abstract
Suppose Est is a randomized estimation algorithm that uses n random bits and outputs values in R^d. We show how to execute Est on k adaptively chosen inputs using only n + O(k log(d + 1)) random bits instead of the trivial nk (at the cost of mild increases in the error and failure probability). Our algorithm combines a variant of the INW pseudorandom generator [Impagliazzo et al., 1994] with a new scheme for shifting and rounding the outputs of Est. We prove that modifying the outputs of Est is necessary in this setting, and furthermore, our algorithm's randomness complexity is near-optimal in the case d <= O(1). As an application, we give a randomness-efficient version of the Goldreich-Levin algorithm; our algorithm finds all Fourier coefficients with absolute value at least theta of a function F: {0, 1}^n -> {-1, 1} using O(n log n) * poly(1/theta) queries to F and O(n) random bits (independent of theta), improving previous work by Bshouty et al. [Bshouty et al., 2004].
Cite as
William M. Hoza and Adam R. Klivans. Preserving Randomness for Adaptive Algorithms. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 43:1-43:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{hoza_et_al:LIPIcs.APPROX-RANDOM.2018.43,
author = {Hoza, William M. and Klivans, Adam R.},
title = {{Preserving Randomness for Adaptive Algorithms}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {43:1--43:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.43},
URN = {urn:nbn:de:0030-drops-94477},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.43},
annote = {Keywords: pseudorandomness, adaptivity, estimation}
}
Document
Commutative Algorithms Approximate the LLL-distribution
Abstract
Following the groundbreaking Moser-Tardos algorithm for the Lovász Local Lemma (LLL), a series of works have exploited a key ingredient of the original analysis, the witness tree lemma, in order to: derive deterministic, parallel and distributed algorithms for the LLL, to estimate the entropy of the output distribution, to partially avoid bad events, to deal with super-polynomially many bad events, and even to devise new algorithmic frameworks. Meanwhile, a parallel line of work has established tools for analyzing stochastic local search algorithms motivated by the LLL that do not fall within the Moser-Tardos framework. Unfortunately, the aforementioned results do not transfer to these more general settings. Mainly, this is because the witness tree lemma, provably, does not longer hold. Here we prove that for commutative algorithms, a class recently introduced by Kolmogorov and which captures the vast majority of LLL applications, the witness tree lemma does hold. Armed with this fact, we extend the main result of Haeupler, Saha, and Srinivasan to commutative algorithms, establishing that the output of such algorithms well-approximates the LLL-distribution, i.e., the distribution obtained by conditioning on all bad events being avoided, and give several new applications. For example, we show that the recent algorithm of Molloy for list coloring number of sparse, triangle-free graphs can output exponential many list colorings of the input graph.
Cite as
Fotis Iliopoulos. Commutative Algorithms Approximate the LLL-distribution. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 44:1-44:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{iliopoulos:LIPIcs.APPROX-RANDOM.2018.44,
author = {Iliopoulos, Fotis},
title = {{Commutative Algorithms Approximate the LLL-distribution}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {44:1--44:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.44},
URN = {urn:nbn:de:0030-drops-94487},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.44},
annote = {Keywords: Lovasz Local Lemma, Local Search, Commutativity, LLL-distribution, Coloring Triangle-free Graphs}
}
Document
The Cover Time of a Biased Random Walk on a Random Regular Graph of Odd Degree
Abstract
We consider a random walk process, introduced by Orenshtein and Shinkar [Tal Orenshtein and Igor Shinkar, 2014], which prefers to visit previously unvisited edges, on the random r-regular graph G_r for any odd r >= 3. We show that this random walk process has asymptotic vertex and edge cover times 1/(r-2)n log n and r/(2(r-2))n log n, respectively, generalizing the result from [Cooper et al., to appear] from r = 3 to any larger odd r. This completes the study of the vertex cover time for fixed r >= 3, with [Petra Berenbrink et al., 2015] having previously shown that G_r has vertex cover time asymptotic to rn/2 when r >= 4 is even.
Cite as
Tony Johansson. The Cover Time of a Biased Random Walk on a Random Regular Graph of Odd Degree. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 45:1-45:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{johansson:LIPIcs.APPROX-RANDOM.2018.45,
author = {Johansson, Tony},
title = {{The Cover Time of a Biased Random Walk on a Random Regular Graph of Odd Degree}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {45:1--45:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.45},
URN = {urn:nbn:de:0030-drops-94494},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.45},
annote = {Keywords: Random walk, random regular graph, cover time}
}
Document
Satisfiability and Derandomization for Small Polynomial Threshold Circuits
Abstract
A polynomial threshold function (PTF) is defined as the sign of a polynomial p : {0,1}^n ->R. A PTF circuit is a Boolean circuit whose gates are PTFs. We study the problems of exact and (promise) approximate counting for PTF circuits of constant depth.
- Satisfiability (#SAT). We give the first zero-error randomized algorithm faster than exhaustive search that counts the number of satisfying assignments of a given constant-depth circuit with a super-linear number of wires whose gates are s-sparse PTFs, for s almost quadratic in the input size of the circuit; here a PTF is called s-sparse if its underlying polynomial has at most s monomials. More specifically, we show that, for any large enough constant c, given a depth-d circuit with (n^{2-1/c})-sparse PTF gates that has at most n^{1+epsilon_d} wires, where epsilon_d depends only on c and d, the number of satisfying assignments of the circuit can be computed in randomized time 2^{n-n^{epsilon_d}} with zero error. This generalizes the result by Chen, Santhanam and Srinivasan (CCC, 2016) who gave a SAT algorithm for constant-depth circuits of super-linear wire complexity with linear threshold function (LTF) gates only.
- Quantified derandomization. The quantified derandomization problem, introduced by Goldreich and Wigderson (STOC, 2014), asks to compute the majority value of a given Boolean circuit, under the promise that the minority-value inputs to the circuit are very few. We give a quantified derandomization algorithm for constant-depth PTF circuits with a super-linear number of wires that runs in quasi-polynomial time. More specifically, we show that for any sufficiently large constant c, there is an algorithm that, given a degree-Delta PTF circuit C of depth d with n^{1+1/c^d} wires such that C has at most 2^{n^{1-1/c}} minority-value inputs, runs in quasi-polynomial time exp ((log n)^{O (Delta^2)}) and determines the majority value of C. (We obtain a similar quantified derandomization result for PTF circuits with n^{Delta}-sparse PTF gates.) This extends the recent result of Tell (STOC, 2018) for constant-depth LTF circuits of super-linear wire complexity.
- Pseudorandom generators. We show how the classical Nisan-Wigderson (NW) generator (JCSS, 1994) yields a nontrivial pseudorandom generator for PTF circuits (of unrestricted depth) with sub-linearly many gates. As a corollary, we get a PRG for degree-Delta PTFs with the seed length exp (sqrt{Delta * log n})* log^2(1/epsilon).
Cite as
Valentine Kabanets and Zhenjian Lu. Satisfiability and Derandomization for Small Polynomial Threshold Circuits. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 46:1-46:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{kabanets_et_al:LIPIcs.APPROX-RANDOM.2018.46,
author = {Kabanets, Valentine and Lu, Zhenjian},
title = {{Satisfiability and Derandomization for Small Polynomial Threshold Circuits}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {46:1--46:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.46},
URN = {urn:nbn:de:0030-drops-94507},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.46},
annote = {Keywords: constant-depth circuits, polynomial threshold functions, circuit analysis algorithms, SAT, derandomization, quantified derandomization, pseudorandom generators.}
}
Document
High Order Random Walks: Beyond Spectral Gap
Abstract
We study high order random walks on high dimensional expanders on simplicial complexes (i.e., hypergraphs). These walks walk from a k-face (i.e., a k-hyperedge) to a k-face if they are both contained in a k+1-face (i.e, a k+1 hyperedge). This naturally generalizes the random walks on graphs that walk from a vertex (0-face) to a vertex if they are both contained in an edge (1-face).
Recent works have studied the spectrum of high order walks operators and deduced fast mixing. However, the spectral gap of high order walks operators is inherently small, due to natural obstructions (called coboundaries) that do not happen for walks on expander graphs.
In this work we go beyond spectral gap, and relate the expansion of a function on k-faces (called k-cochain, for k=0, this is a function on vertices) to its structure.
We show a Decomposition Theorem: For every k-cochain defined on high dimensional expander, there exists a decomposition of the cochain into i-cochains such that the square norm of the k-cochain is a sum of the square norms of the i-chains and such that the more weight the k-cochain has on higher levels of the decomposition the better is its expansion, or equivalently, the better is its shrinkage by the high order random walk operator.
The following corollaries are implied by the Decomposition Theorem:
- We characterize highly expanding k-cochains as those whose mass is concentrated on the highest levels of the decomposition that we construct. For example, a function on edges (i.e. a 1-cochain) which is locally thin (i.e. it contains few edges through every vertex) is highly expanding, while a function on edges that contains all edges through a single vertex is not highly expanding.
- We get optimal mixing for high order random walks on Ramanujan complexes. Ramanujan complexes are recently discovered bounded degree high dimensional expanders. The optimality in their mixing that we prove here, enable us to get from them more efficient Two-Layer-Samplers than those presented by the previous work of Dinur and Kaufman.
Cite as
Tali Kaufman and Izhar Oppenheim. High Order Random Walks: Beyond Spectral Gap. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 47:1-47:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{kaufman_et_al:LIPIcs.APPROX-RANDOM.2018.47,
author = {Kaufman, Tali and Oppenheim, Izhar},
title = {{High Order Random Walks: Beyond Spectral Gap}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {47:1--47:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.47},
URN = {urn:nbn:de:0030-drops-94516},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.47},
annote = {Keywords: High Dimensional Expanders, Simplicial Complexes, Random Walk}
}
Document
Improved Composition Theorems for Functions and Relations
Abstract
One of the central problems in complexity theory is to prove super-logarithmic depth bounds for circuits computing a problem in P, i.e., to prove that P is not contained in NC^1. As an approach for this question, Karchmer, Raz and Wigderson [Mauricio Karchmer et al., 1995] proposed a conjecture called the KRW conjecture, which if true, would imply that P is not cotained in NC^{1}.
Since proving this conjecture is currently considered an extremely difficult problem, previous works by Edmonds, Impagliazzo, Rudich and Sgall [Edmonds et al., 2001], Håstad and Wigderson [Johan Håstad and Avi Wigderson, 1990] and Gavinsky, Meir, Weinstein and Wigderson [Dmitry Gavinsky et al., 2014] considered weaker variants of the conjecture. In this work we significantly improve the parameters in these variants, achieving almost tight lower bounds.
Cite as
Sajin Koroth and Or Meir. Improved Composition Theorems for Functions and Relations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 48:1-48:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{koroth_et_al:LIPIcs.APPROX-RANDOM.2018.48,
author = {Koroth, Sajin and Meir, Or},
title = {{Improved Composition Theorems for Functions and Relations}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {48:1--48:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.48},
URN = {urn:nbn:de:0030-drops-94525},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.48},
annote = {Keywords: circuit complexity, communication complexity, KRW conjecture, composition}
}
Document
Round Complexity Versus Randomness Complexity in Interactive Proofs
Abstract
Consider an interactive proof system for some set S that has randomness complexity r(n) for instances of length n, and arbitrary round complexity. We show a public-coin interactive proof system for S of round complexity O(r(n)/log n). Furthermore, the randomness complexity is preserved up to a constant factor, and the resulting interactive proof system has perfect completeness.
Cite as
Maya Leshkowitz. Round Complexity Versus Randomness Complexity in Interactive Proofs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 49:1-49:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{leshkowitz:LIPIcs.APPROX-RANDOM.2018.49,
author = {Leshkowitz, Maya},
title = {{Round Complexity Versus Randomness Complexity in Interactive Proofs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {49:1--49:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.49},
URN = {urn:nbn:de:0030-drops-94530},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.49},
annote = {Keywords: Interactive Proofs}
}
Document
Improved List-Decodability of Random Linear Binary Codes
Abstract
There has been a great deal of work establishing that random linear codes are as list-decodable as uniformly random codes, in the sense that a random linear binary code of rate 1 - H(p) - epsilon is (p,O(1/epsilon))-list-decodable with high probability. In this work, we show that such codes are (p, H(p)/epsilon + 2)-list-decodable with high probability, for any p in (0, 1/2) and epsilon > 0. In addition to improving the constant in known list-size bounds, our argument - which is quite simple - works simultaneously for all values of p, while previous works obtaining L = O(1/epsilon) patched together different arguments to cover different parameter regimes.
Our approach is to strengthen an existential argument of (Guruswami, Håstad, Sudan and Zuckerman, IEEE Trans. IT, 2002) to hold with high probability. To complement our upper bound for random linear binary codes, we also improve an argument of (Guruswami, Narayanan, IEEE Trans. IT, 2014) to obtain a tight lower bound of 1/epsilon on the list size of uniformly random binary codes; this implies that random linear binary codes are in fact more list-decodable than uniformly random binary codes, in the sense that the list sizes are strictly smaller.
To demonstrate the applicability of these techniques, we use them to (a) obtain more information about the distribution of list sizes of random linear binary codes and (b) to prove a similar result for random linear rank-metric codes.
Cite as
Ray Li and Mary Wootters. Improved List-Decodability of Random Linear Binary Codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 50:1-50:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{li_et_al:LIPIcs.APPROX-RANDOM.2018.50,
author = {Li, Ray and Wootters, Mary},
title = {{Improved List-Decodability of Random Linear Binary Codes}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {50:1--50:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.50},
URN = {urn:nbn:de:0030-drops-94547},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.50},
annote = {Keywords: List-decoding, Random linear codes, Rank-metric codes}
}
Document
Sunflowers and Quasi-Sunflowers from Randomness Extractors
Abstract
The Erdös-Rado sunflower theorem (Journal of Lond. Math. Soc. 1960) is a fundamental result in combinatorics, and the corresponding sunflower conjecture is a central open problem. Motivated by applications in complexity theory, Rossman (FOCS 2010) extended the result to quasi-sunflowers, where similar conjectures emerge about the optimal parameters for which it holds.
In this work, we exhibit a surprising connection between the existence of sunflowers and quasi-sunflowers in large enough set systems, and the problem of constructing (or existing) certain randomness extractors. This allows us to re-derive the known results in a systematic manner, and to reduce the relevant conjectures to the problem of obtaining improved constructions of the randomness extractors.
Cite as
Xin Li, Shachar Lovett, and Jiapeng Zhang. Sunflowers and Quasi-Sunflowers from Randomness Extractors. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 51:1-51:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{li_et_al:LIPIcs.APPROX-RANDOM.2018.51,
author = {Li, Xin and Lovett, Shachar and Zhang, Jiapeng},
title = {{Sunflowers and Quasi-Sunflowers from Randomness Extractors}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {51:1--51:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.51},
URN = {urn:nbn:de:0030-drops-94555},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.51},
annote = {Keywords: Sunflower conjecture, Quasi-sunflowers, Randomness Extractors}
}
Document
Torpid Mixing of Markov Chains for the Six-vertex Model on Z^2
Abstract
In this paper, we study the mixing time of two widely used Markov chain algorithms for the six-vertex model, Glauber dynamics and the directed-loop algorithm, on the square lattice Z^2. We prove, for the first time that, on finite regions of the square lattice these Markov chains are torpidly mixing under parameter settings in the ferroelectric phase and the anti-ferroelectric phase.
Cite as
Tianyu Liu. Torpid Mixing of Markov Chains for the Six-vertex Model on Z^2. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 52:1-52:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{liu:LIPIcs.APPROX-RANDOM.2018.52,
author = {Liu, Tianyu},
title = {{Torpid Mixing of Markov Chains for the Six-vertex Model on Z^2}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {52:1--52:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.52},
URN = {urn:nbn:de:0030-drops-94568},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.52},
annote = {Keywords: the six-vertex model, Eulerian orientations, square lattice, torpid mixing}
}
Document
On the Testability of Graph Partition Properties
Abstract
In this work we study the testability of a family of graph partition properties that generalizes a family previously studied by Goldreich, Goldwasser, and Ron (Journal of the ACM, 1998 ). While the family studied by Goldreich, Goldwasser, and Ron includes a variety of natural properties, such as k-colorability and containing a large cut, it does not include other properties of interest, such as split graphs, and more generally (p,q)-colorable graphs. The generalization we consider allows us to impose constraints on the edge-densities within and between parts (relative to the sizes of the parts). We denote the family studied in this work by GPP.
We first show that all properties in GPP have a testing algorithm whose query complexity is polynomial in 1/epsilon, where epsilon is the given proximity parameter (and there is no dependence on the size of the graph). As the testing algorithm has two-sided error, we next address the question of which properties in GPP can be tested with one-sided error and query complexity polynomial in 1/epsilon. We answer this question by establishing a characterization result. Namely, we define a subfamily GPP_{0,1} of GPP and show that a property P in GPP is testable by a one-sided error algorithm that has query complexity poly(1/epsilon) if and only if P in GPP_{0,1}.
Cite as
Yonatan Nakar and Dana Ron. On the Testability of Graph Partition Properties. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 53:1-53:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{nakar_et_al:LIPIcs.APPROX-RANDOM.2018.53,
author = {Nakar, Yonatan and Ron, Dana},
title = {{On the Testability of Graph Partition Properties}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {53:1--53:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.53},
URN = {urn:nbn:de:0030-drops-94572},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.53},
annote = {Keywords: Graph Partition Properties}
}
Document
On Closeness to k-Wise Uniformity
Abstract
A probability distribution over {-1, 1}^n is (epsilon, k)-wise uniform if, roughly, it is epsilon-close to the uniform distribution when restricted to any k coordinates. We consider the problem of how far an (epsilon, k)-wise uniform distribution can be from any globally k-wise uniform distribution. We show that every (epsilon, k)-wise uniform distribution is O(n^{k/2}epsilon)-close to a k-wise uniform distribution in total variation distance. In addition, we show that this bound is optimal for all even k: we find an (epsilon, k)-wise uniform distribution that is Omega(n^{k/2}epsilon)-far from any k-wise uniform distribution in total variation distance. For k=1, we get a better upper bound of O(epsilon), which is also optimal.
One application of our closeness result is to the sample complexity of testing whether a distribution is k-wise uniform or delta-far from k-wise uniform. We give an upper bound of O(n^{k}/delta^2) (or O(log n/delta^2) when k = 1) on the required samples. We show an improved upper bound of O~(n^{k/2}/delta^2) for the special case of testing fully uniform vs. delta-far from k-wise uniform. Finally, we complement this with a matching lower bound of Omega(n/delta^2) when k = 2.
Our results improve upon the best known bounds from [Alon et al., 2007], and have simpler proofs.
Cite as
Ryan O'Donnell and Yu Zhao. On Closeness to k-Wise Uniformity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 54:1-54:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{odonnell_et_al:LIPIcs.APPROX-RANDOM.2018.54,
author = {O'Donnell, Ryan and Zhao, Yu},
title = {{On Closeness to k-Wise Uniformity}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {54:1--54:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.54},
URN = {urn:nbn:de:0030-drops-94581},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.54},
annote = {Keywords: k-wise independence, property testing, Fourier analysis, Boolean function}
}
Document
Pseudo-Derandomizing Learning and Approximation
Abstract
We continue the study of pseudo-deterministic algorithms initiated by Gat and Goldwasser [Eran Gat and Shafi Goldwasser, 2011]. A pseudo-deterministic algorithm is a probabilistic algorithm which produces a fixed output with high probability. We explore pseudo-determinism in the settings of learning and approximation. Our goal is to simulate known randomized algorithms in these settings by pseudo-deterministic algorithms in a generic fashion - a goal we succinctly term pseudo-derandomization. Learning. In the setting of learning with membership queries, we first show that randomized learning algorithms can be derandomized (resp. pseudo-derandomized) under the standard hardness assumption that E (resp. BPE) requires large Boolean circuits. Thus, despite the fact that learning is an algorithmic task that requires interaction with an oracle, standard hardness assumptions suffice to (pseudo-)derandomize it. We also unconditionally pseudo-derandomize any {quasi-polynomial} time learning algorithm for polynomial size circuits on infinitely many input lengths in sub-exponential time.
Next, we establish a generic connection between learning and derandomization in the reverse direction, by showing that deterministic (resp. pseudo-deterministic) learning algorithms for a concept class C imply hitting sets against C that are computable deterministically (resp. pseudo-deterministically). In particular, this suggests a new approach to constructing hitting set generators against AC^0[p] circuits by giving a deterministic learning algorithm for AC^0[p]. Approximation. Turning to approximation, we unconditionally pseudo-derandomize any poly-time randomized approximation scheme for integer-valued functions infinitely often in subexponential time over any samplable distribution on inputs. As a corollary, we get that the (0,1)-Permanent has a fully pseudo-deterministic approximation scheme running in sub-exponential time infinitely often over any samplable distribution on inputs.
Finally, we {investigate} the notion of approximate canonization of Boolean circuits. We use a connection between pseudodeterministic learning and approximate canonization to show that if BPE does not have sub-exponential size circuits infinitely often, then there is a pseudo-deterministic approximate canonizer for AC^0[p] computable in quasi-polynomial time.
Cite as
Igor Carboni Oliveira and Rahul Santhanam. Pseudo-Derandomizing Learning and Approximation. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 55:1-55:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{carbonioliveira_et_al:LIPIcs.APPROX-RANDOM.2018.55,
author = {Carboni Oliveira, Igor and Santhanam, Rahul},
title = {{Pseudo-Derandomizing Learning and Approximation}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {55:1--55:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.55},
URN = {urn:nbn:de:0030-drops-94598},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.55},
annote = {Keywords: derandomization, learning, approximation, boolean circuits}
}
Document
Luby-Velickovic-Wigderson Revisited: Improved Correlation Bounds and Pseudorandom Generators for Depth-Two Circuits
Abstract
We study correlation bounds and pseudorandom generators for depth-two circuits that consist of a SYM-gate (computing an arbitrary symmetric function) or THR-gate (computing an arbitrary linear threshold function) that is fed by S {AND} gates. Such circuits were considered in early influential work on unconditional derandomization of Luby, Velickovi{c}, and Wigderson [Michael Luby et al., 1993], who gave the first non-trivial PRG with seed length 2^{O(sqrt{log(S/epsilon)})} that epsilon-fools these circuits.
In this work we obtain the first strict improvement of [Michael Luby et al., 1993]'s seed length: we construct a PRG that epsilon-fools size-S {SYM,THR} oAND circuits over {0,1}^n with seed length 2^{O(sqrt{log S})} + polylog(1/epsilon), an exponential (and near-optimal) improvement of the epsilon-dependence of [Michael Luby et al., 1993]. The above PRG is actually a special case of a more general PRG which we establish for constant-depth circuits containing multiple SYM or THR gates, including as a special case {SYM,THR} o AC^0 circuits. These more general results strengthen previous results of Viola [Viola, 2006] and essentially strengthen more recent results of Lovett and Srinivasan [Lovett and Srinivasan, 2011].
Our improved PRGs follow from improved correlation bounds, which are transformed into PRGs via the Nisan-Wigderson "hardness versus randomness" paradigm [Nisan and Wigderson, 1994]. The key to our improved correlation bounds is the use of a recent powerful multi-switching lemma due to Håstad [Johan Håstad, 2014].
Cite as
Rocco A. Servedio and Li-Yang Tan. Luby-Velickovic-Wigderson Revisited: Improved Correlation Bounds and Pseudorandom Generators for Depth-Two Circuits. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 56:1-56:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{servedio_et_al:LIPIcs.APPROX-RANDOM.2018.56,
author = {Servedio, Rocco A. and Tan, Li-Yang},
title = {{Luby-Velickovic-Wigderson Revisited: Improved Correlation Bounds and Pseudorandom Generators for Depth-Two Circuits}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {56:1--56:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.56},
URN = {urn:nbn:de:0030-drops-94601},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.56},
annote = {Keywords: Pseudorandom generators, correlation bounds, constant-depth circuits}
}
Document
Randomly Coloring Graphs of Logarithmically Bounded Pathwidth
Abstract
We consider the problem of sampling a proper k-coloring of a graph of maximal degree Delta uniformly at random. We describe a new Markov chain for sampling colorings, and show that it mixes rapidly on graphs of logarithmically bounded pathwidth if k >=(1+epsilon)Delta, for any epsilon>0, using a hybrid paths argument.
Cite as
Shai Vardi. Randomly Coloring Graphs of Logarithmically Bounded Pathwidth. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 57:1-57:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{vardi:LIPIcs.APPROX-RANDOM.2018.57,
author = {Vardi, Shai},
title = {{Randomly Coloring Graphs of Logarithmically Bounded Pathwidth}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {57:1--57:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.57},
URN = {urn:nbn:de:0030-drops-94618},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.57},
annote = {Keywords: Random coloring, Glauber dynamics, Markov-chain Monte Carlo}
}
Document
Explicit Strong LTCs with Inverse Poly-Log Rate and Constant Soundness
Abstract
An error-correcting code C subseteq F^n is called (q,epsilon)-strong locally testable code (LTC) if there exists a tester that makes at most q queries to the input word. This tester accepts all codewords with probability 1 and rejects all non-codewords x not in C with probability at least epsilon * delta(x,C), where delta(x,C) denotes the relative Hamming distance between the word x and the code C. The parameter q is called the query complexity and the parameter epsilon is called soundness.
Goldreich and Sudan (J.ACM 2006) asked about the existence of strong LTCs with constant query complexity, constant relative distance, constant soundness and inverse polylogarithmic rate. They also asked about the explicit constructions of these codes.
Strong LTCs with the required range of parameters were obtained recently in the works of Viderman (CCC 2013, FOCS 2013) based on the papers of Meir (SICOMP 2009) and Dinur (J.ACM 2007). However, the construction of these codes was probabilistic.
In this work we show that codes presented in the works of Dinur (J.ACM 2007) and Ben-Sasson and Sudan (SICOMP 2005) provide the explicit construction of strong LTCs with the above range of parameters. Previously, such codes were proven to be weak LTCs. Using the results of Viderman (CCC 2013, FOCS 2013) we prove that such codes are, in fact, strong LTCs.
Cite as
Michael Viderman. Explicit Strong LTCs with Inverse Poly-Log Rate and Constant Soundness. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 58:1-58:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
@InProceedings{viderman:LIPIcs.APPROX-RANDOM.2018.58,
author = {Viderman, Michael},
title = {{Explicit Strong LTCs with Inverse Poly-Log Rate and Constant Soundness}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {58:1--58:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.58},
URN = {urn:nbn:de:0030-drops-94620},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.58},
annote = {Keywords: Error-Correcting Codes, Tensor Products, Locally Testable Codes}
}