Volume
LIPIcs, Volume 81
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)
Publication Details
- published at: 2017-08-11
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-044-6
- DBLP: db/conf/approx/approx2017
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Complete Volume
LIPIcs, Volume 81, APPROX/RANDOM'17, Complete Volume
Abstract
LIPIcs, Volume 81, APPROX/RANDOM'17, Complete Volume
Cite as
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@Proceedings{jansen_et_al:LIPIcs.APPROX-RANDOM.2017,
title = {{LIPIcs, Volume 81, APPROX/RANDOM'17, Complete Volume}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017},
URN = {urn:nbn:de:0030-drops-77101},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017},
annote = {Keywords: Network Architecture and Design, Coding and Information Theory, Error Control Codes, Modes of Computation: Online computation, Complexity Measures and Classes, Analysis of Algorithms and Problem Complexity, Numerical Algorithms and Problems, Nonnumerical Algorithms and Problems}
}
Document
Front Matter
Frontmatter, Table of Contents, Preface, Organization, External Reviewers, List of Authors
Abstract
Frontmatter, Table of Contents, Preface, Organization, External Reviewers, List of Authors
Cite as
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, p. 0:i, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{jansen_et_al:LIPIcs.APPROX-RANDOM.2017.0,
author = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
title = {{Frontmatter, Table of Contents, Preface, Organization, External Reviewers, List of Authors}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {0:i--0:i},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.0},
URN = {urn:nbn:de:0030-drops-75493},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.0},
annote = {Keywords: Frontmatter, Table of Contents, Preface, Organization, External Reviewers, List of Authors}
}
Document
Min-Cost Bipartite Perfect Matching with Delays
Abstract
In the min-cost bipartite perfect matching with delays (MBPMD) problem, requests arrive online at points of a finite metric space. Each request is either positive or negative and has to be matched to a request of opposite polarity. As opposed to traditional online matching problems, the algorithm does not have to serve requests as they arrive, and may choose to match them later at a cost. Our objective is to minimize the sum of the distances between matched pairs of requests (the connection cost) and the sum of the waiting times of the requests (the delay cost). This objective exhibits a natural tradeoff between minimizing the distances and the cost of waiting for better matches. This tradeoff appears in many real-life scenarios, notably, ride-sharing platforms. MBPMD is related to its non-bipartite variant, min-cost perfect matching with delays (MPMD), in which each request can be matched to any other request. MPMD was introduced by Emek et al. (STOC'16), who showed an O(log^2(n)+log(Delta))-competitive randomized algorithm on n-point metric spaces with aspect ratio Delta.
Our contribution is threefold. First, we present a new lower bound construction for MPMD and MBPMD. We get a lower bound of Omega(sqrt(log(n)/log(log(n)))) on the competitive ratio of any randomized algorithm for MBPMD. For MPMD, we improve the lower bound from Omega(sqrt(log(n))) (shown by Azar et al., SODA'17) to Omega(log(n)/log(log(n))), thus, almost matching their upper bound of O(log(n)). Second, we adapt the algorithm of Emek et al. to the bipartite case, and provide a simplified analysis that improves the competitive ratio to O(log(n)). The key ingredient of the algorithm is an O(h)-competitive randomized algorithm for MBPMD on weighted trees of height h. Third, we provide an O(h)-competitive deterministic algorithm for MBPMD on weighted trees of height h. This algorithm is obtained by adapting the algorithm for MPMD by Azar et al. to the apparently more complicated bipartite setting.
Cite as
Itai Ashlagi, Yossi Azar, Moses Charikar, Ashish Chiplunkar, Ofir Geri, Haim Kaplan, Rahul Makhijani, Yuyi Wang, and Roger Wattenhofer. Min-Cost Bipartite Perfect Matching with Delays. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 1:1-1:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{ashlagi_et_al:LIPIcs.APPROX-RANDOM.2017.1,
author = {Ashlagi, Itai and Azar, Yossi and Charikar, Moses and Chiplunkar, Ashish and Geri, Ofir and Kaplan, Haim and Makhijani, Rahul and Wang, Yuyi and Wattenhofer, Roger},
title = {{Min-Cost Bipartite Perfect Matching with Delays}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {1:1--1:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.1},
URN = {urn:nbn:de:0030-drops-75509},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.1},
annote = {Keywords: online algorithms with delayed service, bipartite matching, competitive analysis}
}
Document
Global and Fixed-Terminal Cuts in Digraphs
Abstract
The computational complexity of multicut-like problems may vary significantly depending on whether the terminals are fixed or not. In this work we present a comprehensive study of this phenomenon in two types of cut problems in directed graphs: double cut and bicut.
1. Fixed-terminal edge-weighted double cut is known to be solvable efficiently. We show that fixed-terminal node-weighted double cut cannot be approximated to a factor smaller than 2 under the Unique Games Conjecture (UGC), and we also give a 2-approximation algorithm. For the global version of the problem, we prove an inapproximability bound of 3/2 under UGC.
2. Fixed-terminal edge-weighted bicut is known to have an approximability factor of 2 that is tight under UGC. We show that the global edge-weighted bicut is approximable to
a factor strictly better than 2, and that the global node-weighted bicut cannot be approximated to a factor smaller than 3/2 under UGC.
3. In relation to these investigations, we also prove two results on undirected graphs which are of independent interest. First, we show NP-completeness and a tight inapproximability bound of 4/3 for the node-weighted 3-cut problem under UGC. Second, we show that for constant k, there exists an efficient algorithm to solve the minimum {s,t}-separating k-cut problem.
Our techniques for the algorithms are combinatorial, based on LPs and based on the enumeration of approximate min-cuts. Our hardness results are based on combinatorial reductions and integrality gap instances.
Cite as
Kristóf Bérczi, Karthekeyan Chandrasekaran, Tamás Király, Euiwoong Lee, and Chao Xu. Global and Fixed-Terminal Cuts in Digraphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 2:1-2:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{berczi_et_al:LIPIcs.APPROX-RANDOM.2017.2,
author = {B\'{e}rczi, Krist\'{o}f and Chandrasekaran, Karthekeyan and Kir\'{a}ly, Tam\'{a}s and Lee, Euiwoong and Xu, Chao},
title = {{Global and Fixed-Terminal Cuts in Digraphs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {2:1--2:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.2},
URN = {urn:nbn:de:0030-drops-75511},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.2},
annote = {Keywords: Directed Graphs, Arborescence, Graph Cuts, Hardness of Approximation}
}
Document
A PTAS for Three-Edge-Connected Survivable Network Design in Planar Graphs
Abstract
We consider the problem of finding the minimum-weight subgraph that satisfies given connectivity requirements. Specifically, given a requirement r in {0, 1, 2, 3} for every vertex, we seek the minimum-weight subgraph that contains, for every pair of vertices u and v, at least min{r(v), r(u)} edge-disjoint u-to-v paths. We give a polynomial-time approximation scheme (PTAS) for this problem when the input graph is planar and the subgraph may use multiple copies of any given edge (paying for each edge separately). This generalizes an earlier result for r in {0, 1, 2}. In order to achieve this PTAS, we prove some properties of triconnected planar graphs that may be of independent interest.
Cite as
Glencora Borradaile and Baigong Zheng. A PTAS for Three-Edge-Connected Survivable Network Design in Planar Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 3:1-3:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{borradaile_et_al:LIPIcs.APPROX-RANDOM.2017.3,
author = {Borradaile, Glencora and Zheng, Baigong},
title = {{A PTAS for Three-Edge-Connected Survivable Network Design in Planar Graphs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {3:1--3:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.3},
URN = {urn:nbn:de:0030-drops-75525},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.3},
annote = {Keywords: Three-Edge Connectivity, Polynomial-Time Approximation Schemes, Planar Graphs}
}
Document
The Quest for Strong Inapproximability Results with Perfect Completeness
Abstract
The Unique Games Conjecture (UGC) has pinned down the approximability of all constraint satisfaction problems (CSPs), showing that a natural semidefinite programming relaxation offers the optimal worst-case approximation ratio for any CSP. This elegant picture, however, does not apply for CSP instances that are perfectly satisfiable, due to the imperfect completeness inherent in the UGC. For the important case when the input CSP instance admits a satisfying assignment, it therefore remains wide open to understand how well it can be approximated.
This work is motivated by the pursuit of a better understanding of the inapproximability of perfectly satisfiable instances of CSPs. Our main conceptual contribution is the formulation of a (hypergraph) version of Label Cover which we call "V label cover." Assuming a conjecture concerning the inapproximability of V label cover on perfectly satisfiable instances, we prove the following implications:
* There is an absolute constant c0 such that for k >= 3, given a satisfiable instance of Boolean k-CSP, it is hard to find an assignment satisfying more than c0 k^2/2^k fraction of the constraints.
* Given a k-uniform hypergraph, k >= 2, for all epsilon > 0, it is hard to tell if it is q-strongly colorable or has no independent set with an epsilon fraction of vertices, where q = ceiling[k + sqrt(k) - 0.5].
* Given a k-uniform hypergraph, k >= 3, for all epsilon > 0, it is hard to tell if it is (k-1)-rainbow colorable or has no independent set with an epsilon fraction of vertices.
We further supplement the above results with a proof that an ``almost Unique'' version of Label Cover can be approximated within a constant factor on satisfiable instances.
Cite as
Joshua Brakensiek and Venkatesan Guruswami. The Quest for Strong Inapproximability Results with Perfect Completeness. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{brakensiek_et_al:LIPIcs.APPROX-RANDOM.2017.4,
author = {Brakensiek, Joshua and Guruswami, Venkatesan},
title = {{The Quest for Strong Inapproximability Results with Perfect Completeness}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {4:1--4:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.4},
URN = {urn:nbn:de:0030-drops-75537},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.4},
annote = {Keywords: inapproximability, hardness of approximation, dictatorship testing, constraint satisfaction, hypergraph coloring}
}
Document
Scheduling Problems over Network of Machines
Abstract
We consider scheduling problems in which jobs need to be processed through a (shared) network of machines. The network is given in the form of a graph the edges of which represent the machines. We are also given a set of jobs, each specified by its processing time and a path in the graph. Every job needs to be processed in the order of edges specified by its path. We assume that jobs can wait between machines and preemption is not allowed; that is, once a job is started being processed on a machine, it must be completed without interruption. Every machine can only process one job at a time.
The makespan of a schedule is the earliest time by which all the jobs have finished processing. The flow time (a.k.a. the completion time) of a job in a schedule is the difference in time between when it finishes processing on its last machine and when the it begins processing on its first machine. The total flow time (or the sum of completion times) is the sum of flow times (or completion times) of all jobs. Our focus is on finding schedules with the minimum sum of completion times or minimum makespan.
In this paper, we develop several algorithms (both approximate and exact) for the problem both on general graphs and when the underlying graph of machines is a tree. Even in the very special case when the underlying network is a simple star, the problem is very interesting as it models a biprocessor scheduling with applications to data migration.
Cite as
Zachary Friggstad, Arnoosh Golestanian, Kamyar Khodamoradi, Christopher Martin, Mirmahdi Rahgoshay, Mohsen Rezapour, Mohammad R. Salavatipour, and Yifeng Zhang. Scheduling Problems over Network of Machines. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{friggstad_et_al:LIPIcs.APPROX-RANDOM.2017.5,
author = {Friggstad, Zachary and Golestanian, Arnoosh and Khodamoradi, Kamyar and Martin, Christopher and Rahgoshay, Mirmahdi and Rezapour, Mohsen and Salavatipour, Mohammad R. and Zhang, Yifeng},
title = {{Scheduling Problems over Network of Machines}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {5:1--5:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.5},
URN = {urn:nbn:de:0030-drops-75547},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.5},
annote = {Keywords: approximation algorithms, job-shop scheduling, min-sum edge coloring, minimum latency}
}
Document
Approximating Incremental Combinatorial Optimization Problems
Abstract
We consider incremental combinatorial optimization problems, in which a solution is constructed incrementally over time, and the goal is to optimize not the value of the final solution but the average value over all timesteps. We consider a natural algorithm of moving towards a global optimum solution as quickly as possible. We show that this algorithm provides an approximation guarantee of (9+sqrt(21))/15 > 0.9 for a large class of incremental combinatorial optimization problems defined axiomatically, which includes (bipartite and non-bipartite) matchings, matroid intersections, and stable sets in claw-free graphs. Furthermore, our analysis is tight.
Cite as
Michel X. Goemans and Francisco Unda. Approximating Incremental Combinatorial Optimization Problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{goemans_et_al:LIPIcs.APPROX-RANDOM.2017.6,
author = {Goemans, Michel X. and Unda, Francisco},
title = {{Approximating Incremental Combinatorial Optimization Problems}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {6:1--6:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.6},
URN = {urn:nbn:de:0030-drops-75559},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.6},
annote = {Keywords: Approximation algorithm, matching, incremental problems, matroid intersection, integral polytopes, stable sets}
}
Document
Stochastic Unsplittable Flows
Abstract
We consider the stochastic unsplittable flow problem: given a graph with edge-capacities, and source-sink pairs with each pair having a size and a value, the goal is to route the pairs unsplittably while respecting edge capacities to maximize the total value of the routed pairs. However, the size of each pair is a random variable and is revealed only after we decide to route that pair. Which pairs should we route, along which paths, and in what order so as to maximize the expected value?
We present results for several cases of the problem under the no-bottleneck assumption. We show a logarithmic approximation algorithm for the single-sink problem on general graphs, considerably improving on the prior results of Chawla and Roughgarden which worked for planar graphs. We present an approximation to the stochastic unsplittable flow problem on directed acyclic graphs, within less than a logarithmic factor of the best known approximation in the non-stochastic setting. We present a non-adaptive strategy on trees that is within a constant factor of the best adaptive strategy, asymptotically matching the best results for the non-stochastic unsplittable flow problem on trees. Finally, we give results for the stochastic unsplittable flow problem on general graphs.
Our techniques include using edge-confluent flows for the single-sink problem in order to control the interaction between flow-paths, and a reduction from general scheduling policies to "safe" ones (i.e., those guaranteeing no capacity violations), which may be of broader interest.
Cite as
Anupam Gupta and Archit Karandikar. Stochastic Unsplittable Flows. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 7:1-7:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{gupta_et_al:LIPIcs.APPROX-RANDOM.2017.7,
author = {Gupta, Anupam and Karandikar, Archit},
title = {{Stochastic Unsplittable Flows}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {7:1--7:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.7},
URN = {urn:nbn:de:0030-drops-75569},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.7},
annote = {Keywords: Approximation Algorithms, Stochastic optimization, confluent flows, unsplittable flows}
}
Document
Streaming Complexity of Approximating Max 2CSP and Max Acyclic Subgraph
Abstract
We study the complexity of estimating the optimum value of a Boolean 2CSP (arity two constraint satisfaction problem) in the single-pass streaming setting, where the algorithm is presented the constraints in an arbitrary order. We give a streaming algorithm to estimate the optimum within a factor approaching 2/5 using logarithmic space, with high probability. This beats the trivial factor 1/4 estimate obtained by simply outputting 1/4-th of the total number of constraints.
The inspiration for our work is a lower bound of Kapralov, Khanna, and Sudan (SODA'15) who showed that a similar trivial estimate (of factor 1/2) is the best one can do for Max CUT. This lower bound implies that beating a factor 1/2 for Max DICUT (a special case of Max 2CSP), in particular, to distinguish between the case when the optimum is m/2 versus when it is at most (1/4+eps)m, where m is the total number of edges, requires polynomial space. We complement this hardness result by showing that for DICUT, one can distinguish between the case in which the optimum exceeds (1/2+eps)m and the case in which it is close to m/4.
We also prove that estimating the size of the maximum acyclic subgraph of a directed graph, when its edges are presented in a single-pass stream, within a factor better than 7/8 requires polynomial space.
Cite as
Venkatesan Guruswami, Ameya Velingker, and Santhoshini Velusamy. Streaming Complexity of Approximating Max 2CSP and Max Acyclic Subgraph. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 8:1-8:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{guruswami_et_al:LIPIcs.APPROX-RANDOM.2017.8,
author = {Guruswami, Venkatesan and Velingker, Ameya and Velusamy, Santhoshini},
title = {{Streaming Complexity of Approximating Max 2CSP and Max Acyclic Subgraph}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {8:1--8:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.8},
URN = {urn:nbn:de:0030-drops-75570},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.8},
annote = {Keywords: approximation algorithms, constraint satisfaction problems, optimization, hardness of approximation, maximum acyclic subgraph}
}
Document
Symmetric Interdiction for Matching Problems
Abstract
Motivated by denial-of-service network attacks, we introduce the symmetric interdiction model, where both the interdictor and the optimizer are subject to the same constraints of the underlying optimization problem. We give a general framework that relates optimization to symmetric interdiction for a broad class of optimization problems. We then study the symmetric matching interdiction problem - with applications in traffic engineering - in more detail. This problem can be simply stated as follows: find a matching whose removal minimizes the size of the maximum matching in the remaining graph. We show that this problem is APX-hard, and obtain a 3/2-approximation algorithm that improves on the approximation guarantee provided by the general framework.
Cite as
Samuel Haney, Bruce Maggs, Biswaroop Maiti, Debmalya Panigrahi, Rajmohan Rajaraman, and Ravi Sundaram. Symmetric Interdiction for Matching Problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{haney_et_al:LIPIcs.APPROX-RANDOM.2017.9,
author = {Haney, Samuel and Maggs, Bruce and Maiti, Biswaroop and Panigrahi, Debmalya and Rajaraman, Rajmohan and Sundaram, Ravi},
title = {{Symmetric Interdiction for Matching Problems}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {9:1--9:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.9},
URN = {urn:nbn:de:0030-drops-75587},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.9},
annote = {Keywords: Approximation algorithms, matching, interdiction Digital Object}
}
Document
A Lottery Model for Center-Type Problems with Outliers
Abstract
In this paper, we give tight approximation algorithms for the k-center and matroid center problems with outliers. Unfairness arises naturally in this setting: certain clients could always be considered as outliers. To address this issue, we introduce a lottery model in which each client is allowed to submit a parameter indicating the lower-bound on the probability that it should be covered and we look for a random solution that satisfies all the given requests. Out techniques include a randomized rounding procedure to round a point inside a matroid intersection polytope to a basis plus at most one extra item such that all marginal probabilities are preserved and such that a certain linear function of the variables does not decrease in the process with probability one.
Cite as
David G. Harris, Thomas Pensyl, Aravind Srinivasan, and Khoa Trinh. A Lottery Model for Center-Type Problems with Outliers. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 10:1-10:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{harris_et_al:LIPIcs.APPROX-RANDOM.2017.10,
author = {Harris, David G. and Pensyl, Thomas and Srinivasan, Aravind and Trinh, Khoa},
title = {{A Lottery Model for Center-Type Problems with Outliers}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {10:1--10:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.10},
URN = {urn:nbn:de:0030-drops-75596},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.10},
annote = {Keywords: approximation algorithms, randomized rounding, clustering problems}
}
Document
Streaming Algorithms for Maximizing Monotone Submodular Functions under a Knapsack Constraint
Abstract
In this paper, we consider the problem of maximizing a monotone submodular function subject to a knapsack constraint in the streaming setting. In particular, the elements arrive sequentially and at any point of time, the algorithm has access only to a small fraction of the data stored in primary memory. For this problem, we propose a (0.363-epsilon)-approximation algorithm, requiring only a single pass through the data; moreover, we propose a (0.4-epsilon)-approximation algorithm requiring a constant number of passes through the data. The required memory space of both algorithms depends only on the size of the knapsack capacity and epsilon.
Cite as
Chien-Chung Huang, Naonori Kakimura, and Yuichi Yoshida. Streaming Algorithms for Maximizing Monotone Submodular Functions under a Knapsack Constraint. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 11:1-11:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{huang_et_al:LIPIcs.APPROX-RANDOM.2017.11,
author = {Huang, Chien-Chung and Kakimura, Naonori and Yoshida, Yuichi},
title = {{Streaming Algorithms for Maximizing Monotone Submodular Functions under a Knapsack Constraint}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {11:1--11:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.11},
URN = {urn:nbn:de:0030-drops-75602},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.11},
annote = {Keywords: submodular functions, single-pass streaming, multiple-pass streaming, constant approximation}
}
Document
Fractional Set Cover in the Streaming Model
Abstract
We study the Fractional Set Cover problem in the streaming model. That is, we consider the relaxation of the set cover problem over a universe of n elements and a collection of m sets, where each set can be picked fractionally, with a value in [0,1]. We present a randomized (1+a)-approximation algorithm that makes p passes over the data, and uses O(polylog(m,n,1/a) (mn^(O(1/(pa)))+n)) memory space. The algorithm works in both the set arrival and the edge arrival models. To the best of our knowledge, this is the first streaming result for the fractional set cover problem. We obtain our results by employing the multiplicative weights update framework in the streaming settings.
Cite as
Piotr Indyk, Sepideh Mahabadi, Ronitt Rubinfeld, Jonathan Ullman, Ali Vakilian, and Anak Yodpinyanee. Fractional Set Cover in the Streaming Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 12:1-12:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{indyk_et_al:LIPIcs.APPROX-RANDOM.2017.12,
author = {Indyk, Piotr and Mahabadi, Sepideh and Rubinfeld, Ronitt and Ullman, Jonathan and Vakilian, Ali and Yodpinyanee, Anak},
title = {{Fractional Set Cover in the Streaming Model}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {12:1--12:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.12},
URN = {urn:nbn:de:0030-drops-75613},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.12},
annote = {Keywords: Streaming Algorithms, Fractional Set Cover, LP relaxation, Multiplicative Weight Update}
}
Document
Online Strip Packing with Polynomial Migration
Abstract
We consider the relaxed online strip packing problem, where rectangular items arrive online and have to be packed into a strip of fixed width such that the packing height is minimized. Thereby, repacking of previously packed items is allowed. The amount of repacking is measured by the migration factor, defined as the total size of repacked items divided by the size of the arriving item. First, we show that no algorithm with constant migration factor can produce solutions with asymptotic ratio better than 4/3. Against this background, we allow amortized migration, i.e. to save migration for a later time step. As a main result, we present an AFPTAS with asymptotic ratio 1 + O(epsilon) for any epsilon > 0 and amortized migration factor polynomial in 1/epsilon. To our best knowledge, this is the first algorithm for online strip packing considered in a repacking model.
Cite as
Klaus Jansen, Kim-Manuel Klein, Maria Kosche, and Leon Ladewig. Online Strip Packing with Polynomial Migration. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{jansen_et_al:LIPIcs.APPROX-RANDOM.2017.13,
author = {Jansen, Klaus and Klein, Kim-Manuel and Kosche, Maria and Ladewig, Leon},
title = {{Online Strip Packing with Polynomial Migration}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {13:1--13:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.13},
URN = {urn:nbn:de:0030-drops-75620},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.13},
annote = {Keywords: strip packing, bin packing, online algorithms, migration factor}
}
Document
Density Independent Algorithms for Sparsifying k-Step Random Walks
Abstract
We give faster algorithms for producing sparse approximations of the transition matrices of k-step random walks on undirected and weighted graphs. These transition matrices also form graphs, and arise as intermediate objects in a variety of graph algorithms. Our improvements are based on a better understanding of processes that sample such walks, as well as tighter bounds on key weights underlying these sampling processes. On a graph with n vertices and m edges, our algorithm produces a graph with about nlog(n) edges that approximates the k-step random walk graph in about m + k^2 nlog^4(n) time. In order to obtain this runtime bound, we also revisit "density independent" algorithms for sparsifying graphs whose runtime overhead is expressed only in terms of the number of vertices.
Cite as
Gorav Jindal, Pavel Kolev, Richard Peng, and Saurabh Sawlani. Density Independent Algorithms for Sparsifying k-Step Random Walks. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{jindal_et_al:LIPIcs.APPROX-RANDOM.2017.14,
author = {Jindal, Gorav and Kolev, Pavel and Peng, Richard and Sawlani, Saurabh},
title = {{Density Independent Algorithms for Sparsifying k-Step Random Walks}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {14:1--14:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.14},
URN = {urn:nbn:de:0030-drops-75638},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.14},
annote = {Keywords: random walks, graph sparsification, spectral graph theory, effective resistances}
}
Document
Maximum Matching in Two, Three, and a Few More Passes Over Graph Streams
Abstract
We consider the maximum matching problem in the semi-streaming model formalized by Feigenbaum, Kannan, McGregor, Suri, and Zhang that is inspired by giant graphs of today. As our main result, we give a two-pass (1/2 + 1/16)-approximation algorithm for triangle-free graphs and a two-pass (1/2 + 1/32)-approximation algorithm for general graphs; these improve the approximation ratios of 1/2 + 1/52 for bipartite graphs and 1/2 + 1/140 for general graphs by Konrad, Magniez, and Mathieu. In three passes, we achieve approximation ratios of 1/2 + 1/10 for triangle-free graphs and 1/2 + 1/19.753 for general graphs. We also give a multi-pass algorithm where we bound the number of passes precisely - we give a (2/3 - epsilon)-approximation algorithm that uses 2/(3 epsilon) passes for triangle-free graphs and 4/(3 epsilon) passes for general graphs. Our algorithms are simple and combinatorial, use O(n log(n)) space, and have O(1) update time per edge.
For general graphs, our multi-pass algorithm improves the best known deterministic algorithms in terms of the number of passes:
* Ahn and Guha give a (2/3 - epsilon)-approximation algorithm that uses O(log(1/epsilon)/epsilon^2) passes, whereas our (2/3 - epsilon)-approximation algorithm uses 4/(epsilon) passes;
* they also give a (1 - epsilon)-approximation algorithm that uses O(log(n) poly(1/epsilon)) passes, where n is the number of vertices of the input graph; although our algorithm is (2/3 - epsilon)-approximation, our number of passes do not depend on n.
Earlier multi-pass algorithms either have a large constant inside big-O notation for the number of passes or the constant cannot be determined due to the involved analysis, so our multi-pass algorithm should use much fewer passes for approximation ratios bounded slightly below 2/3.
Cite as
Sagar Kale and Sumedh Tirodkar. Maximum Matching in Two, Three, and a Few More Passes Over Graph Streams. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 15:1-15:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{kale_et_al:LIPIcs.APPROX-RANDOM.2017.15,
author = {Kale, Sagar and Tirodkar, Sumedh},
title = {{Maximum Matching in Two, Three, and a Few More Passes Over Graph Streams}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {15:1--15:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.15},
URN = {urn:nbn:de:0030-drops-75645},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.15},
annote = {Keywords: Semi Streaming, Maximum Matching}
}
Document
Submodular Secretary Problems: Cardinality, Matching, and Linear Constraints
Abstract
We study various generalizations of the secretary problem with submodular objective functions. Generally, a set of requests is revealed step-by-step to an algorithm in random order. For each request, one option has to be selected so as to maximize a monotone submodular function while ensuring feasibility. For our results, we assume that we are given an offline algorithm computing an alpha-approximation for the respective problem. This way, we separate computational limitations from the ones due to the online nature. When only focusing on the online aspect, we can assume alpha = 1.
In the submodular secretary problem, feasibility constraints are cardinality constraints, or equivalently, sets are feasible if and only if they are independent sets of a k-uniform matroid. That is, out of a randomly ordered stream of entities, one has to select a subset of size k. For this problem, we present a 0.31alpha-competitive algorithm for all k, which asymptotically reaches competitive ratio alpha/e for large k. In submodular secretary matching, one side of a bipartite graph is revealed online. Upon arrival, each node has to be matched permanently to an offline node or discarded irrevocably. We give a 0.207alpha-competitive algorithm. This also covers the problem, in which sets of entities are feasible if and only if they are independent with respect to a transversal matroid. In both cases, we improve over previously best known competitive ratios, using a generalization of the algorithm for the classic secretary problem.
Furthermore, we give an O(alpha d^(-2/(B-1)))-competitive algorithm for submodular function maximization subject to linear packing constraints. Here, d is the column sparsity, that is the maximal number of none-zero entries in a column of the constraint matrix, and B is the minimal capacity of the constraints. Notably, this bound is independent of the total number of constraints. We improve the algorithm to be O(alpha d^(-1/(B-1)))-competitive if both d and B are known to the algorithm beforehand.
Cite as
Thomas Kesselheim and Andreas Tönnis. Submodular Secretary Problems: Cardinality, Matching, and Linear Constraints. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 16:1-16:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{kesselheim_et_al:LIPIcs.APPROX-RANDOM.2017.16,
author = {Kesselheim, Thomas and T\"{o}nnis, Andreas},
title = {{Submodular Secretary Problems: Cardinality, Matching, and Linear Constraints}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {16:1--16:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.16},
URN = {urn:nbn:de:0030-drops-75657},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.16},
annote = {Keywords: Secretary Problem, Online Algorithms, Submodular Maximization}
}
Document
On the Integrality Gap of the Prize-Collecting Steiner Forest LP
Abstract
In the prize-collecting Steiner forest (PCSF) problem, we are given an undirected graph G=(V,E), nonnegative edge costs {c_e} for e in E, terminal pairs {(s_i,t_i)} for i=1,...,k, and penalties {pi_i} for i=1,...,k for each terminal pair; the goal is to find a forest F to minimize c(F) + sum{ pi_i: (s_i,t_i) is not connected in F }. The Steiner forest problem can be viewed as the special case where pi_i are infinite for all i. It was widely believed that the integrality gap of the natural (and well-studied) linear-programming (LP) relaxation for PCSF (PCSF-LP) is at most 2. We dispel this belief by showing that the integrality gap of this LP is at least 9/4 even if the input instance is planar. We also show that using this LP, one cannot devise a Lagrangian-multiplier-preserving (LMP) algorithm with approximation guarantee better than 4. Our results thus show a separation between the integrality gaps of the LP-relaxations for prize-collecting and non-prize-collecting (i.e., standard) Steiner forest, as well as the approximation ratios achievable relative to the optimal LP solution by LMP- and non-LMP-approximation algorithms for PCSF. For the special case of prize-collecting Steiner tree (PCST), we prove that the natural LP relaxation admits basic feasible solutions with all coordinates of value at most 1/3 and all edge variables positive. Thus, we rule out the possibility of approximating PCST with guarantee better than 3 using a direct iterative rounding method.
Cite as
Jochen Könemann, Neil Olver, Kanstantsin Pashkovich, R. Ravi, Chaitanya Swamy, and Jens Vygen. On the Integrality Gap of the Prize-Collecting Steiner Forest LP. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 17:1-17:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{konemann_et_al:LIPIcs.APPROX-RANDOM.2017.17,
author = {K\"{o}nemann, Jochen and Olver, Neil and Pashkovich, Kanstantsin and Ravi, R. and Swamy, Chaitanya and Vygen, Jens},
title = {{On the Integrality Gap of the Prize-Collecting Steiner Forest LP}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {17:1--17:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.17},
URN = {urn:nbn:de:0030-drops-75665},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.17},
annote = {Keywords: Integrality gap, Steiner tree, Steiner forest, prize-collecting, Lagrangianmultiplier- preserving}
}
Document
Approximating Unique Games Using Low Diameter Graph Decomposition
Abstract
We design approximation algorithms for Unique Gmeas when the constraint graph admits good low diameter graph decomposition. For the M2Lin(k) problem in K(r)-minor free graphs, when there is an assignment satisfying 1-eps fraction of constraints, we present an algorithm that produces an assignment satisfying 1-O(r*eps) fraction of constraints, with the approximation ratio independent of the alphabet size. A corollary is an improved approximation algorithm for the Min-UnCut problem for K(r)-minor free graphs. For general Unique Games in K(r)-minor free graphs, we provide another algorithm that produces an assignment satisfying 1-O(r *sqrt(eps)) fraction of constraints.
Our approach is to round a linear programming relaxation to find a minimum subset of edges that intersects all the inconsistent cycles. We show that it is possible to apply the low diameter graph decomposition technique on the constraint graph directly, rather than to work on the label extended graph as in previous algorithms for Unique Games. The same approach applies when the constraint graph is of genus g, and we get similar results with r replaced by log g in the M2Lin(k) problem and by sqrt(log g) in the general problem. The former result generalizes the result of Gupta-Talwar for Unique Games in the M2Lin(k) case, and the latter result generalizes the result of Trevisan for general Unique Games.
Cite as
Vedat Levi Alev and Lap Chi Lau. Approximating Unique Games Using Low Diameter Graph Decomposition. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{levialev_et_al:LIPIcs.APPROX-RANDOM.2017.18,
author = {Levi Alev, Vedat and Lau, Lap Chi},
title = {{Approximating Unique Games Using Low Diameter Graph Decomposition}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {18:1--18:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.18},
URN = {urn:nbn:de:0030-drops-75676},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.18},
annote = {Keywords: Unique Games, Low Diameter Graph Decomposition, Bounded Genus Graphs, Fixed Minor Free Graphs, Approximation Algorithms, Linear Programming}
}
Document
Greedy Minimization of Weakly Supermodular Set Functions
Abstract
Many problems in data mining and unsupervised machine learning take the form of minimizing a set function with cardinality constraints. More explicitly, denote by [n] the set {1,...,n} and let f(S) be a function from 2^[n] to R+. Our goal is to minimize f(S) subject to |S| <= k. These problems include clustering and covering problems as well as sparse regression, matrix approximation problems and many others. These combinatorial problems are hard to minimize in general. Finding good (e.g. constant factor) approximate solutions for them requires significant sophistication and highly specialized algorithms.
In this paper we analyze the behavior of the greedy algorithm to all of these problems. We start by claiming that the functions above are special. A trivial observation is that they are non-negative and non-increasing, that is, f(S) >= f(union(S,T)) >= 0 for any S and T. This immediately shows that expanding solution sets is (at least potentially) beneficial in terms of reducing the function value. But, monotonicity is not sufficient to ensure that any number of greedy extensions of a given solution would significantly reduce the objective function.
Cite as
Edo Liberty and Maxim Sviridenko. Greedy Minimization of Weakly Supermodular Set Functions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 19:1-19:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{liberty_et_al:LIPIcs.APPROX-RANDOM.2017.19,
author = {Liberty, Edo and Sviridenko, Maxim},
title = {{Greedy Minimization of Weakly Supermodular Set Functions}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {19:1--19:11},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.19},
URN = {urn:nbn:de:0030-drops-75682},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.19},
annote = {Keywords: Weak Supermodularity, Greedy Algorithms, Machine Learning, Data Mining}
}
Document
Renyi Entropy Estimation Revisited
Abstract
We revisit the problem of estimating entropy of discrete distributions from independent samples, studied recently by Acharya, Orlitsky, Suresh and Tyagi (SODA 2015), improving their upper and lower bounds on the necessary sample size n. For estimating Renyi entropy of order alpha, up to constant accuracy and error probability, we show the following
* Upper bounds n = O(1) 2^{(1-1/alpha)H_alpha} for integer alpha>1, as the worst case over distributions with Renyi entropy equal to H_alpha.
* Lower bounds n = Omega(1) K^{1-1/alpha} for any real alpha>1, with the constant being an inverse polynomial of the accuracy, as the worst case over all distributions on K elements.
Our upper bounds essentially replace the alphabet size by a factor exponential in the entropy, which offers improvements especially in low or medium entropy regimes (interesting for example in anomaly detection). As for the lower bounds, our proof explicitly shows how the complexity depends on both alphabet and accuracy, partially solving the open problem posted in previous works. The argument for upper bounds derives a clean identity for the variance of falling-power sum of a multinomial distribution. Our approach for lower bounds utilizes convex optimization to find a distribution with possibly worse estimation performance, and may be of independent interest as a tool to work with Le Cam’s two point method.
Cite as
Maciej Obremski and Maciej Skorski. Renyi Entropy Estimation Revisited. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 20:1-20:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{obremski_et_al:LIPIcs.APPROX-RANDOM.2017.20,
author = {Obremski, Maciej and Skorski, Maciej},
title = {{Renyi Entropy Estimation Revisited}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {20:1--20:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.20},
URN = {urn:nbn:de:0030-drops-75699},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.20},
annote = {Keywords: Renyi entropy, entropy estimation, sample complexity, convex optimization}
}
Document
Approximating Sparsest Cut in Low Rank Graphs via Embeddings from Approximately Low Dimensional Spaces
Abstract
We consider the problem of embedding a finite set of points x_1, ... , x_n in R^d that satisfy l_2^2 triangle inequalities into l_1, when the points are approximately low-dimensional. Goemans (unpublished, appears in a work of Magen and Moharammi (2008) ) showed that such points residing in exactly d dimensions can be embedded into l_1 with distortion at most sqrt{d}. We prove the following robust analogue of this statement: if there exists a r-dimensional subspace Pi such that the projections onto this subspace satisfy sum_{i,j in [n]} norm{Pi x_i - Pi x_j}_2^2 >= Omega(1) * sum_{i,j \in [n]} norm{x_i - x_j}_2^2, then there is an embedding of the points into l_1 with O(sqrt{r}) average distortion. A consequence of this result is that the integrality gap of the well-known Goemans-Linial SDP relaxation for the Uniform Sparsest Cut problem is O(sqrt{r}) on graphs G whose r-th smallest normalized eigenvalue of the Laplacian satisfies lambda_r(G)/n >= Omega(1)*Phi_{SDP}(G). Our result improves upon the previously known bound of O(r) on the average distortion, and the integrality gap of the Goemans-Linial SDP under the same preconditions, proven in [Deshpande and Venkat, 2014], and [Deshpande, Harsha and Venkat 2016].
Cite as
Yuval Rabani and Rakesh Venkat. Approximating Sparsest Cut in Low Rank Graphs via Embeddings from Approximately Low Dimensional Spaces. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{rabani_et_al:LIPIcs.APPROX-RANDOM.2017.21,
author = {Rabani, Yuval and Venkat, Rakesh},
title = {{Approximating Sparsest Cut in Low Rank Graphs via Embeddings from Approximately Low Dimensional Spaces}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {21:1--21:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.21},
URN = {urn:nbn:de:0030-drops-75705},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.21},
annote = {Keywords: Metric Embeddings, Sparsest Cut, Negative type metrics, Approximation Algorithms}
}
Document
When Are Welfare Guarantees Robust?
Abstract
Computational and economic results suggest that social welfare maximization and combinatorial auction design are much easier when bidders' valuations satisfy the "gross substitutes" condition. The goal of this paper is to evaluate rigorously the folklore belief that the main take-aways from these results remain valid in settings where the gross substitutes condition holds only approximately. We show that for valuations that pointwise approximate a gross substitutes valuation (in fact even a linear valuation), optimal social welfare cannot be approximated to within a subpolynomial factor and demand oracles cannot be simulated using a subexponential number of value queries. We then provide several positive results by imposing additional structure on the valuations (beyond gross substitutes), using a more stringent notion of approximation, and/or using more powerful oracle access to the valuations. For example, we prove that the performance of the greedy algorithm degrades gracefully for near-linear valuations with approximately decreasing marginal values; that with demand queries, approximate welfare guarantees for XOS valuations degrade gracefully for valuations that are pointwise close to XOS; and that the performance of the Kelso-Crawford auction degrades gracefully for valuations that are close to various subclasses of gross substitutes valuations.
Cite as
Tim Roughgarden, Inbal Talgam-Cohen, and Jan Vondrák. When Are Welfare Guarantees Robust?. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 22:1-22:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{roughgarden_et_al:LIPIcs.APPROX-RANDOM.2017.22,
author = {Roughgarden, Tim and Talgam-Cohen, Inbal and Vondr\'{a}k, Jan},
title = {{When Are Welfare Guarantees Robust?}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {22:1--22:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.22},
URN = {urn:nbn:de:0030-drops-75714},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.22},
annote = {Keywords: Valuation (set) functions, gross substitutes, linearity, approximation}
}
Document
Glauber Dynamics for Ising Model on Convergent Dense Graph Sequences
Abstract
We study the Glauber dynamics for Ising model on (sequences of) dense graphs. We view the dense graphs through the lens of graphons. For the ferromagnetic Ising model with inverse temperature beta on a convergent sequence of graphs G_n with limit graphon W we show fast mixing of the Glauber dynamics if beta * lambda_1(W) < 1$ and slow (torpid) mixing if beta * lambda_1(W) > 1 (where lambda_1(W)is the largest eigenvalue of the graphon). We also show that in the case beta * lambda_1(W) = 1 there is insufficient information to determine the mixing time (it can be either fast or slow).
Cite as
Rupam Acharyya and Daniel Stefankovic. Glauber Dynamics for Ising Model on Convergent Dense Graph Sequences. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 23:1-23:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{acharyya_et_al:LIPIcs.APPROX-RANDOM.2017.23,
author = {Acharyya, Rupam and Stefankovic, Daniel},
title = {{Glauber Dynamics for Ising Model on Convergent Dense Graph Sequences}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {23:1--23:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.23},
URN = {urn:nbn:de:0030-drops-75724},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.23},
annote = {Keywords: Spin systems, Glauber dynamics, Ising model, graphons}
}
Document
On the Expansion of Group-Based Lifts
Abstract
A k-lift of an n-vertex base graph G is a graph H on nxk vertices, where each vertex v of G is replaced by k vertices v_1,...,v_k and each edge uv in G is replaced by a matching representing a bijection pi_{uv} so that the edges of H are of the form (u_i,v_{pi_{uv}(i)}). Lifts have been investigated as a means to efficiently construct expanders. In this work, we study lifts obtained from groups and group actions. We derive the spectrum of such lifts via the representation theory principles of the underlying group. Our main results are:
1. A uniform random lift by a cyclic group of order k of any n-vertex d-regular base graph G, with the nontrivial eigenvalues of the adjacency matrix of G bounded by lambda in magnitude, has the new nontrivial eigenvalues bounded by lambda+O(sqrt{d}) in magnitude with probability 1-ke^{-Omega(n/d^2)}. The probability bounds as well as the dependency on lambda are almost optimal. As a special case, we obtain that there is a constant c_1 such that for every k<=2^{c_1n/d^2}, there exists a lift H of every Ramanujan graph by a cyclic group of order k such that H is almost Ramanujan (nontrivial eigenvalues of the adjacency matrix at most O(sqrt{d}) in magnitude). We also show how this result leads to a quasi-polynomial time deterministic algorithm to construct almost Ramanujan expanders.
2. There is a constant c_2 such that for every k>=2^{c_2nd}, there does not exist an abelian k-lift H of any n-vertex d-regular base graph such that H is almost Ramanujan. This can be viewed as an analogue of the well-known no-expansion result for constant degree abelian Cayley graphs.
Suppose k_0 is the order of the largest abelian group that produces expanding lifts. Our two results highlight lower and upper bounds on k_0 that are tight upto a factor of d^3 in the exponent, thus suggesting a threshold phenomenon.
Cite as
Naman Agarwal, Karthekeyan Chandrasekaran, Alexandra Kolla, and Vivek Madan. On the Expansion of Group-Based Lifts. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 24:1-24:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{agarwal_et_al:LIPIcs.APPROX-RANDOM.2017.24,
author = {Agarwal, Naman and Chandrasekaran, Karthekeyan and Kolla, Alexandra and Madan, Vivek},
title = {{On the Expansion of Group-Based Lifts}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {24:1--24:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.24},
URN = {urn:nbn:de:0030-drops-75739},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.24},
annote = {Keywords: Expanders, Lifts, Spectral Graph Theory}
}
Document
Efficient Removal Lemmas for Matrices
Abstract
The authors and Fischer recently proved that any hereditary property of two-dimensional matrices (where the row and column order is not ignored) over a finite alphabet is testable with a constant number of queries, by establishing an (ordered) matrix removal lemma, which states the following: If a matrix is far from satisfying some hereditary property, then a large enough constant-size random submatrix of it does not satisfy the property with probability at least 9/10. Here being far from the property means that one needs to modify a constant fraction of the entries of the matrix to make it satisfy the property.
However, in the above general removal lemma, the required size of the random submatrix grows very fast as a function of the distance of the matrix from satisfying the property. In this work we establish much more efficient removal lemmas for several special cases of the above problem. In particular, we show the following: If an epsilon-fraction of the entries of a binary matrix M can be covered by pairwise-disjoint copies of some (s x t) matrix A, then a delta-fraction of the (s x t)-submatrices of M are equal to A, where delta is polynomial in epsilon.
We generalize the work of Alon, Fischer and Newman [SICOMP'07] and make progress towards proving one of their conjectures. The proofs combine their efficient conditional regularity lemma for matrices with additional combinatorial and probabilistic ideas.
Cite as
Noga Alon and Omri Ben-Eliezer. Efficient Removal Lemmas for Matrices. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 25:1-25:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{alon_et_al:LIPIcs.APPROX-RANDOM.2017.25,
author = {Alon, Noga and Ben-Eliezer, Omri},
title = {{Efficient Removal Lemmas for Matrices}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {25:1--25:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.25},
URN = {urn:nbn:de:0030-drops-75744},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.25},
annote = {Keywords: Property Testing, Removal Lemma, Matrix Regularity Lemma, Binary Matrix}
}
Document
The String of Diamonds Is Tight for Rumor Spreading
Abstract
For a rumor spreading protocol, the spread time is defined as the first time that everyone learns the rumor. We compare the synchronous push&pull rumor spreading protocol with its asynchronous variant, and show that for any n-vertex graph and any starting vertex, the ratio between their expected spread times is bounded by O(n^{1/3} log^{2/3} n). This improves the O(sqrt n) upper bound of Giakkoupis, Nazari, and Woelfel (in Proceedings of ACM Symposium on Principles of Distributed Computing, 2016). Our bound is tight up to a factor of O(log n), as illustrated by the string of diamonds graph.
Cite as
Omer Angel, Abbas Mehrabian, and Yuval Peres. The String of Diamonds Is Tight for Rumor Spreading. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 26:1-26:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{angel_et_al:LIPIcs.APPROX-RANDOM.2017.26,
author = {Angel, Omer and Mehrabian, Abbas and Peres, Yuval},
title = {{The String of Diamonds Is Tight for Rumor Spreading}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {26:1--26:9},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.26},
URN = {urn:nbn:de:0030-drops-75754},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.26},
annote = {Keywords: randomized rumor spreading, push\&pull protocol, asynchronous time model, string of diamonds}
}
Document
Sharper Bounds for Regularized Data Fitting
Abstract
We study matrix sketching methods for regularized variants of linear regression, low rank approximation, and canonical correlation analysis. Our main focus is on sketching techniques which preserve the objective function value for regularized problems, which is an area that has remained largely unexplored. We study regularization both in a fairly broad setting, and in the specific context of the popular and widely used technique of ridge regularization; for the latter, as applied to each of these problems, we show algorithmic resource bounds in which the statistical dimension appears in places where in previous bounds the rank would appear. The statistical dimension is always smaller than the rank, and decreases as the amount of regularization increases. In particular we show this for the ridge low-rank approximation problem as well as regularized low-rank approximation problems in a much more general setting, where the regularizing function satisfies some very general conditions (chiefly, invariance under orthogonal transformations).
Cite as
Haim Avron, Kenneth L. Clarkson, and David P. Woodruff. Sharper Bounds for Regularized Data Fitting. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 27:1-27:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{avron_et_al:LIPIcs.APPROX-RANDOM.2017.27,
author = {Avron, Haim and Clarkson, Kenneth L. and Woodruff, David P.},
title = {{Sharper Bounds for Regularized Data Fitting}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {27:1--27:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.27},
URN = {urn:nbn:de:0030-drops-75761},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.27},
annote = {Keywords: Matrices, Regression, Low-rank approximation, Regularization, Canonical Correlation Analysis}
}
Document
The Lovász Theta Function for Random Regular Graphs and Community Detection in the Hard Regime
Abstract
We derive upper and lower bounds on the degree d for which the Lovasz theta function, or equivalently sum-of-squares proofs with degree two, can refute the existence of a k-coloring in random regular graphs G(n,d). We show that this type of refutation fails well above the k-colorability transition, and in particular everywhere below the Kesten-Stigum threshold. This is consistent with the conjecture that refuting k-colorability, or distinguishing G(n,d) from the planted coloring model, is hard in this region. Our results also apply to the disassortative case of the stochastic block model, adding evidence to the conjecture that there is a regime where community detection is computationally hard even though it is information-theoretically possible. Using orthogonal polynomials, we also provide explicit upper bounds on the theta function for regular graphs of a given girth, which may be of independent interest.
Cite as
Jess Banks, Robert Kleinberg, and Cristopher Moore. The Lovász Theta Function for Random Regular Graphs and Community Detection in the Hard Regime. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 28:1-28:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{banks_et_al:LIPIcs.APPROX-RANDOM.2017.28,
author = {Banks, Jess and Kleinberg, Robert and Moore, Cristopher},
title = {{The Lov\'{a}sz Theta Function for Random Regular Graphs and Community Detection in the Hard Regime}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {28:1--28:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.28},
URN = {urn:nbn:de:0030-drops-75771},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.28},
annote = {Keywords: Lov\'{a}sz Theta Function, Random Regular Graphs, Sum of Squares, Orthogonal Polynomials}
}
Document
Cutoff for a Stratified Random Walk on the Hypercube
Abstract
We consider the random walk on the hypercube which moves by picking an ordered pair (i,j) of distinct coordinates uniformly at random and adding the bit at location i to the bit at location j, modulo 2. We show that this Markov chain has cutoff at time (3/2)n*log(n) with window of size n, solving a question posed by Chung and Graham (1997).
Cite as
Anna Ben-Hamou and Yuval Peres. Cutoff for a Stratified Random Walk on the Hypercube. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 29:1-29:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{benhamou_et_al:LIPIcs.APPROX-RANDOM.2017.29,
author = {Ben-Hamou, Anna and Peres, Yuval},
title = {{Cutoff for a Stratified Random Walk on the Hypercube}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {29:1--29:10},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.29},
URN = {urn:nbn:de:0030-drops-75787},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.29},
annote = {Keywords: Mixing times, cutoff, hypercube}
}
Document
Lower Bounds for 2-Query LCCs over Large Alphabet
Abstract
A locally correctable code (LCC) is an error correcting code that allows correction of any arbitrary coordinate of a corrupted codeword by querying only a few coordinates. We show that any 2-query locally correctable code C:{0,1}^k -> Sigma^n that can correct a constant fraction of corrupted symbols must have n >= exp(k/\log|Sigma|) under the assumption that the LCC is zero-error. We say that an LCC is zero-error if there exists a non-adaptive corrector algorithm that succeeds with probability 1 when the input is an uncorrupted codeword. All known constructions of LCCs are zero-error.
Our result is tight upto constant factors in the exponent. The only previous lower bound on the length of 2-query LCCs over large alphabet was Omega((k/log|\Sigma|)^2) due to Katz and Trevisan (STOC 2000). Our bound implies that zero-error LCCs cannot yield 2-server private information retrieval (PIR) schemes with sub-polynomial communication. Since there exists a 2-server PIR scheme with sub-polynomial communication (STOC 2015) based on a zero-error 2-query locally decodable code (LDC), we also obtain a separation between LDCs and LCCs over large alphabet.
Cite as
Arnab Bhattacharyya, Sivakanth Gopi, and Avishay Tal. Lower Bounds for 2-Query LCCs over Large Alphabet. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 30:1-30:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{bhattacharyya_et_al:LIPIcs.APPROX-RANDOM.2017.30,
author = {Bhattacharyya, Arnab and Gopi, Sivakanth and Tal, Avishay},
title = {{Lower Bounds for 2-Query LCCs over Large Alphabet}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {30:1--30:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.30},
URN = {urn:nbn:de:0030-drops-75792},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.30},
annote = {Keywords: Locally correctable code, Private information retrieval, Szemer\'{e}di regularity lemma}
}
Document
Sum-of-Squares Certificates for Maxima of Random Tensors on the Sphere
Abstract
For an n-variate order-d tensor A, define A_{max} := sup_{||x||_2 = 1} <A,x^(otimes d)>, to be the maximum value taken by the tensor on the unit sphere. It is known that for a random tensor with i.i.d. +1/-1 entries, A_{max} <= sqrt(n.d.log(d)) w.h.p. We study the problem of efficiently certifying upper bounds on A_{max} via the natural relaxation from the Sum of Squares (SoS) hierarchy. Our results include:
* When A is a random order-q tensor, we prove that q levels of SoS certifies an upper bound B on A_{max} that satisfies B <= A_{max} * (n/q^(1-o(1)))^(q/4-1/2) w.h.p. Our upper bound improves a result of Montanari and Richard (NIPS 2014) when q is large.
* We show the above bound is the best possible up to lower order terms, namely the optimum of the level-q SoS relaxation is at least
A_{max} * (n/q^(1+o(1)))^(q/4-1/2).
* When A is a random order-d tensor, we prove that q levels of SoS certifies an upper bound B on A_{max} that satisfies B <= A_{max} * (n*polylog/q)^(d/4 - 1/2) w.h.p. For growing q, this improves upon the bound certified by constant levels of SoS. This answers in part, a question posed by Hopkins, Shi, and Steurer (COLT 2015), who tightly characterized constant levels of SoS.
Cite as
Vijay Bhattiprolu, Venkatesan Guruswami, and Euiwoong Lee. Sum-of-Squares Certificates for Maxima of Random Tensors on the Sphere. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 31:1-31:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{bhattiprolu_et_al:LIPIcs.APPROX-RANDOM.2017.31,
author = {Bhattiprolu, Vijay and Guruswami, Venkatesan and Lee, Euiwoong},
title = {{Sum-of-Squares Certificates for Maxima of Random Tensors on the Sphere}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {31:1--31:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.31},
URN = {urn:nbn:de:0030-drops-75808},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.31},
annote = {Keywords: Sum-of-Squares, Optimization over Sphere, Random Polynomials}
}
Document
Continuous Monitoring of l_p Norms in Data Streams
Abstract
In insertion-only streaming, one sees a sequence of indices a_1, a_2, ..., a_m in [n]. The stream defines a sequence of m frequency vectors x(1), ..., x(m) each in R^n, where x(t) is the frequency vector of items after seeing the first t indices in the stream. Much work in the streaming literature focuses on estimating some function f(x(m)). Many applications though require obtaining estimates at time t of f(x(t)), for every t in [m]. Naively this guarantee is obtained by devising an algorithm with failure probability less than 1/m, then performing a union bound over all stream updates to guarantee that all m estimates are simultaneously accurate with good probability. When f(x) is some l_p norm of x, recent works have shown that this union bound is wasteful and better space complexity is possible for the continuous monitoring problem, with the strongest known results being for p=2. In this work, we improve the state of the art for all 0<p<2, which we obtain via a novel analysis of Indyk's p-stable sketch.
Cite as
Jaroslaw Blasiok, Jian Ding, and Jelani Nelson. Continuous Monitoring of l_p Norms in Data Streams. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 32:1-32:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{blasiok_et_al:LIPIcs.APPROX-RANDOM.2017.32,
author = {Blasiok, Jaroslaw and Ding, Jian and Nelson, Jelani},
title = {{Continuous Monitoring of l\underlinep Norms in Data Streams}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {32:1--32:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.32},
URN = {urn:nbn:de:0030-drops-75816},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.32},
annote = {Keywords: data streams, continuous monitoring, moment estimation}
}
Document
Vertex Isoperimetry and Independent Set Stability for Tensor Powers of Cliques
Abstract
The tensor power of the clique on t vertices (denoted by K_t^n) is the graph on vertex set {1, ..., t}^n such that two vertices x, y in {1, ..., t}^n are connected if and only if x_i != y_i for all i in {1, ..., n}. Let the density of a subset S of K_t^n to be mu(S) := |S|/t^n. Also let the vertex boundary of a set S to be the vertices of the graph, including those of S, which are incident to some vertex of S. We investigate two similar problems on such graphs.
First, we study the vertex isoperimetry problem. Given a density nu in [0, 1] what is the smallest possible density of the vertex boundary of a subset of K_t^n of density nu? Let Phi_t(nu) be the infimum of these minimum densities as n -> infinity. We find a recursive relation allows one to compute Phi_t(nu) in time polynomial to the number of desired bits of precision.
Second, we study given an independent set I of K_t^n of density mu(I) = (1-epsilon)/t, how close it is to a maximum-sized independent set J of density 1/t. We show that this deviation (measured by mu(I\J)) is at most 4 epsilon^{(log t)/(log t - log(t-1))} as long as epsilon < 1 - 3/t + 2/t^2. This substantially improves on results of Alon, Dinur, Friedgut, and Sudakov (2004) and Ghandehari and Hatami (2008) which had an O(epsilon) upper bound. We also show the exponent (log t)/(log t - log(t-1)) is optimal assuming n tending to infinity and epsilon tending to 0. The methods have similarity to recent work by Ellis, Keller, and Lifshitz (2016) in the context of Kneser graphs and other settings.
The author hopes that these results have potential applications in hardness of approximation, particularly in approximate graph coloring and independent set problems.
Cite as
Joshua Brakensiek. Vertex Isoperimetry and Independent Set Stability for Tensor Powers of Cliques. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{brakensiek:LIPIcs.APPROX-RANDOM.2017.33,
author = {Brakensiek, Joshua},
title = {{Vertex Isoperimetry and Independent Set Stability for Tensor Powers of Cliques}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {33:1--33:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.33},
URN = {urn:nbn:de:0030-drops-75828},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.33},
annote = {Keywords: extremal combinatorics, independent sets, isoperimetry, stability}
}
Document
Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings
Abstract
We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall, and Spencer in 2002. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2^{-s}, (a+1)2^{-s}] x [b2^{-t}, (b+1)2^{-t}] for a,b,s,t nonnegative integers. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n^{4.09}), which implies that the mixing time is at most O(n^{5.09}). We complement this by showing that the relaxation time is at least Omega(n^{1.38}), improving upon the previously best lower bound of Omega(n*log n) coming from the diameter of the chain.
Cite as
Sarah Cannon, David A. Levin, and Alexandre Stauffer. Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 34:1-34:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{cannon_et_al:LIPIcs.APPROX-RANDOM.2017.34,
author = {Cannon, Sarah and Levin, David A. and Stauffer, Alexandre},
title = {{Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {34:1--34:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.34},
URN = {urn:nbn:de:0030-drops-75830},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.34},
annote = {Keywords: Random dyadic tilings, spectral gap, rapid mixing}
}
Document
Agnostic Learning from Tolerant Natural Proofs
Abstract
We generalize the "learning algorithms from natural properties" framework of [CIKK16] to get agnostic learning algorithms from natural properties with extra features. We show that if a natural property (in the sense of Razborov and Rudich [RR97]) is useful also against functions that are close to the class of "easy" functions, rather than just against "easy" functions, then it can be used to get an agnostic learning algorithm over the uniform distribution with membership queries.
* For AC0[q], any prime q (constant-depth circuits of polynomial size, with AND, OR, NOT, and MODq gates of unbounded fanin), which happens to have a natural property with the requisite extra feature by [Raz87, Smo87, RR97], we obtain the first agnostic learning algorithm for AC0[q], for every prime q. Our algorithm runs in randomized quasi-polynomial time, uses membership queries, and outputs a circuit for a given Boolean function f that agrees with f on all but at most polylog(n)*opt fraction of inputs, where opt is the relative distance between f and the closest function h in the class AC0[q].
* For the ideal case, a natural proof of strongly exponential correlation circuit lower bounds against a circuit class C containing AC0[2] (i.e., circuits of size exp(Omega(n)) cannot compute some n-variate function even with exp(-Omega(n)) advantage over random guessing) would yield a polynomial-time query agnostic learning algorithm for C with the approximation error O(opt).
Cite as
Marco L. Carmosino, Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova. Agnostic Learning from Tolerant Natural Proofs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 35:1-35:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{carmosino_et_al:LIPIcs.APPROX-RANDOM.2017.35,
author = {Carmosino, Marco L. and Impagliazzo, Russell and Kabanets, Valentine and Kolokolova, Antonina},
title = {{Agnostic Learning from Tolerant Natural Proofs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {35:1--35:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.35},
URN = {urn:nbn:de:0030-drops-75842},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.35},
annote = {Keywords: agnostic learning, natural proofs, circuit lower bounds, meta-algorithms, AC0\lbrackq\rbrack, Nisan-Wigderson generator}
}
Document
On the Complexity of Constrained Determinantal Point Processes
Abstract
Determinantal Point Processes (DPPs) are probabilistic models that arise in quantum physics and random matrix theory and have recently found numerous applications in theoretical computer science and machine learning. DPPs define probability distributions over subsets of a given ground set, they exhibit interesting properties such as negative correlation, and, unlike other models of negative correlation such as Markov random fields, have efficient algorithms for sampling. When applied to kernel methods in machine learning, DPPs favor subsets of the given data with more diverse features. However, many real-world applications require efficient algorithms to sample from DPPs with additional constraints on the sampled subset, e.g., partition or matroid constraints that are important from the viewpoint of ensuring priors, resource or fairness constraints on the sampled subset. Whether one can efficiently sample from DPPs in such constrained settings is an important problem that was first raised in a survey of DPPs for machine learning by Kulesza and Taskar and studied in some recent works.
The main contribution of this paper is the first resolution of the complexity of sampling from DPPs with constraints. On the one hand, we give exact efficient algorithms for sampling from constrained DPPs when the description of the constraints is in unary; this includes special cases of practical importance such as a small number of partition, knapsack or budget constraints. On the other hand, we prove that when the constraints are specified in binary, this problem is #P-hard via a reduction from the problem of computing mixed discriminants; implying that it may be unlikely that there is an FPRAS. Technically, our algorithmic result benefits from viewing the constrained sampling problem via the lens of polynomials and we obtain our complexity results by providing an equivalence between computing mixed discriminants and sampling from partition constrained DPPs. As a consequence, we obtain a few corollaries of independent interest: 1) An algorithm to count, sample (and, hence, optimize) over the base polytope of regular matroids when there are additional (succinct) budget constraints and,
2) An algorithm to evaluate and compute mixed characteristic polynomials, that played a central role in the resolution of the Kadison-Singer problem, for certain special cases.
Cite as
L. Elisa Celis, Amit Deshpande, Tarun Kathuria, Damian Straszak, and Nisheeth K. Vishnoi. On the Complexity of Constrained Determinantal Point Processes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 36:1-36:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{celis_et_al:LIPIcs.APPROX-RANDOM.2017.36,
author = {Celis, L. Elisa and Deshpande, Amit and Kathuria, Tarun and Straszak, Damian and Vishnoi, Nisheeth K.},
title = {{On the Complexity of Constrained Determinantal Point Processes}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {36:1--36:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.36},
URN = {urn:nbn:de:0030-drops-75851},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.36},
annote = {Keywords: determinantal point processes, constraints, matroids, sampling and counting, polynomials, mixed discriminant}
}
Document
Sample-Based High-Dimensional Convexity Testing
Abstract
In the problem of high-dimensional convexity testing, there is an unknown set S in the n-dimensional Euclidean space which is promised to be either convex or c-far from every convex body with respect to the standard multivariate normal distribution. The job of a testing algorithm is then to distinguish between these two cases while making as few inspections of the set S as possible.
In this work we consider sample-based testing algorithms, in which the testing algorithm only has access to labeled samples (x,S(x)) where each x is independently drawn from the normal distribution. We give nearly matching sample complexity upper and lower bounds for both one-sided and two-sided convexity testing algorithms in this framework. For constant c, our results show that the sample complexity of one-sided convexity testing is exponential in n, while for two-sided convexity testing it is exponential in the square root of n.
Cite as
Xi Chen, Adam Freilich, Rocco A. Servedio, and Timothy Sun. Sample-Based High-Dimensional Convexity Testing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 37:1-37:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{chen_et_al:LIPIcs.APPROX-RANDOM.2017.37,
author = {Chen, Xi and Freilich, Adam and Servedio, Rocco A. and Sun, Timothy},
title = {{Sample-Based High-Dimensional Convexity Testing}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {37:1--37:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.37},
URN = {urn:nbn:de:0030-drops-75867},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.37},
annote = {Keywords: Property testing, convexity, sample-based testing}
}
Document
Adaptivity Is Exponentially Powerful for Testing Monotonicity of Halfspaces
Abstract
We give a poly(log(n),1/epsilon)-query adaptive algorithm for testing whether an unknown Boolean function f:{-1, 1}^n -> {-1, 1}, which is promised to be a halfspace, is monotone versus epsilon-far from monotone. Since non-adaptive algorithms are known to require almost Omega(n^{1/2}) queries to test whether an unknown halfspace is monotone versus far from monotone, this shows that adaptivity enables an exponential improvement in the query complexity of monotonicity testing for halfspaces.
Cite as
Xi Chen, Rocco A. Servedio, Li-Yang Tan, and Erik Waingarten. Adaptivity Is Exponentially Powerful for Testing Monotonicity of Halfspaces. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 38:1-38:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{chen_et_al:LIPIcs.APPROX-RANDOM.2017.38,
author = {Chen, Xi and Servedio, Rocco A. and Tan, Li-Yang and Waingarten, Erik},
title = {{Adaptivity Is Exponentially Powerful for Testing Monotonicity of Halfspaces}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {38:1--38:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.38},
URN = {urn:nbn:de:0030-drops-75877},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.38},
annote = {Keywords: property testing, linear threshold functions, monotonicity, adaptivity}
}
Document
On Axis-Parallel Tests for Tensor Product Codes
Abstract
Many low-degree tests examine the input function via its restrictions to random hyperplanes of a certain dimension. Examples include the line-vs-line (Arora, Sudan 2003), plane-vs-plane (Raz, Safra 1997), and cube-vs-cube (Bhangale, Dinur, Livni 2017) tests.
In this paper we study tests that only consider restrictions along axis-parallel hyperplanes, which have been studied by Polishchuk and Spielman (1994) and Ben-Sasson and Sudan (2006). While such tests are necessarily "weaker", they work for a more general class of codes, namely tensor product codes. Moreover, axis-parallel tests play a key role in constructing LTCs with inverse polylogarithmic rate and short PCPs (Polishchuk, Spielman 1994; Ben-Sasson, Sudan 2008; Meir 2010). We present two results on axis-parallel tests.
(1) Bivariate low-degree testing with low-agreement. We prove an analogue of the Bivariate Low-Degree Testing Theorem of Polishchuk and Spielman in the low-agreement regime, albeit with much larger field size. Namely, for the 2-wise tensor product of the Reed-Solomon code, we prove that for sufficiently large fields, the 2-query variant of the axis-parallel line test (row-vs-column test) works for arbitrarily small agreement. Prior analyses of axis-parallel tests assumed high agreement, and no results for such tests in the low-agreement regime were known.
Our proof technique deviates significantly from that of Polishchuk and Spielman, which relies on algebraic methods such as Bezout's Theorem, and instead leverages a fundamental result in extremal graph theory by Kovari, Sos, and Turan. To our knowledge, this is the first time this result is used in the context of low-degree testing.
(2) Improved robustness for tensor product codes. Robustness is a strengthening of local testability that underlies many applications. We prove that the axis-parallel hyperplane test for the m-wise tensor product of a linear code with block length n and distance d is Omega(d^m/n^m)-robust. This improves on a theorem of Viderman (2012) by a factor of 1/poly(m). While the improvement is not large, we believe that our proof is a notable simplification compared to prior work.
Cite as
Alessandro Chiesa, Peter Manohar, and Igor Shinkar. On Axis-Parallel Tests for Tensor Product Codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 39:1-39:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{chiesa_et_al:LIPIcs.APPROX-RANDOM.2017.39,
author = {Chiesa, Alessandro and Manohar, Peter and Shinkar, Igor},
title = {{On Axis-Parallel Tests for Tensor Product Codes}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {39:1--39:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.39},
URN = {urn:nbn:de:0030-drops-75882},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.39},
annote = {Keywords: tensor product codes, locally testable codes, low-degree testing, extremal graph theory}
}
Document
Charting the Replica Symmetric Phase
Abstract
Random graph models and associated inference problems such as the stochastic block model play an eminent role in computer science, discrete mathematics and statistics. Based on non-rigorous arguments physicists predicted the existence of a generic phase transition that separates a "replica symmetric phase" where statistical inference is impossible from a phase where the detection of the "ground truth" is information-theoretically possible.
In this paper we prove a contiguity result that shows that detectability is indeed impossible within the replica-symmetric phase for a broad class of models. In particular, this implies the detectability conjecture for the disassortative stochastic block model from [Decelle et al.: Phys Rev E 2011]. Additionally, we investigate key features of the replica symmetric phase such as the nature of point-to-set correlations (`reconstruction').
Cite as
Amin Coja-Oghlan, Charilaos Efthymiou, Nor Jaafari, Mihyun Kang, and Tobias Kapetanopoulos. Charting the Replica Symmetric Phase. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 40:1-40:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{cojaoghlan_et_al:LIPIcs.APPROX-RANDOM.2017.40,
author = {Coja-Oghlan, Amin and Efthymiou, Charilaos and Jaafari, Nor and Kang, Mihyun and Kapetanopoulos, Tobias},
title = {{Charting the Replica Symmetric Phase}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {40:1--40:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.40},
URN = {urn:nbn:de:0030-drops-75895},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.40},
annote = {Keywords: Random factor graph, bounds for condensation phase transition, Potts antiferromagnet, diluted k-spin model, stochastic block model}
}
Document
Probabilistic Logarithmic-Space Algorithms for Laplacian Solvers
Abstract
A recent series of breakthroughs initiated by Spielman and Teng culminated in the construction of nearly linear time Laplacian solvers, approximating the solution of a linear system Lx=b, where L is the normalized Laplacian of an undirected graph. In this paper we study the space complexity of the problem. Surprisingly we are able to show a probabilistic, logspace algorithm solving the problem. We further extend the algorithm to other families of graphs like Eulerian graphs (and directed regular graphs) and graphs that mix in polynomial time.
Our approach is to pseudo-invert the Laplacian, by first "peeling-off" the problematic kernel of the operator, and then to approximate the inverse of the remaining part by using a Taylor series. We approximate the Taylor series using a previous work and the special structure of the problem. For directed graphs we exploit in the analysis the Jordan normal form and results from matrix functions.
Cite as
Dean Doron, François Le Gall, and Amnon Ta-Shma. Probabilistic Logarithmic-Space Algorithms for Laplacian Solvers. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 41:1-41:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{doron_et_al:LIPIcs.APPROX-RANDOM.2017.41,
author = {Doron, Dean and Le Gall, Fran\c{c}ois and Ta-Shma, Amnon},
title = {{Probabilistic Logarithmic-Space Algorithms for Laplacian Solvers}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {41:1--41:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.41},
URN = {urn:nbn:de:0030-drops-75908},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.41},
annote = {Keywords: Laplacian solvers, Randomized logspace, Bounded-space complexity classes, Random walks, Matrix computation}
}
Document
Streaming Periodicity with Mismatches
Abstract
We study the problem of finding all k-periods of a length-n string S, presented as a data stream. S is said to have k-period p if its prefix of length n-p differs from its suffix of length n-p in at most k locations.
We give a one-pass streaming algorithm that computes the k-periods of a string S using poly(k, log n) bits of space, for k-periods of length at most n/2. We also present a two-pass streaming algorithm that computes k-periods of S using poly(k, log n) bits of space, regardless of period length. We complement these results with comparable lower bounds.
Cite as
Funda Ergün, Elena Grigorescu, Erfan Sadeqi Azer, and Samson Zhou. Streaming Periodicity with Mismatches. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 42:1-42:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{ergun_et_al:LIPIcs.APPROX-RANDOM.2017.42,
author = {Erg\"{u}n, Funda and Grigorescu, Elena and Sadeqi Azer, Erfan and Zhou, Samson},
title = {{Streaming Periodicity with Mismatches}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {42:1--42:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.42},
URN = {urn:nbn:de:0030-drops-75911},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.42},
annote = {Keywords: String algorithms, Streaming algorithms, Pattern matching, Randomized algorithms, Sublinear algorithms}
}
Document
Locality via Partially Lifted Codes
Abstract
In error-correcting codes, locality refers to several different ways of quantifying how easily a small amount of information can be recovered from encoded data. In this work, we study a notion of locality called the s-Disjoint-Repair-Group Property (s-DRGP). This notion can interpolate between two very different settings in coding theory: that of Locally Correctable Codes (LCCs) when s is large - a very strong guarantee - and Locally Recoverable Codes (LRCs) when s is small - a relatively weaker guarantee. This motivates the study of the s-DRGP for intermediate s, which is the focus of our paper. We construct codes in this parameter regime which have a higher rate than previously known codes. Our construction is based on a novel variant of the lifted codes of Guo, Kopparty and Sudan. Beyond the results on the s-DRGP, we hope that our construction is of independent interest, and will find uses elsewhere.
Cite as
S. Luna Frank-Fischer, Venkatesan Guruswami, and Mary Wootters. Locality via Partially Lifted Codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 43:1-43:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{frankfischer_et_al:LIPIcs.APPROX-RANDOM.2017.43,
author = {Frank-Fischer, S. Luna and Guruswami, Venkatesan and Wootters, Mary},
title = {{Locality via Partially Lifted Codes}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {43:1--43:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.43},
URN = {urn:nbn:de:0030-drops-75922},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.43},
annote = {Keywords: Error correcting codes, locality, lifted codes}
}
Document
Testing Hereditary Properties of Sequences
Abstract
A hereditary property of a sequence is one that is preserved when restricting to subsequences. We show that there exist hereditary properties of sequences that cannot be tested with sublinear queries, resolving an open question posed by Newman et al. This proof relies crucially on an infinite alphabet, however; for finite alphabets, we observe that any hereditary property can be tested with a constant number of queries.
Cite as
Cody R. Freitag, Eric Price, and William J. Swartworth. Testing Hereditary Properties of Sequences. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 44:1-44:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{freitag_et_al:LIPIcs.APPROX-RANDOM.2017.44,
author = {Freitag, Cody R. and Price, Eric and Swartworth, William J.},
title = {{Testing Hereditary Properties of Sequences}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {44:1--44:10},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.44},
URN = {urn:nbn:de:0030-drops-75938},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.44},
annote = {Keywords: Property Testing}
}
Document
Traveling in Randomly Embedded Random Graphs
Abstract
We consider the problem of traveling among random points in Euclidean space, when only a random fraction of the pairs are joined by traversable connections. In particular, we show a threshold for a pair of points to be connected by a geodesic of length arbitrarily close to their Euclidean distance, and analyze the minimum length Traveling Salesperson Tour, extending the Beardwood-Halton-Hammersley theorem to this setting.
Cite as
Alan Frieze and Wesley Pegden. Traveling in Randomly Embedded Random Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 45:1-45:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{frieze_et_al:LIPIcs.APPROX-RANDOM.2017.45,
author = {Frieze, Alan and Pegden, Wesley},
title = {{Traveling in Randomly Embedded Random Graphs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {45:1--45:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.45},
URN = {urn:nbn:de:0030-drops-75949},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.45},
annote = {Keywords: Traveling Salesman, Euclidean, Shortest Path}
}
Document
The Minrank of Random Graphs
Abstract
The minrank of a directed graph G is the minimum rank of a matrix M that can be obtained from the adjacency matrix of G by switching some ones to zeros (i.e., deleting edges) and then setting all diagonal entries to one. This quantity is closely related to the fundamental information-theoretic problems of (linear) index coding (Bar-Yossef et al., FOCS'06), network coding and distributed storage, and to Valiant's approach for proving superlinear circuit lower bounds (Valiant, Boolean Function Complexity '92).
We prove tight bounds on the minrank of directed Erdos-Renyi random graphs G(n,p) for all regimes of 0<p<1. In particular, for any constant p, we show that minrk(G) = Theta(n/log n) with high probability, where G is chosen from G(n,p). This bound gives a near quadratic improvement over the previous best lower bound of Omega(sqrt{n}) (Haviv and Langberg, ISIT'12), and partially settles an open problem raised by Lubetzky and Stav (FOCS '07). Our lower bound matches the well-known upper bound obtained by the "clique covering" solution, and settles the linear index coding problem for random graphs.
Finally, our result suggests a new avenue of attack, via derandomization, on Valiant's approach for proving superlinear lower bounds for logarithmic-depth semilinear circuits.
Cite as
Alexander Golovnev, Oded Regev, and Omri Weinstein. The Minrank of Random Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 46:1-46:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{golovnev_et_al:LIPIcs.APPROX-RANDOM.2017.46,
author = {Golovnev, Alexander and Regev, Oded and Weinstein, Omri},
title = {{The Minrank of Random Graphs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {46:1--46:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.46},
URN = {urn:nbn:de:0030-drops-75953},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.46},
annote = {Keywords: circuit complexity, index coding, information theory}
}
Document
Efficiently Decodable Codes for the Binary Deletion Channel
Abstract
In the random deletion channel, each bit is deleted independently with probability p. For the random deletion channel, the existence of codes of rate (1-p)/9, and thus bounded away from 0 for any p < 1, has been known. We give an explicit construction with polynomial time encoding and deletion correction algorithms with rate c_0 (1-p) for an absolute constant c_0 > 0.
Cite as
Venkatesan Guruswami and Ray Li. Efficiently Decodable Codes for the Binary Deletion Channel. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 47:1-47:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{guruswami_et_al:LIPIcs.APPROX-RANDOM.2017.47,
author = {Guruswami, Venkatesan and Li, Ray},
title = {{Efficiently Decodable Codes for the Binary Deletion Channel}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {47:1--47:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.47},
URN = {urn:nbn:de:0030-drops-75964},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.47},
annote = {Keywords: Coding theory, Combinatorics, Synchronization errors, Channel capacity}
}
Document
On Some Computations on Sparse Polynomials
Abstract
In arithmetic circuit complexity the standard operations are +,x. Yet, in some scenarios exponentiation gates are considered as well. In this paper we study the question of efficiently evaluating a polynomial given an oracle access to its power. Among applications, we show that:
* A reconstruction algorithm for a circuit class c can be extended to handle f^e for f in C.
* There exists an efficient deterministic algorithm for factoring sparse multiquadratic polynomials.
* There is a deterministic algorithm for testing a factorization of sparse polynomials, with constant individual degrees, into sparse irreducible factors. That is, testing if f = g_1 x ... x g_m when f has constant individual degrees and g_i-s are irreducible.
* There is a deterministic reconstruction algorithm for multilinear depth-4 circuits with two multiplication gates.
* There exists an efficient deterministic algorithm for testing whether two powers of sparse polynomials are equal. That is, f^d = g^e when f and g are sparse.
Cite as
Ilya Volkovich. On Some Computations on Sparse Polynomials. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 48:1-48:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{volkovich:LIPIcs.APPROX-RANDOM.2017.48,
author = {Volkovich, Ilya},
title = {{On Some Computations on Sparse Polynomials}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {48:1--48:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.48},
URN = {urn:nbn:de:0030-drops-75976},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.48},
annote = {Keywords: Derandomization, Arithmetic Circuits, Reconstruction}
}
Document
Communication Complexity of Statistical Distance
Abstract
We prove nearly matching upper and lower bounds on the randomized communication complexity of the following problem: Alice and Bob are each given a probability distribution over $n$ elements, and they wish to estimate within +-epsilon the statistical (total variation) distance between their distributions. For some range of parameters, there is up to a log(n) factor gap between the upper and lower bounds, and we identify a barrier to using information complexity techniques to improve the lower bound in this case. We also prove a side result that we discovered along the way: the randomized communication complexity of n-bit Majority composed with n-bit Greater-Than is Theta(n log n).
Cite as
Thomas Watson. Communication Complexity of Statistical Distance. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 49:1-49:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
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@InProceedings{watson:LIPIcs.APPROX-RANDOM.2017.49,
author = {Watson, Thomas},
title = {{Communication Complexity of Statistical Distance}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {49:1--49:10},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.49},
URN = {urn:nbn:de:0030-drops-75984},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.49},
annote = {Keywords: Communication, complexity, statistical, distance}
}