9 Search Results for "Anari, Nima"

Document
DOI: 10.4230/LIPIcs.ITCS.2022.5
Domain Sparsification of Discrete Distributions Using Entropic Independence

Authors: Nima Anari, Michał Dereziński, Thuy-Duong Vuong, and Elizabeth Yang

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
We present a framework for speeding up the time it takes to sample from discrete distributions μ defined over subsets of size k of a ground set of n elements, in the regime where k is much smaller than n. We show that if one has access to estimates of marginals P_{S∼ μ} {i ∈ S}, then the task of sampling from μ can be reduced to sampling from related distributions ν supported on size k subsets of a ground set of only n^{1-α}⋅ poly(k) elements. Here, 1/α ∈ [1, k] is the parameter of entropic independence for μ. Further, our algorithm only requires sparsified distributions ν that are obtained by applying a sparse (mostly 0) external field to μ, an operation that for many distributions μ of interest, retains algorithmic tractability of sampling from ν. This phenomenon, which we dub domain sparsification, allows us to pay a one-time cost of estimating the marginals of μ, and in return reduce the amortized cost needed to produce many samples from the distribution μ, as is often needed in upstream tasks such as counting and inference. For a wide range of distributions where α = Ω(1), our result reduces the domain size, and as a corollary, the cost-per-sample, by a poly(n) factor. Examples include monomers in a monomer-dimer system, non-symmetric determinantal point processes, and partition-constrained Strongly Rayleigh measures. Our work significantly extends the reach of prior work of Anari and Dereziński who obtained domain sparsification for distributions with a log-concave generating polynomial (corresponding to α = 1). As a corollary of our new analysis techniques, we also obtain a less stringent requirement on the accuracy of marginal estimates even for the case of log-concave polynomials; roughly speaking, we show that constant-factor approximation is enough for domain sparsification, improving over O(1/k) relative error established in prior work.
Cite as

Nima Anari, Michał Dereziński, Thuy-Duong Vuong, and Elizabeth Yang. Domain Sparsification of Discrete Distributions Using Entropic Independence. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 5:1-5:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{anari_et_al:LIPIcs.ITCS.2022.5,
  author =	{Anari, Nima and Derezi\'{n}ski, Micha{\l} and Vuong, Thuy-Duong and Yang, Elizabeth},
  title =	{{Domain Sparsification of Discrete Distributions Using Entropic Independence}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{5:1--5:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.5},
  URN =		{urn:nbn:de:0030-drops-156013},
  doi =		{10.4230/LIPIcs.ITCS.2022.5},
  annote =	{Keywords: Domain Sparsification, Markov Chains, Sampling, Entropic Independence}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.32
From Coupling to Spectral Independence and Blackbox Comparison with the Down-Up Walk

Authors: Kuikui Liu

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
We show that the existence of a "good" coupling w.r.t. Hamming distance for any local Markov chain on a discrete product space implies rapid mixing of the Glauber dynamics in a blackbox fashion. More specifically, we only require the expected distance between successive iterates under the coupling to be summable, as opposed to being one-step contractive in the worst case. Combined with recent local-to-global arguments [Chen et al., 2021], we establish asymptotically optimal lower bounds on the standard and modified log-Sobolev constants for the Glauber dynamics for sampling from spin systems on bounded-degree graphs when a curvature condition [Ollivier, 2009] is satisfied. To achieve this, we use Stein’s method for Markov chains [Bresler and Nagaraj, 2019; Reinert and Ross, 2019] to show that a "good" coupling for a local Markov chain yields strong bounds on the spectral independence of the distribution in the sense of [Anari et al., 2020]. Our primary application is to sampling proper list-colorings on bounded-degree graphs. In particular, combining the coupling for the flip dynamics given by [Vigoda, 2000; Chen et al., 2019] with our techniques, we show optimal O(nlog n) mixing for the Glauber dynamics for sampling proper list-colorings on any bounded-degree graph with maximum degree Δ whenever the size of the color lists are at least ({11/6 - ε}) Δ, where ε ≈ 10^{-5} is small constant. While O(n²) mixing was already known before, our approach additionally yields Chernoff-type concentration bounds for Hamming Lipschitz functions in this regime, which was not known before. Our approach is markedly different from prior works establishing spectral independence for spin systems using spatial mixing [Anari et al., 2020; Z. {Chen} et al., 2020; Chen et al., 2021; Feng et al., 2021], which crucially is still open in this regime for proper list-colorings.
Cite as

Kuikui Liu. From Coupling to Spectral Independence and Blackbox Comparison with the Down-Up Walk. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 32:1-32:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{liu:LIPIcs.APPROX/RANDOM.2021.32,
  author =	{Liu, Kuikui},
  title =	{{From Coupling to Spectral Independence and Blackbox Comparison with the Down-Up Walk}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{32:1--32:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.32},
  URN =		{urn:nbn:de:0030-drops-147259},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.32},
  annote =	{Keywords: Markov chains, Approximate counting, Spectral independence}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.41
Matroid Intersection: A Pseudo-Deterministic Parallel Reduction from Search to Weighted-Decision

Authors: Sumanta Ghosh and Rohit Gurjar

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
We study the matroid intersection problem from the parallel complexity perspective. Given two matroids over the same ground set, the problem asks to decide whether they have a common base and its search version asks to find a common base, if one exists. Another widely studied variant is the weighted decision version where with the two matroids, we are given small weights on the ground set elements and a target weight W, and the question is to decide whether there is a common base of weight at least W. From the perspective of parallel complexity, the relation between the search and the decision versions is not well understood. We make a significant progress on this question by giving a pseudo-deterministic parallel (NC) algorithm for the search version that uses an oracle access to the weighted decision. The notion of pseudo-deterministic NC was recently introduced by Goldwasser and Grossman [Shafi Goldwasser and Ofer Grossman, 2017], which is a relaxation of NC. A pseudo-deterministic NC algorithm for a search problem is a randomized NC algorithm that, for a given input, outputs a fixed solution with high probability. In case the given matroids are linearly representable, our result implies a pseudo-deterministic NC algorithm (without the weighted decision oracle). This resolves an open question posed by Anari and Vazirani [Nima Anari and Vijay V. Vazirani, 2020].
Cite as

Sumanta Ghosh and Rohit Gurjar. Matroid Intersection: A Pseudo-Deterministic Parallel Reduction from Search to Weighted-Decision. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 41:1-41:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ghosh_et_al:LIPIcs.APPROX/RANDOM.2021.41,
  author =	{Ghosh, Sumanta and Gurjar, Rohit},
  title =	{{Matroid Intersection: A Pseudo-Deterministic Parallel Reduction from Search to Weighted-Decision}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{41:1--41:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.41},
  URN =		{urn:nbn:de:0030-drops-147342},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.41},
  annote =	{Keywords: Linear Matroid, Matroid Intersection, Parallel Complexity, Pseudo-deterministic NC}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.83
Sampling Arborescences in Parallel

Authors: Nima Anari, Nathan Hu, Amin Saberi, and Aaron Schild

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
We study the problem of sampling a uniformly random directed rooted spanning tree, also known as an arborescence, from a possibly weighted directed graph. Classically, this problem has long been known to be polynomial-time solvable; the exact number of arborescences can be computed by a determinant [Tutte, 1948], and sampling can be reduced to counting [Jerrum et al., 1986; Jerrum and Sinclair, 1996]. However, the classic reduction from sampling to counting seems to be inherently sequential. This raises the question of designing efficient parallel algorithms for sampling. We show that sampling arborescences can be done in RNC. For several well-studied combinatorial structures, counting can be reduced to the computation of a determinant, which is known to be in NC [Csanky, 1975]. These include arborescences, planar graph perfect matchings, Eulerian tours in digraphs, and determinantal point processes. However, not much is known about efficient parallel sampling of these structures. Our work is a step towards resolving this mystery.
Cite as

Nima Anari, Nathan Hu, Amin Saberi, and Aaron Schild. Sampling Arborescences in Parallel. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 83:1-83:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{anari_et_al:LIPIcs.ITCS.2021.83,
  author =	{Anari, Nima and Hu, Nathan and Saberi, Amin and Schild, Aaron},
  title =	{{Sampling Arborescences in Parallel}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{83:1--83:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.83},
  URN =		{urn:nbn:de:0030-drops-136225},
  doi =		{10.4230/LIPIcs.ITCS.2021.83},
  annote =	{Keywords: parallel algorithms, arborescences, spanning trees, random sampling}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.56
An Extension of Plücker Relations with Applications to Subdeterminant Maximization

Authors: Nima Anari and Thuy-Duong Vuong

Published in: LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

Abstract
Given a matrix A and k ≥ 0, we study the problem of finding the k × k submatrix of A with the maximum determinant in absolute value. This problem is motivated by the question of computing the determinant-based lower bound of cite{LSV86} on hereditary discrepancy, which was later shown to be an approximate upper bound as well [Matoušek, 2013]. The special case where k coincides with one of the dimensions of A has been extensively studied. Nikolov gave a 2^{O(k)}-approximation algorithm for this special case, matching known lower bounds; he also raised as an open problem the question of designing approximation algorithms for the general case. We make progress towards answering this question by giving the first efficient approximation algorithm for general k× k subdeterminant maximization with an approximation ratio that depends only on k. Our algorithm finds a k^{O(k)}-approximate solution by performing a simple local search. Our main technical contribution, enabling the analysis of the approximation ratio, is an extension of Plücker relations for the Grassmannian, which may be of independent interest; Plücker relations are quadratic polynomial equations involving the set of k× k subdeterminants of a k× n matrix. We find an extension of these relations to k× k subdeterminants of general m× n matrices.
Cite as

Nima Anari and Thuy-Duong Vuong. An Extension of Plücker Relations with Applications to Subdeterminant Maximization. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 56:1-56:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{anari_et_al:LIPIcs.APPROX/RANDOM.2020.56,
  author =	{Anari, Nima and Vuong, Thuy-Duong},
  title =	{{An Extension of Pl\"{u}cker Relations with Applications to Subdeterminant Maximization}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{56:1--56:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-164-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{176},
  editor =	{Byrka, Jaros{\l}aw and Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.56},
  URN =		{urn:nbn:de:0030-drops-126596},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.56},
  annote =	{Keywords: Pl\"{u}cker relations, determinant maximization, local search, exchange property, discrete concavity, discrepancy}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2020.54
Matching Is as Easy as the Decision Problem, in the NC Model

Authors: Nima Anari and Vijay V. Vazirani

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Abstract
Is matching in NC, i.e., is there a deterministic fast parallel algorithm for it? This has been an outstanding open question in TCS for over three decades, ever since the discovery of randomized NC matching algorithms [Karp et al., 1985; Mulmuley et al., 1987]. Over the last five years, the theoretical computer science community has launched a relentless attack on this question, leading to the discovery of several powerful ideas. We give what appears to be the culmination of this line of work: An NC algorithm for finding a minimum-weight perfect matching in a general graph with polynomially bounded edge weights, provided it is given an oracle for the decision problem. Consequently, for settling the main open problem, it suffices to obtain an NC algorithm for the decision problem. We believe this new fact has qualitatively changed the nature of this open problem. All known efficient matching algorithms for general graphs follow one of two approaches: given by [Edmonds, 1965] and [Lovász, 1979]. Our oracle-based algorithm follows a new approach and uses many of ideas discovered in the last five years. The difficulty of obtaining an NC perfect matching algorithm led researchers to study matching vis-a-vis clever relaxations of the class NC. In this vein, recently [Goldwasser and Grossman, 2015] gave a pseudo-deterministic RNC algorithm for finding a perfect matching in a bipartite graph, i.e., an RNC algorithm with the additional requirement that on the same graph, it should return the same (i.e., unique) perfect matching for almost all choices of random bits. A corollary of our reduction is an analogous algorithm for general graphs.
Cite as

Nima Anari and Vijay V. Vazirani. Matching Is as Easy as the Decision Problem, in the NC Model. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 54:1-54:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{anari_et_al:LIPIcs.ITCS.2020.54,
  author =	{Anari, Nima and Vazirani, Vijay V.},
  title =	{{Matching Is as Easy as the Decision Problem, in the NC Model}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{54:1--54:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.54},
  URN =		{urn:nbn:de:0030-drops-117399},
  doi =		{10.4230/LIPIcs.ITCS.2020.54},
  annote =	{Keywords: Parallel Algorithm, Pseudo-Deterministic, Perfect Matching, Tutte Matrix}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2018.21
Planar Maximum Matching: Towards a Parallel Algorithm

Authors: Samir Datta, Raghav Kulkarni, Ashish Kumar, and Anish Mukherjee

Published in: LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Abstract
Perfect matchings in planar graphs have been extensively studied and understood in the context of parallel complexity [P W Kastelyn, 1967; Vijay Vazirani, 1988; Meena Mahajan and Kasturi R. Varadarajan, 2000; Datta et al., 2010; Nima Anari and Vijay V. Vazirani, 2017]. However, corresponding results for maximum matchings have been elusive. We partly bridge this gap by proving: 1) An SPL upper bound for planar bipartite maximum matching search. 2) Planar maximum matching search reduces to planar maximum matching decision. 3) Planar maximum matching count reduces to planar bipartite maximum matching count and planar maximum matching decision. The first bound improves on the known [Thanh Minh Hoang, 2010] bound of L^{C_=L} and is adaptable to any special bipartite graph class with non-zero circulation such as bounded genus graphs, K_{3,3}-free graphs and K_5-free graphs. Our bounds and reductions non-trivially combine techniques like the Gallai-Edmonds decomposition [L. Lovász and M.D. Plummer, 1986], deterministic isolation [Datta et al., 2010; Samir Datta et al., 2012; Rahul Arora et al., 2016], and the recent breakthroughs in the parallel search for planar perfect matchings [Nima Anari and Vijay V. Vazirani, 2017; Piotr Sankowski, 2018].
Cite as

Samir Datta, Raghav Kulkarni, Ashish Kumar, and Anish Mukherjee. Planar Maximum Matching: Towards a Parallel Algorithm. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 21:1-21:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{datta_et_al:LIPIcs.ISAAC.2018.21,
  author =	{Datta, Samir and Kulkarni, Raghav and Kumar, Ashish and Mukherjee, Anish},
  title =	{{Planar Maximum Matching: Towards a Parallel Algorithm}},
  booktitle =	{29th International Symposium on Algorithms and Computation (ISAAC 2018)},
  pages =	{21:1--21:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-094-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{123},
  editor =	{Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.21},
  URN =		{urn:nbn:de:0030-drops-99695},
  doi =		{10.4230/LIPIcs.ISAAC.2018.21},
  annote =	{Keywords: maximum matching, planar graphs, parallel complexity, reductions}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2018.41
Graph Clustering using Effective Resistance

Authors: Vedat Levi Alev, Nima Anari, Lap Chi Lau, and Shayan Oveis Gharan

Published in: LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

Abstract
We design a polynomial time algorithm that for any weighted undirected graph G = (V, E, w) and sufficiently large \delta > 1, partitions V into subsets V(1),..., V(h) for some h>= 1, such that at most \delta^{-1} fraction of the weights are between clusters, i.e. sum(i < j) |E(V(i), V(j)| < w(E)/\delta and the effective resistance diameter of each of the induced subgraphs G[V(i)] is at most \delta^3 times the inverse of the average weighted degree, i.e. max{ Reff(u, v) : u, v \in V(i)} < \delta^3 · |V|/w(E) for all i = 1,..., h. In particular, it is possible to remove one percent of weight of edges of any given graph such that each of the resulting connected components has effective resistance diameter at most the inverse of the average weighted degree. Our proof is based on a new connection between effective resistance and low conductance sets. We show that if the effective resistance between two vertices u and v is large, then there must be a low conductance cut separating u from v. This implies that very mildly expanding graphs have constant effective resistance diameter. We believe that this connection could be of independent interest in algorithm design.
Cite as

Vedat Levi Alev, Nima Anari, Lap Chi Lau, and Shayan Oveis Gharan. Graph Clustering using Effective Resistance. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 41:1-41:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{alev_et_al:LIPIcs.ITCS.2018.41,
  author =	{Alev, Vedat Levi and Anari, Nima and Lau, Lap Chi and Oveis Gharan, Shayan},
  title =	{{Graph Clustering using Effective Resistance}},
  booktitle =	{9th Innovations in Theoretical Computer Science Conference (ITCS 2018)},
  pages =	{41:1--41:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-060-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{94},
  editor =	{Karlin, Anna R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.41},
  URN =		{urn:nbn:de:0030-drops-83696},
  doi =		{10.4230/LIPIcs.ITCS.2018.41},
  annote =	{Keywords: Electrical Flows, Effective Resistance, Conductance, Graph Partitioning}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2017.36
Nash Social Welfare, Matrix Permanent, and Stable Polynomials

Authors: Nima Anari, Shayan Oveis Gharan, Amin Saberi, and Mohit Singh

Published in: LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

Abstract
We study the problem of allocating m items to n agents subject to maximizing the Nash social welfare (NSW) objective. We write a novel convex programming relaxation for this problem, and we show that a simple randomized rounding algorithm gives a 1/e approximation factor of the objective, breaking the 1/2e^(1/e) approximation factor of Cole and Gkatzelis. Our main technical contribution is an extension of Gurvits's lower bound on the coefficient of the square-free monomial of a degree m-homogeneous stable polynomial on m variables to all homogeneous polynomials. We use this extension to analyze the expected welfare of the allocation returned by our randomized rounding algorithm.
Cite as

Nima Anari, Shayan Oveis Gharan, Amin Saberi, and Mohit Singh. Nash Social Welfare, Matrix Permanent, and Stable Polynomials. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 36:1-36:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{anari_et_al:LIPIcs.ITCS.2017.36,
  author =	{Anari, Nima and Oveis Gharan, Shayan and Saberi, Amin and Singh, Mohit},
  title =	{{Nash Social Welfare, Matrix Permanent, and Stable Polynomials}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{36:1--36:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-029-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{67},
  editor =	{Papadimitriou, Christos H.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.36},
  URN =		{urn:nbn:de:0030-drops-81489},
  doi =		{10.4230/LIPIcs.ITCS.2017.36},
  annote =	{Keywords: Nash Welfare, Permanent, Matching, Stable Polynomial, Randomized Algorithm, Saddle Point}
}
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