Bipartite Perfect Matching is in Quasi-NC

Fenner, Stephen A.; Gurjar, Rohit; Thierauf, Thomas

doi:10.1137/16m1097870

Cited by 9 publications

(5 citation statements)

References 33 publications

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“…Several nice algorithmic ideas have been discovered in these works and our algorithm has benefited from some of these; in turn, it will not be surprising if some of our ideas turn out to be useful in the resolution of the main open problem. First, Fenner, Gurjar, and Thierauf gave a quasi-NC algorithm for perfect matching in bipartite graphs [FGT16], followed by the algorithm of Svensson and Tarnawski for general graphs [ST17]. Algorithms were also found for the generalization of bipartite matching to the linear matroid intersection problem by Gurjar and Thierauf [GT16], and to a further generalization of finding a vertex of a polytope with faces given by totally unimodular constraints, by Gurjar, Thierauf, and Vishnoi [GTV17].…”

Section: Introductionmentioning

confidence: 99%

Planar Graph Perfect Matching Is in NC

Anari

Vazirani

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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Is perfect matching in NC? That is, is there a deterministic fast parallel algorithm for it? This has been an outstanding open question in theoretical computer science for over three decades, ever since the discovery of RNC matching algorithms. Within this question, the case of planar graphs has remained an enigma: On the one hand, counting the number of perfect matchings is far harder than finding one (the former is #P-complete and the latter is in P), and on the other, for planar graphs, counting has long been known to be in NC whereas finding one has resisted a solution.In this paper, we give an NC algorithm for finding a perfect matching in a planar graph. Our algorithm uses the above-stated fact about counting matchings in a crucial way. Our main new idea is an NC algorithm for finding a face of the perfect matching polytope at which Ω(n) new conditions, involving constraints of the polytope, are simultaneously satisfied. Several other ideas are also needed, such as finding a point in the interior of the minimum weight face of this polytope and finding a balanced tight odd set in NC.

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Section: Introductionmentioning

confidence: 99%

Planar Graph Perfect Matching Is in NC

Anari

Vazirani

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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“…We note that the analysis of Gopalan et al [5] can be extended to non-binary alphabets F p , in which case their combinatorial characterization extends to the one above with H = F p . As a side remark, we note that the study of graphs with nonzero circulations was instrumental in the recent construction of a deterministic quasi-polynomial algorithm for perfect matching in NC [4]. However, beyond some superficial similarities, the setup seems inherently different than ours.…”

Section: Maximally Recoverable Codesmentioning

confidence: 88%

The Independence Number of the Birkhoff Polytope Graph, and Applications to Maximally Recoverable Codes

Kane¹,

Lovett²,

Rao³

2019

SIAM J. Comput.

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Maximally recoverable codes are codes designed for distributed storage which combine quick recovery from single node failure and optimal recovery from catastrophic failure. Gopalan et al [SODA 2017] studied the alphabet size needed for such codes in grid topologies and gave a combinatorial characterization for it.Consider a labeling of the edges of the complete bipartite graph K n,n with labels coming from F d 2 , that satisfies the following condition: for any simple cycle, the sum of the labels over its edges is nonzero. The minimal d where this is possible controls the alphabet size needed for maximally recoverable codes in n × n grid topologies.Prior to the current work, it was known that d is between (log n) 2 and n log n. We improve both bounds and show that d is linear in n. The upper bound is a recursive construction which beats the random construction. The lower bound follows by first relating the problem to the independence number of the Birkhoff polytope graph, and then providing tight bounds for it using the representation theory of the symmetric group. *

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“…Another related problem had been recently studied in [FGT16] in the context of derandomizing parallel algorithms for matching. The authors also consider the problem of assigning weights to edges of a graph, so that simple cycles carry non-zero weight.…”

Section: A Related Workmentioning

confidence: 99%

“…The authors also consider the problem of assigning weights to edges of a graph, so that simple cycles carry non-zero weight. The key differences from our setting are: we need a single assignment while [FGT16] may have multiple assignments; we care about all simple cycles, while [FGT16] only needed non-zero weights on short cycles; we are interested in fields of characteristic 2 while [FGT16] work in characteristic zero.…”

Section: A Related Workmentioning

confidence: 99%

Maximally Recoverable Codes for Grid-like Topologies

Gopalan¹,

Hu²,

Kopparty³

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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The explosion in the volumes of data being stored online has resulted in distributed storage systems transitioning to erasure coding based schemes. Yet, the codes being deployed in practice are fairly short. In this work, we address what we view as the main coding theoretic barrier to deploying longer codes in storage: at large lengths, failures are not independent and correlated failures are inevitable. This motivates designing codes that allow quick data recovery even after large correlated failures, and which have efficient encoding and decoding.We propose that code design for distributed storage be viewed as a two step process. The first step is choose a topology of the code, which incorporates knowledge about the correlate d failures that need to be handled, and ensures local recovery from such failures. In the second step one specifies a code with the chosen topology by choosing coefficients from a finite field Fq. In this step, one tries to balance reliability (which is better over larger fields) with encoding and decoding efficiency (which is better over smaller fields).This work initiates an in-depth study of this reliability/efficiency tradeoff. We consider the field-size needed for achieving maximal recoverability: the strongest reliability possible with a given topology. We propose a family of topologies called grid-like topologies which unify a number of topologies considered both in theory and practice, and prove the following results about codes for such topologies:The first super-polynomial lower bound on the field size needed for achieving maximal recoverability in a simple grid-like topology. To our knowledge, there was no super-linear lower bound known before, for any topology. A combinatorial characterization of erasure patterns correctable by Maximally Recoverable codes for a topology which corresponds to tensoring MDS codes with a parity check code. This topology is used in practice (for instance see [MLR + 14]). We conjecture a similar characterization for Maximally Recoverable codes instantiating arbitrary tensor product topologies.

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Bipartite Perfect Matching is in Quasi-NC

Cited by 9 publications

References 33 publications

Planar Graph Perfect Matching Is in NC

Planar Graph Perfect Matching Is in NC

The Independence Number of the Birkhoff Polytope Graph, and Applications to Maximally Recoverable Codes

Maximally Recoverable Codes for Grid-like Topologies

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