A new NC-algorithm for finding a perfect matching in bipartite planar and small genus graphs (extended abstract)

Mahajan, Meena; Varadarajan, Kasturi

doi:10.1145/335305.335346

Cited by 28 publications

(33 citation statements)

References 27 publications

(27 reference statements)

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“…Mahajan and Varadarajan used lemma 2 to find linearly many edge-disjoint cycles in bipartite planar graphs [MV00]. We use a similar step, but instead of cycles we have to work with even walks, i.e., cycles with possibly repeated edges.…”

Section: Lemma 2 ([Lub86]mentioning

confidence: 99%

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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: Lemma 2 ([Lub86]mentioning

confidence: 99%

“…Lemma 18 (Adapted from [MV00]). There is an NC algorithm that returns |E|/288 edge-disjoint planar faces of a graph satisfying the assumptions of lemma 3.…”

Section: Finding Linearly Many Edge-disjoint Even Walksmentioning

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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“…Thus, it is of great interest to find specific graphs for which the maximum matching problem can be solved exactly [3]. In the past decades, the problems related maximum matchings have attracted considerable attention from the community of mathematics and theoretical computer science [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28].…”

Section: Introductionmentioning

confidence: 99%

Maximum matchings in scale-free networks with identical degree distribution

Zhang

2017

Theoretical Computer Science

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The size and number of maximum matchings in a network have found a large variety of applications in many fields. As a ubiquitous property of diverse real systems, power-law degree distribution was shown to have a profound influence on size of maximum matchings in scale-free networks, where the size of maximum matchings is small and a perfect matching often does not exist. In this paper, we study analytically the maximum matchings in two scale-free networks with identical degree sequence, and show that the first network has no perfect matchings, while the second one has many. For the first network, we determine explicitly the size of maximum matchings, and provide an exact recursive solution for the number of maximum matchings. For the second one, we design an orientation and prove that it is Pfaffian, based on which we derive a closed-form expression for the number of perfect matchings. Moreover, we demonstrate that the entropy for perfect matchings is equal to that corresponding to the extended Sierpiński graph with the same average degree as both studied scale-free networks. Our results indicate that power-law degree distribution alone is not sufficient to characterize the size and number of maximum matchings in scale-free networks.

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“…They obtained an NC algorithm for finding a maximum flow from a set of sources to a set of sinks in a planar network; as a corollary, they obtained an NC algorithm for finding a perfect matching in bipartite planar graphs. In 2000, Mahajan and Varadarajan gave an elegant way of using the NC algorithm for counting perfect matchings to find one, hence giving a different NC algorithm for bipartite planar graphs [28]; as is well known, Kasteleyn's algorithm for counting the number of perfect matchings in a planar graph [22] can be easily made into an NC algorithm for counting matchings by using Csanky's NC algorithm for the determinant of a matrix [7].…”

Section: History and Related Resultsmentioning

confidence: 99%

NC Algorithms for Computing a Perfect Matching, the Number of Perfect Matchings, and a Maximum Flow in One-Crossing-Minor-Free Graphs

Eppstein

Vazirani

2019

The 31st ACM Symposium on Parallelism in Algorithms and Architectures

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In 1988, Vazirani gave an NC algorithm for computing the number of perfect matchings in K 3,3 -minor-free graphs by building on Kasteleyn's scheme for planar graphs, and stated that this "opens up the possibility of obtaining an NC algorithm for finding a perfect matching in K 3,3 -free graphs." In this paper, we finally settle this 30-year-old open problem. Building on the recent breakthrough result of Anari and Vazirani giving an NC algorithm for finding a perfect matching in planar graphs and graphs of bounded genus, we also obtain NC algorithms for any minor-closed graph family that forbids a one-crossing graph. The class contains several well-studied graph families including the K 3,3 -minor-free graphs and K 5 -minor-free graphs. Graphs in these classes not only have unbounded genus, but also can have genus as high as O(n). In particular, we obtain NC algorithms for:• Determining whether a one-crossing-minor-free graph has a perfect matching and if so, finding one.• Finding a minimum weight perfect matching in a one-crossing-minor-free graph, assuming that the edge weights are polynomially bounded.• Finding a maximum st-flow in a one-crossing-minor-free flow network, with arbitrary capacities.The main new idea enabling our results is the definition and use of matching-mimicking networks, small replacement networks that behave the same, with respect to matching problems involving a fixed set of terminals, as the larger network they replace.The K 3,3 -minor-free graphs are particularly attractive as a target for this problem because they form a natural extreme case for certain approaches. In particular, they are known to have Pfaffian orientations, by which their matchings can be counted using matrix determinants [22,25], while for K 3,3 itself and for any minor-free family that does not forbid it, this tool is unavailable. However, our result breaks through this barrier: we give an NC algorithm for finding a perfect matching in graphs belonging to any one-crossing-minor-free class of graphs. That is, if H is any graph that can be drawn in the plane with only one crossing pair of edges, then we can find perfect matchings in the H-minor-free graphs in NC. Because K 3,3 can be drawn with one crossing, our result includes in particular the K 3,3 -minor-free graphs. More generally, we can find in NC a perfect matching of minimum weight, in the same families of graphs, when the weights are polynomially-bounded integers.In another direction, we obtain an NC algorithm for finding a maximum st-flow in any flow network whose underlying undirected graph belongs to a one-crossing-minor-free family. This generalizes Johnson's 1987 result [19], showing that maximum st-flow in a planar network is in NC. Technical ideasOur main new technical idea is that of a matching-mimicking network. Given a graph G and a set T of terminal vertices, a matching-mimicking network is a graph G , containing T , that has the same pattern of matchings: every matching of G that covers G \ T corresponds to a matching of G that covers G \ T and v...

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A new NC-algorithm for finding a perfect matching in bipartite planar and small genus graphs (extended abstract)

Cited by 28 publications

References 27 publications

Planar Graph Perfect Matching Is in NC

Planar Graph Perfect Matching Is in NC

Maximum matchings in scale-free networks with identical degree distribution

NC Algorithms for Computing a Perfect Matching, the Number of Perfect Matchings, and a Maximum Flow in One-Crossing-Minor-Free Graphs

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