Planar Graph Perfect Matching Is in NC

Anari, Nima; Vazirani, Vijay V.

doi:10.1109/focs.2018.00068

Cited by 10 publications

(7 citation statements)

References 29 publications

(27 reference statements)

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“…A number of new insights into matching were obtained in these works and several of them found their way, implicitly or explicitly, into the work of [1]. In a similar vein, we believe that results such as ours, which extend the frontier of NC matching algorithms, are likely to play a critical role towards the resolution of the full problem.…”

Section: History and Related Resultssupporting

confidence: 55%

“…• Because Anari and Vazirani [1] show how to find perfect matchings in bounded-genus graphs, our method immediately extends to facial 3-clique-sums of bounded-genus graphs and boundedtreewidth graphs. However, it is not clear what happens when bounded-genus pieces are glued by clique-sums on triangles that are not faces.…”

Section: Discussion and Open Problemsmentioning

confidence: 98%

“…• We need to be able to construct minimum-weight perfect matchings in the pieces of the structural decomposition, namely planar graphs and graphs of bounded treewidth. For planar graphs a minimum-weight perfect matching algorithm in NC (restricted to polynomiallybounded integer weights) was given by Anari and Vazirani [1]. For bounded treewidth graphs it appears that the log-space version of Courcelle's theorem [9] does not support optimization of structures expressible in monadic second-order logic, so it does not directly provide an algorithm for minimum-weight perfect matching in bounded-treewidth graphs.…”

Section: Minimum Weight Perfect Matching Algorithmmentioning

confidence: 99%

“…Obtaining an NC algorithm for matching has been an outstanding open question in theoretical computer science for over three decades, ever since the discovery of RNC matching algorithms [21,30]. In a recent breakthrough result, Anari and Vazirani gave an NC algorithm for finding a perfect matching in planar graphs [1]. By using a reduction from flow problems on other surfaces to planar flow [3], they also extended their result to graphs of bounded genus.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: History and Related Resultssupporting

confidence: 55%

Section: Discussion and Open Problemsmentioning

confidence: 98%

Section: Minimum Weight Perfect Matching Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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“…Our result extends this observation to planar graphs (see Appendix B for a proof of this reduction). Many graph problems are easier to solve for planar graphs than for general graphs; in particular, we note the NC algorithm for counting perfect matchings based on the work of Kasteleyn [Kas67] and Csanky [Csa75], and its remarkable recent application by Anari & Vazirani [AV18] (see also [San18]) to find perfect matchings in planar graphs. It is interesting that the lower bound for the Weighted Graph Matching problem derived by Mulmuley & Shah continues to hold even when the input is restricted to be planar.…”

Section: Pram Lower Boundsmentioning

confidence: 99%

Parametric Shortest Paths in Planar Graphs

Gajjar

Radhakrishnan

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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We construct a family of planar graphs {G n } n≥4 , where G n has n vertices including a source vertex s and a sink vertex t, and edge weights that change linearly with a parameter λ such that, as λ varies in (−∞, +∞), the piece-wise linear cost of the shortest path from s to t has n Ω(log n) pieces. This shows that lower bounds obtained earlier by Carstensen (1983) and Mulmuley & Shah (2001) for general graphs also hold for planar graphs, thereby refuting a conjecture of Nikolova (2009).Gusfield (1980) and Dean (2009) showed that the number of pieces for every n-vertex graph with linear edge weights is n log n+O(1) . We generalize this result in two ways. (i) If the edge weights vary as a polynomial of degree at most d, then the number of pieces is n log n+(α(n)+O(1)) d , where α(n) is the slow growing inverse Ackermann function. (ii) If the edge weights are linear forms of three parameters, then the number of pieces, appropriately defined for R 3 , is n (log n) 2 +O(log n) .

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On the Parallel Complexity of Constrained Read-Once Refutations in UTVPI Constraint Systems

Subramani

Wojciechowski²

2022

Lecture Notes in Computer Science

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Planar Graph Perfect Matching Is in NC

Cited by 10 publications

References 29 publications

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

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

Parametric Shortest Paths in Planar Graphs

On the Parallel Complexity of Constrained Read-Once Refutations in UTVPI Constraint Systems

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