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The directed planar reachability problem
 In Proc. 25th annual Conference on Foundations of Software Technology and Theoretical Computer Science (FST&TCS), number 1373 in Lecture Notes in Computer Science
, 2005
"... Abstract. We investigate the stconnectivity problem for directed planar graphs, which is hard for L and is contained in NL but is not known to be complete. We show that this problem is logspacereducible to its complement, and we show that the problem of searching graphs of genus 1 reduces to the ..."
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Cited by 20 (8 self)
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Abstract. We investigate the stconnectivity problem for directed planar graphs, which is hard for L and is contained in NL but is not known to be complete. We show that this problem is logspacereducible to its complement, and we show that the problem of searching graphs of genus 1 reduces to the planar case. We also consider a previouslystudied subclass of planar graphs known as grid graphs. We show that the directed planar stconnectivity problem reduces to the reachability problem for directed grid graphs. A special case of the gridgraph reachability problem where no edges are directed from right to left is known as the “acyclic grid graph reachability problem”. We show that this problem lies in the complexity class UL. 1
DETERMINISTICALLY ISOLATING A PERFECT MATCHING IN BIPARTITE PLANAR GRAPHS
"... Abstract. We present a deterministic way of assigning small (log bit) weights to the edges of a bipartite planar graph so that the minimum weight perfect matching becomes unique. The isolation lemma as described in [MVV87] achieves the same for general graphs using a randomized weighting scheme, whe ..."
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Cited by 3 (2 self)
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Abstract. We present a deterministic way of assigning small (log bit) weights to the edges of a bipartite planar graph so that the minimum weight perfect matching becomes unique. The isolation lemma as described in [MVV87] achieves the same for general graphs using a randomized weighting scheme, whereas we can do it deterministically when restricted to bipartite planar graphs. As a consequence, we reduce both decision and construction versions of the matching problem to testing whether a matrix is singular, under the promise that its determinant is 0 or 1, thus obtaining a highly parallel SPL algorithm for bipartite planar graphs. This improves the earlier known bounds of nonuniform SPL by [ARZ99] and NC 2 by [MN95, MV00]. It also rekindles the hope of obtaining a deterministic parallel algorithm for constructing a perfect matching in nonbipartite planar graphs, which has been open for a long time. Our techniques are elementary and simple. 1.
Seeking a vertex of the planar matching polytope in nc
 In Proceedings of the 12th European Symposium on Algorithms ESA, LNCS
, 2004
"... Abstract. For planar graphs, counting the number of perfect matchings (and hence determining whether there exists a perfect matching) can be done in NC [4, 10]. For planar bipartite graphs, finding a perfect matching when one exists can also be done in NC [8, 7]. However in general planar graphs (wh ..."
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Cited by 2 (2 self)
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Abstract. For planar graphs, counting the number of perfect matchings (and hence determining whether there exists a perfect matching) can be done in NC [4, 10]. For planar bipartite graphs, finding a perfect matching when one exists can also be done in NC [8, 7]. However in general planar graphs (when the bipartite condition is removed), no NC algorithm for constructing a perfect matching is known. We address a relaxation of this problem. We consider the fractional matching polytope P(G) of a planar graph G. Each vertex of this polytope is either a perfect matching, or a halfintegral solution: an assignment of weights from the set {0, 1/2, 1} to each edge of G so that the weights of edges incident on each vertex of G add up to 1 [6]. We show that a vertex of this polytope can be found in NC, provided G has at least one perfect matching to begin with. If, furthermore, the graph is bipartite, then all vertices are integral, and thus our procedure actually finds a perfect matching without explicitly exploiting the bipartiteness of G. 1
Some perfect matchings and perfect halfintegral matchings in NC ∗
, 2008
"... We show that for any class of bipartite graphs which is closed under edge deletion and where the number of perfect matchings can be counted in NC, there is a deterministic NC algorithm for finding a perfect matching. In particular, a perfect matching can be found in NC for planar bipartite graphs an ..."
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Cited by 2 (2 self)
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We show that for any class of bipartite graphs which is closed under edge deletion and where the number of perfect matchings can be counted in NC, there is a deterministic NC algorithm for finding a perfect matching. In particular, a perfect matching can be found in NC for planar bipartite graphs and K3,3free bipartite graphs via this approach. A crucial ingredient is part of an interiorpoint algorithm due to Goldberg, Plotkin, Shmoys and Tardos. An easy observation allows this approach to handle regular bipartite graphs as well. We show, by a careful analysis of the polynomial time algorithm due to Galluccio and Loebl, that the number of perfect matchings in a graph of small (O(log n)) genus can be counted in NC. So perfect matchings in small genus bipartite graphs can also be found via this approach. We then present a different algorithm for finding a perfect matching in a planar bipartite graph. This algorithm is substantially different from the algorithm described above, and also from the algorithm of Miller and Naor, which predates the approach of Goldberg et al. and tackles the same problem. Our new algorithm extends to small genus bipartite graphs, but not to K3,3free bipartite graphs. We next show that a nontrivial extension of this algorithm allows
COMPLEXITY THEORETIC ASPECTS OF PLANAR RESTRICTIONS AND OBLIVIOUSNESS
, 2006
"... In this thesis, we deal largely with complexity theoretic aspects in planar restrictions and obliviousness. Our main motivation was to identify problems for which the planar restriction is much easier, computationally, than the unrestricted version. First, we study constant width polynomialsized ci ..."
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In this thesis, we deal largely with complexity theoretic aspects in planar restrictions and obliviousness. Our main motivation was to identify problems for which the planar restriction is much easier, computationally, than the unrestricted version. First, we study constant width polynomialsized circuits of low (polylogarithmic) genus; we show how such circuits characterize exactly the wellknown circuit complexity class ACC0 (given that the unrestricted version captures the whole of NC1). We also give a new circuit characterization of the class NC1. Shifting our focus from circuits to graphs, we look at different notions of connectivity. We investigate the directed planar graph reachability problem, as a possibly more tractable special case of the arbitrary graph reachability problem (which is NLcomplete). We prove that this problem logspacereduces to its complement, and also that reachability questions on genus 1 graphs reduce to that in planar graphs. We also prove that reachability in a particularly simple class of planar graphs (namely, grid graphs) is no easier than the general directed planar reachability question. We then proceed to isolate to several large classes of planar graphs for which the reachability questions are solvable in deterministic logspace. Counting the number of spanning trees in a graph is a useful extension of the task of determining
Improved Bounds for Bipartite Matching on Surfaces
"... We exhibit the following new upper bounds on the space complexity and the parallel complexity of the Bipartite Perfect Matching (BPM) problem for graphs of small genus: (1) BPM in planar graphs is in UL (improves upon the SPL bound from Datta et. al. [7]); (2) BPM in constant genus graphs is in NL ( ..."
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We exhibit the following new upper bounds on the space complexity and the parallel complexity of the Bipartite Perfect Matching (BPM) problem for graphs of small genus: (1) BPM in planar graphs is in UL (improves upon the SPL bound from Datta et. al. [7]); (2) BPM in constant genus graphs is in NL (orthogonal to the SPL bound from Datta et. al. [8]); (3) BPM in polylogarithmic genus graphs is in NC; (extends the NC bound for O(log n) genus graphs from Mahajan and Varadarajan [22], and Kulkarni et. al. [19]. For Part (1) we combine the flow technique of Miller and Naor [23] with the double counting technique of Reinhardt and Allender [27]. For Part (2) and (3) we extend [23] to higher genus surfaces in the spirit of Chambers, Erickson and Nayyeri [4].