Planarity, Determinants, Permanents, and (Unique) Matchings

Datta, Samir; Kulkarni, Raghav; Limaye, Nutan; Mahajan, Meena

doi:10.1145/1714450.1714453

Cited by 14 publications

(10 citation statements)

References 39 publications

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“…Limaye et al [2009] have shown that the longest path problem for planar DAGs also is solvable in UL. Finally, Datta et al [2007; use similar weighting techniques to establish improved upper bounds for bipartite planar matching problems.…”

Section: Resultsmentioning

confidence: 99%

Directed Planar Reachability Is in Unambiguous Log-Space

Bourke

Tewari

Vinodchandran

2009

ACM Trans. Comput. Theory

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We make progress in understanding the complexity of the graph reachability problem in the context of unambiguous logarithmic space computation; a restricted form of nondeterminism. As our main result, we show a new upper bound on the directed planar reachability problem by showing that it can be decided in the class unambiguous logarithmic space (UL). We explore the possibility of showing the same upper bound for the general graph reachability problem. We give a simple reduction showing that the reachability problem for directed graphs with thickness two is complete for the class nondeterministic logarithmic space (NL). Hence an extension of our results to directed graphs with thickness two will unconditionally collapse NL to UL.

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Section: Resultsmentioning

confidence: 99%

Directed Planar Reachability Is in Unambiguous Log-Space

Bourke

Tewari

Vinodchandran

2009

ACM Trans. Comput. Theory

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“…We know that the reachability in directed planar graphs reduces to bipartite planar matching while the reachability in layered grid graphs reduces to the UPM question in the same [DKLM07]. Note that the isolation step in our algorithm works in Logspace.…”

Section: Other Variationsmentioning

confidence: 99%

“…Proof. This is described in [DKLM07]. It follows closely the procedure for embedding a planar graph into a grid, described by [ABCDR06].…”

Section: Planar Matching and Grid Graphsmentioning

confidence: 99%

Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs

2009

Self Cite

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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 non-uniform 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 non-bipartite planar graphs, which has been open for a long time. Our techniques are elementary and simple.This work was done while the second author was visiting Chennai Mathematical Institute.

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“…The idea of graph embedding into a grid is quite standard. In this context of domino tilings it was used in [DKLM,Val2], and for other small sets of tiles in [BNRR, MR, PY]. Of course, the technical details are somewhat different in each case.…”

Section: Final Remarks and Open Problemsmentioning

confidence: 99%