Derandomizing Isolation Lemma for $K_{3,3}$-free and $K_5$-free Bipartite Graphs

Rahul Arora; Ashu Gupta; Rohit Gurjar; Raghunath Tewari

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arXiv · 1411.7614

Derandomizing Isolation Lemma for $K_{3,3}$-free and $K_5$-free Bipartite Graphs

Rahul Arora·Ashu Gupta·Rohit Gurjar·Raghunath Tewari

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Abstract

The perfect matching problem has a randomized NC algorithm, using the celebrated Isolation Lemma of Mulmuley, Vazirani and Vazirani. The Isolation Lemma states that giving a random weight assignment to the edges of a graph, ensures that it has a unique minimum weight perfect matching, with a good probability. We derandomize this lemma for $K_{3,3}$-free and $K_5$-free bipartite graphs, i.e. we give a deterministic log-space construction of such a weight assignment for these graphs. Such a construction was known previously for planar bipartite graphs. Our result implies that the perfect matching problem for $K_{3,3}$-free and $K_5$-free bipartite graphs is in SPL. It also gives an alternate proof for an already known result -- reachability for $K_{3,3}$-free and $K_5$-free graphs is in UL.

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Rahul Arora, Ashu Gupta, Rohit Gurjar, Raghunath Tewari. 2014-11-27. Derandomizing Isolation Lemma for $K_{3,3}$-free and $K_5$-free Bipartite Graphs. https://arxiv.org/abs/1411.7614

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