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

Rahul Arora; Ashu Gupta; Rohit Gurjar; Raghunath Tewari

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

Computational Complexity 2014-12-01 v1 Data Structures and Algorithms

Authors: Rahul Arora , Ashu Gupta , Rohit Gurjar , Raghunath Tewari

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.

Keywords

isolation lemma derandomization perfect matching K_{3,3}-free bipartite graphs K_5-free bipartite graphs log-space construction SPL

Cite

@article{arxiv.1411.7614,
  title  = {Derandomizing Isolation Lemma for $K_{3,3}$-free and $K_5$-free Bipartite Graphs},
  author = {Rahul Arora and Ashu Gupta and Rohit Gurjar and Raghunath Tewari},
  journal= {arXiv preprint arXiv:1411.7614},
  year   = {2014}
}

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

Abstract

Keywords

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