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

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.