Space complexity of perfect matching in bounded genus bipartite graphs

摘要

We investigate the space complexity of certain perfect matching problems over bipartite graphs embedded on surfaces of constant genus (orientable or non-orientable). We show that the problems of deciding whether such graphs have (1) a perfect matching or not and (2) a unique perfect matching or not, are in the log-space complexity class Stoic Probabilistic Log-space (SPL). Since SPL is contained in the log-space counting classes ⊕L (in fact in ModkL for all k⩾2), C=L, and PL, our upper bound places the above-mentioned matching problems in these counting classes as well. We also show that the search version, computing a perfect matching, for this class of graphs can be performed by a log-space transducer with an SPL oracle. Our results extend the same upper bounds for these problems over bipartite planar graphs known earlier. As our main technical result, we design a log-space computable and polynomially bounded weight function which isolates a minimum weight perfect matching in bipartite graphs embedded on surfaces of constant genus. We use results from algebraic topology for proving the correctness of the weight function.