Space Complexity of Perfect Matching in Bounded Genus Bipartite Graphs

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 logspace complexity class SPL. Since SPL is contained in the logspace 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 is in FL. Our results extend the same upper bounds for these problems over bipartite planar graphs known earlier. As our main technical result, we design a logspace 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. ∗Chennai Mathematical Institute, India: email:[email protected] †University of Chicago: email:[email protected] ‡University of Nebraska-Lincoln: email:[email protected]. Research supported in part by NSF grants CCF-0830730 and CCF-0916525. §University of Nebraska-Lincoln: email:[email protected]. Research supported in part by NSF grants CCF-0830730 and CCF-0916525. 1 ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 79 (2010)