The precise complexity of finding rainbow even matchings

Abstract

A progress in complexity lower bounds might be achieved by studying problems where a very precise complexity is conjectured. In this note we propose one such problem: Given a planar graph on n vertices and disjoint pairs of its edges p1,...,pg, perfect matching M is Rainbow Even Matching (REM) if (Formula Presented) is even for each i = 1,...,g. A straightforward algorithm finds a REM or asserts that no REM exists in (Formula Presented) steps and we conjecture that no deterministic or randomised algorithm has complexity asymptotically smaller than 2g. Our motivation is also to pinpoint the curse of dimensionality of the Max-Cut problem for graphs embedded into orientable surfaces: a basic problem of statistical physics.