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 p(1), ... , p(g), perfect matching M is Rainbow Even Matching (REM) if vertical bar M boolean AND p(i)vertical bar is even for each i = 1, ... , g.
A straightforward algorithm finds a REM or asserts that no REM exists in 2(g) x poly(n) steps and we conjecture that no deterministic or randomised algorithm has complexity asymptotically smaller than 2(g). 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.