Faster Exponential-Time Algorithms in Graphs of Bounded Average Degree

We first show that the Traveling Salesman Problem in an n-vertex graph with average degree bounded by d can be solved in O*(2((1-epsilon d)n)) time(1) and exponential space for a constant epsilon(d) depending only on d. Thus, we generalize the recent results of Bjorklund et al. [TALG 2012] on graphs of bounded degree. Then, we move to the problem of counting perfect matchings in a graph. We first present a simple algorithm for counting perfect matchings in an n-vertex graph in O*(2(n/2)) time and polynomial space; our algorithm matches the complexity bounds of the algorithm of Bjorklund [SODA 2012], but relies on inclusion-exclusion principle instead of algebraic transformations. Building upon this result, we show that the number of perfect matchings in an n-vertex graph with average degree bounded by d can be computed in O*(2((1-epsilon 2d)n/2)) time and exponential space, where epsilon(2d) is the constant obtained by us for the Traveling Salesman Problem in graphs of average degree at most 2d. Moreover we obtain a simple algorithm that computes a permanent of an n x n matrix over an arbitrary commutative ring with at most dn nonzero entries using O*(2((1-1/(3.55d))n)) time and ring operations, improving and simplifying the recent result of Izumi and Wadayama [FOGS 2012].