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(2d) time and exponential space for a constant εd depending only on d. Thus, we generalize the recent results of Björklund 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) time and polynomial space; our algorithm matches the complexity bounds of the algorithm of Björklund [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(22d) time and exponential space, where ε2d 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 counts the number of perfect matchings in an n-vertex bipartite graph of average degree at most d in O(2) time, improving and simplifying the recent result of Izumi and Wadayama [FOCS 2012].