Faster exponential-time algorithms in graphs of bounded average degree

TL;DR

This paper presents faster exponential-time algorithms for the Traveling Salesman Problem and counting perfect matchings in graphs with bounded average degree, generalizing recent results and improving efficiency.

Contribution

It introduces new algorithms with improved exponential bounds for TSP and perfect matchings in graphs of bounded average degree, extending prior work.

Findings

TSP solvable in O*(2^{(1-ε_d)n}) time for graphs with average degree d

Counting perfect matchings in O*(2^{n/2}) time using inclusion-exclusion

Enhanced algorithms for graphs with bounded average degree with improved exponential bounds

Abstract

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-\eps_d)n}) time and exponential space for a constant \eps_d depending only on d, where the O*-notation suppresses factors polynomial in the input size. 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-\eps_{2d})n/2}) time and exponential space, where \eps_{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 counts the number of perfect matchings in an n-vertex bipartite graph of average degree at most d in O*(2^{(1-1/(3.55d))n/2}) time, improving and simplifying the recent result of Izumi and Wadayama [FOCS 2012].