Faster Exponential-Time Algorithms in Graphs of Bounded Average Degree

Abstract

We present a number of exponential-time algorithms for problems in sparse matrices and graphs of bounded average degree. First, we obtain a simple algorithm that computes a permanent of an n i? n matrix over an arbitrary commutative ring with at most dn non-zero entries using O ? ( 2 ( 1 - 1 / ( 3.55 d ) ) n ) time and ring operations,1 improving and simplifying the recent result of Izumi and Wadayama FOCS 2012].Second, we 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.Third, we show a combinatorial lemma that bounds the number of Hamiltonian-like structures in a graph of bounded average degree. Using this result, we show that1.a minimum weight Hamiltonian cycle in an n-vertex graph with average degree bounded by d can be found in O ? ( 2 ( 1 - e d ) n ) time and exponential space for a constant e d depending only on d;2.the number of perfect matchings in an n-vertex graph with average degree bounded by d can be computed in O ? ( 2 ( 1 - e d ' ) n / 2 ) time and exponential space, for a constant e d ' depending only on d. The algorithm for minimum weight Hamiltonian cycle generalizes the recent results of Bjorklund et al. TALG 2012] on graphs of bounded degree.