Faster exponential-time algorithms in graphs of bounded average degree

• Authors: Marek Cygan;Marcin Pilipczuk
• Venue: ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
• Year: 2013

We first show that the Traveling Salesman Problem in an n-vertex graph with average degree bounded by d can be solved in $\mathcal{O}^\star (2^{(1-\varepsilon _d)n})$ 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 $\mathcal{O}^\star (2^{n/2})$ 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 $\mathcal{O}^\star (2^{(1-\varepsilon _{2d})n/2})$ 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 computes a permanent of an n ×n matrix over an arbitrary commutative ring with at most dn non-zero entries using $\mathcal{O}^\star (2^{(1-1/(3.55 d))n})$ time and ring operations, improving and simplifying the recent result of Izumi and Wadayama [FOCS 2012].