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×n matrix over an arbitrary commutative ring with at most dn non-zero entries using O (2 (1−1/(3.55d))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 Björklund [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 that

Exact Algorithms on Graphs of Bounded Average Degree

“…In 2008, Björklund, Husfeldt, Kaski, and Koivisto [5,6] observed that such an improvement can be made if we restrict ourselves to bounded degree graphs. Further work of Cygan and Pilipczuk [8] and Golovnev, Kulikov, and Mihajlin [9] extended these results to graphs of bounded average degree.…”

“…With this line of reasoning, Cygan and Pilipczuk [8] showed that for every degree bound d , there exists a constant " d > 0 such that traveling salesman problem in graphs of bounded average degree by d can be solved in .2 " d / n n O.1/ time. It should be noted that the constant " d depends here doubly exponentially on d , as opposed to single-exponential dependency in the works for bounded degree graphs.…”

“…The above approach for traveling salesman problem has been generalized to graphs of bounded average degree by Cygan and Pilipczuk [8] using the following observation. Assume a graph G has n vertices and average degree bounded by d .…”