Faster exponential-time algorithms in graphs of bounded average degree

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−ε d )n ) time 1 and exponential space for a constant ε d 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 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 O ⋆ (2 (1−ε 2d )n/2 ) time and exponential space, where ε 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 [3], and breaking these time complexity barriers seems like a very challenging task.From a broader perspective, improving upon a trivial brute-force or a simple dynamic programming algorithm is one of the main goals the field of exponential-time algorithms. Although the last few years brought a number of positive results in that direction, most notably the O ⋆ (1.66 n ) randomized algorithm for finding a Hamiltonian cycle in an undirected graph [2], it is conjectured (the so-called Strong Exponential Time Hypothesis [8]) that the problem of satisfying a general CNF-SAT formulae does not admit any exponentially better algorithm than the trivial brute-force one. A number of lower bounds were proven using this assumption [6,10,11].In 2008 Björklund et al. [5] observed that the classical dynamic programming algorithm for TSP can be trimmed to running time O ⋆ (2 (1−ε∆)n ) in graphs of maximum degree ∆. The cost of this improvement is the use of exponential space, as we can no longer easily translate the dynamic programming algorithm into an inclusion-exclusion formula. The ideas from [5] were also applied to the Fast Subset Convolution algorithm, yielding a similar improvements for the problem of computing the chromatic number in graphs of bounded * Partially supported by NCN grant N206567140 and Foundation for Polish Science.

A Faster Hafnian Formula for Complex Matrices and Its Benchmarking on a Supercomputer

“…Finally, Björklund [29] and later Cygan and Pilipczuk [14] provided O(poly(n)2 n/2 ) time and polynomial space algorithms for the calculation of the general ring hafnian. These algorithms are believed to be close to optimal unless there are surprisingly efficient algorithms for the Permanent.…”

“…In this paper we improve upon recently developed algorithms for counting the number of perfect matchings of undirected graphs [29,14] and the calculation of hafnians. Furthermore these algorithms are generalized to allow for the inclusion of graphs that contain loops which have been recently shown to be linked to the calculation of spectral lines of molecules [3].…”

“…In the following sections we present an algorithm that allows us to count the number of perfect matchings of a graph with n vertices in time O(n 3 2 n/2 ) for (unweighted) graphs with and without loops, and then generalize it to weighted graphs. Our algorithm and its analysis largely follows that of Cygan and Pilipczuk [14] with one crucial exception. Whereas the terms in their formula are computed by an O(n 4 ) dynamic programming tabulation, we reduce the terms to efficiently computable functions of the traces of the first n/2 powers of a matrix.…”

“…In this manuscript we develop a new algorithm to compute hafnians of general complex matrices that runs in O(n 3 2 n/2 ) time where n is the size of the matrix. Our algorithm builds on the hafnian algorithm of Cygan and Pilipczuk [14], here adapted to the field of complex numbers using elementary tools from linear algebra. Compared to the general ring hafnian algorithm of Cygan and Pilipczuk, our algorithm is a factor n faster.…”

Families with Infants: A General Approach to Solve Hard Partition Problems

“…However in recent years it was shown that the bound can be improved for various special cases. In [5,6,8] better upper bounds are proved for graphs of bounded degree.…”