Faster exponential-time algorithms in graphs of bounded average degree

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].Comment: 10 page

Below All Subsets for Some Permutational Counting Problems

We show that the two problems of computing the permanent of an $n\times n$ matrix of $\operatorname{poly}(n)$-bit integers and counting the number of Hamiltonian cycles in a directed $n$-vertex multigraph with $\operatorname{exp}(\operatorname{poly}(n))$ edges can be reduced to relatively few smaller instances of themselves. In effect we derive the first deterministic algorithms for these two problems that run in $o(2^n)$ time in the worst case. Classic $\operatorname{poly}(n)2^n$ time algorithms for the two problems have been known since the early 1960's. Our algorithms run in $2^{n-\Omega(\sqrt{n/\log n})}$ time.Comment: Corrected several technical errors, added comment on how to use the algorithm for ATSP, and changed title slightly to a more adequate on

Space Saving by Dynamic Algebraization

Dynamic programming is widely used for exact computations based on tree decompositions of graphs. However, the space complexity is usually exponential in the treewidth. We study the problem of designing efficient dynamic programming algorithm based on tree decompositions in polynomial space. We show how to construct a tree decomposition and extend the algebraic techniques of Lokshtanov and Nederlof such that the dynamic programming algorithm runs in time $O^*(2^h)$, where $h$ is the maximum number of vertices in the union of bags on the root to leaf paths on a given tree decomposition, which is a parameter closely related to the tree-depth of a graph. We apply our algorithm to the problem of counting perfect matchings on grids and show that it outperforms other polynomial-space solutions. We also apply the algorithm to other set covering and partitioning problems.Comment: 14 pages, 1 figur

GTRACE-RS: Efficient Graph Sequence Mining using Reverse Search

The mining of frequent subgraphs from labeled graph data has been studied extensively. Furthermore, much attention has recently been paid to frequent pattern mining from graph sequences. A method, called GTRACE, has been proposed to mine frequent patterns from graph sequences under the assumption that changes in graphs are gradual. Although GTRACE mines the frequent patterns efficiently, it still needs substantial computation time to mine the patterns from graph sequences containing large graphs and long sequences. In this paper, we propose a new version of GTRACE that enables efficient mining of frequent patterns based on the principle of a reverse search. The underlying concept of the reverse search is a general scheme for designing efficient algorithms for hard enumeration problems. Our performance study shows that the proposed method is efficient and scalable for mining both long and large graph sequence patterns and is several orders of magnitude faster than the original GTRACE

A survey on algorithmic aspects of modular decomposition

The modular decomposition is a technique that applies but is not restricted to graphs. The notion of module naturally appears in the proofs of many graph theoretical theorems. Computing the modular decomposition tree is an important preprocessing step to solve a large number of combinatorial optimization problems. Since the first polynomial time algorithm in the early 70's, the algorithmic of the modular decomposition has known an important development. This paper survey the ideas and techniques that arose from this line of research