Solving Connectivity Problems Parameterized by Treewidth in Single Exponential Time

Abstract

For the vast majority of local problems on graphs of small treewidth (where, by local we mean that a solution can be verified by checking separately the neighbourhood of each vertex), standard dynamic programming techniques give c tw | V | O(1) time algorithms, where tw is the treewidth of the input graph G = ( V,E ) and c is a constant. On the other hand, for problems with a global requirement (usually connectivity) the best–known algorithms were naive dynamic programming schemes running in at least tw tw time. We bridge this gap by introducing a technique we named Cut&Count that allows to produce c tw | V | O(1) time Monte-Carlo algorithms for most connectivity-type problems, including Hamiltonian Path , Steiner Tree , Feedback Vertex Set and Connected Dominating Set . These results have numerous consequences in various fields, like parameterized complexity, exact and approximate algorithms on planar and H -minor-free graphs and exact algorithms on graphs of bounded degree. The constant c in our algorithms is in all cases small, and in several cases we are able to show that improving those constants would cause the Strong Exponential Time Hypothesis to fail. In all these fields we are able to improve the best-known results for some problems. Also, looking from a more theoretical perspective, our results are surprising since the equivalence relation that partitions all partial solutions with respect to extendability to global solutions seems to consist of at least tw tw equivalence classes for all these problems. Our results answer an open problem raised by Lokshtanov, Marx and Saurabh [SODA’11]. In contrast to the problems aimed at minimizing the number of connected components that we solve using Cut&Count as mentioned above, we show that, assuming the Exponential Time Hypothesis, the aforementioned gap cannot be bridged for some problems that aim to maximize the number of connected components like Cycle Packing .

Detours in Directed Graphs

“…Combining this fact with an FPT 3approximation algorithm [11], running in time 2 O(k) •n O (1) , to compute the treewidth of a graph, brings us to a graph of treewidth at most 96k + O (1). Finally, solving Longest Detour on graphs of bounded treewidth by one of the known single-exponential algorithms, see [19,10,27], will result in running time 3 96k • n O (1) . Thus on undirected graphs, our algorithm reduces the constant c in the base of the exponent from 3 96 down to 10.8!…”

Four Shorts Stories on Surprising Algorithmic Uses of Treewidth

“…Let us consider the k-Path problem as our running example: given a (planar) graph G and an integer k, we have to decide if G contains a simple path on k vertices. Let us first note that k-Path is FPT parameterized by the treewidth w of the input graph G. More precisely, standard dynamic programming tech-niques give 2 O(w log w) n O (1) running time, while more sophisticated arguments are needed to obtain 2 O(w) n O (1) time [11,25,33,34,36,37,45] (note that some of these algorithms are randomized and some of these algorithms work only on planar graphs).…”

2-Approximating Feedback Vertex Set in Tournaments

“…In the case of directed graphs, it means G − S is a directed acyclic graph (DAG). In the (Directed) Feedback Vertex Set ((D)FVS) problem we are given as input a (directed) graph G and a weight function w : V (G) → N. The objective is to find a minimum weight feedback vertex set S. Both the directed and undirected version of the problem are NP-complete [14] and have been extensively studied from the perspective of approximation algorithms [1,12], parameterized algorithms [6,8,19], exact exponential time algorithms [23,29] as well as graph theory [11,24].…”