Exact Exponential-Time Algorithms for Domination Problems in Graphs

Abstract

This PhD thesis studies exact exponential-time algorithms for domination problems in graphs. Domination problems in graphs are a special kind of subset problems in graphs. A subset problem in a graph is a problem where one is given a graph G=(V,E), and one is asked whether there exist some subset S of a set of items U in the graph (mostly U is either the vertices V or the edges E) that satisfies certain properties. Domination problems in graphs are subset problems in which there is a domination criterion based on a neighbourhood relation in the graph that decides which elements of U are dominated by a given subset S, and where one of the properties that a solution subset S must satisfy is that S must dominate its complement U\S. The most well-known graph domination problem is the Dominating Set problem where the set U is the set of vertices V of G, a vertex subset S dominates all vertices in G that have a neighbour in S, and one is asked to compute the smallest vertex subset S of V that dominates all vertices in V\S. Other examples of domination problems in graphs are Independent Set, Edge Dominating Set, Total Dominating Set, Red-Blue Dominating Set, Partial Dominating Set, and #Dominating Set. We study exact exponential-time algorithms for these problems. These are algorithms that, when executed, use a number of operations that is exponential in a complexity parameter of the input in the worst case. That is, these algorithms use exponential time. Exact exponential-time algorithms then return an optimal solution to the problem. This in contrast to other fields of algorithm design where one sometimes trades running time for other properties of the algorithm or the returned solution. In this thesis, we also study parameterised algorithms for domination problems in graph on graph decompositions. These are algorithms whose worst-case running times are polynomial in the size of the graph and exponential in the graph-width parameter associated to the graph decomposition. Our study led to faster exact exponential-time algorithms for many well-known graph domination problems. This includes an O(1.4969^n)-time algorithm for Dominating Set, an O(1.2114^n)-time algorithm for Independent Set, an O(1.3226^n)-time algorithm for Edge Dominating Set, an O(1.4969^n)-time algorithm for Total Dominating Set, an O(1.2252^n)-time algorithm for Red-Blue Dominating Set, an O(1.5014^n)-time algorithm for Partial Dominating Set, and an O(1.5002^n)-time algorithm for #Dominating Set. We also obtained faster algorithms for these and many other graph domination problems on some prominent types of graph decompositions: tree decompositions, branch decompositions, and, for some problems, clique decompositions (also called k-expressions). A series of interesting new insights and techniques arose from this study. We mention the techniques of inclusion/exclusion-based branching and extended inclusion/exclusion-based branching. We also mention our generalisation of the fast subset convolution algorithm, which we translated to the setting of state-based dynamic programming algorithms on graph decompositions. This thesis also contains an accessible introduction to the field of exact exponential-time algorithms.