The Power of Data Reduction : Kernels for Fundamental Graph Problems

Abstract

The purpose of this thesis is to give a mathematical analysis of the power of data reduction for dealing with fundamental NP-hard graph problems. It has often been observed that the use of heuristic reduction rules in a preprocessing phase gives significant performance gains when solving such problems. However, there is little scientific explanation for these empirically observed successes. We use the concept of kernelization, developed within the field of parameterized complexity theory, to give a mathematical analysis of the power of such data reduction techniques. A kernelization, or kernel, is a polynomial-time preprocessing algorithm that transforms an instance of a parameterized problem into an equivalent instance whose size depends only on the parameter. The concept of kernelization therefore formalizes efficient and provably effective preprocessing. In our analysis of fundamental graph problems we utilize various structural measures of graphs as the complexity parameter; these include the vertex cover number, the feedback vertex number, the treewidth, and the vertex-deletion distance to various well-studied graph classes. We parameterize four fundamental classes of graph problems by such graph-structural measures. We determine which of these parameterizations admit kernelizations for which the size of the output is bounded by a polynomial in the parameter. Towards this end, we also develop technical tools to prove that a parameterized problem does not admit a kernel of polynomial size, subject to certain complexity-theoretic assumptions. The four fundamental problems we study are Vertex Cover, Treewidth, Graph Coloring, and Longest Path. For the Vertex Cover problem we introduce novel reduction rules that provably reduce the size of an instance to at most O(k^3) vertices in polynomial time, where k is the size of a feedback vertex set of the input graph. We also prove that the existence of a kernel for the parameterization by the vertex-deletion distance to an outerplanar graph or a clique, leads to a collapse of the polynomial hierarchy and is therefore unlikely. In our analysis of the Treewidth problem, we prove that preprocessing rules that were initially developed for heuristic algorithms, lead to a polynomial kernel for Treewidth parameterized by the vertex cover number. By developing additional rules that eliminate almost-simplicial vertices and shrink clique-seeing paths, we obtain a polynomial kernel parameterized by the feedback vertex number. Finally, we prove that Treewidth and Pathwidth do not admit polynomial kernels parameterized by the vertex-deletion distance to a clique, unless the polynomial hierarchy collapses. We analyze the kernelization complexity of graph coloring problems with respect to parameterizations that measure the vertex-deletion to graph classes such as cographs and co-chordal graphs. We show that the existence of polynomial kernels is determined by the extremal properties of No-instances of the List Coloring problem on such graph classes. Finally, we investigate Longest Path and related problems, with structural parameterizations. We obtain polynomial kernels for parameterizations by the vertex cover number, the max leaf number, and the vertex-deletion distance to a cluster graph. These results are complemented by a lower bound for the parameterization by the deletion distance to an outerplanar graph