The Bidimensionality Theory and Its Algorithmic Applications

Abstract

Our newly developing theory of bidimensional graph problems provides general techniques for designing efficient fixed-parameter algorithms and approximation algorithms for NP-hard graph problems in broad classes of graphs. This theory applies to graph problems that are bidimensional in the sense that (1) the solution value for the k × k grid graph (and similar graphs) grows with k, typically as Ω(k2), and (2) the solution value goes down when contracting edges and optionally when deleting edges. Examples of such problems include feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex-removal parameters, dominating set, edge dominating set, r-dominating set, connected dominating set, connected edge dominating set, connected r-dominating set, and unweighted TSP tour (a walk in the graph visiting all vertices). Bidimensional problems have many structural properties; for example, any graph embeddable in a surface of bounded genus has treewidth bounded above by the square root of the problem's solution value. These properties lead to efficient---often subexponential---fixed-parameter algorithms, as well as polynomial-time approximation schemes, for many minor-closed graph classes. One type of minor-closed graph class of particular relevance has bounded local treewidth, in the sense that the treewidth of a graph is bounded above in terms of the diameter; indeed, we show that such a bound is always at most linear. The bidimensionality theory unifies and improves several previous results. The theory is based on algorithmic and combinatorial extensions to parts of the Robertson-Seymour Graph Minor Theory, in particular initiating a parallel theory of graph contractions. The foundation of this work is the topological theory of drawings of graphs on surfaces and our results regarding the relation (the linearity) of the size of the largest grid minor in terms of treewidth in bounded-genus graphs and more generally in graphs excluding a fixed graph H as a minor. In this thesis, we also develop the algorithmic theory of vertex separators, and its relation to the embeddings of certain metric spaces. Unlike in the edge case, we show that embeddings into L1 (and even Euclidean embeddings) are insufficient, but that the additional structure provided by many embedding theorems does suffice for our purposes. We obtain an O( logn ) approximation for min-ratio vertex cuts in general graphs, based on a new semidefinite relaxation of the problem, and a tight analysis of the integrality gap which is shown to be Θ( logn ). We also prove various approximate max-flow/min-vertex-cut theorems, which in particular give a constant-factor approximation for min-ratio vertex cuts in any excluded-minor family of graphs. Previously, this was known only for planar graphs, and for general excluded-minor families the best-known ratio was O(log n). These results have a number of applications. We exhibit an O( logn ) pseudo-approximation for finding balanced vertex separators in general graphs. Furthermore, we obtain improved approximation ratios for treewidth: In any graph of treewidth k, we show how to find a tree decomposition of width at most O(k logk ), whereas previous algorithms yielded O( k log k). For graphs excluding a fixed graph as a minor, we give a constant-factor approximation for the treewidth; this via the bidimensionality theory can be used to obtain the first polynomial-time approximation schemes for problems like minimum feedback vertex set and minimum connected dominating set in such graphs. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)