In this work we consider two two-criteria optimization problems: given an input graph, the goal is to find its interval (or chordal) supergraph that minimizes the number of edges and its clique number simultaneously. For the interval supergraph, the problem can be restated as simultaneous minimization of the pathwidth @math and the profile @math of the input graph @math . We prove that for an arbitrary graph @math and an integer @math , there exists an interval supergraph @math of @math such that for its clique number it holds @math and the number of its edges is bounded by @math . In other words, the pathwidth and the profile of a graph can be simultaneously minimized within the factors of @math (plus a small constant) and @math , respectively. Note that for a fixed @math , both upper bounds provide constant factor approximations. On the negative side, we show an example that proves that, for some graphs, there is no solution in which both parameters are optimal. In case of finding a chordal supergraph, the two corresponding graph parameters that reflect its clique size and number of edges are the treewidth and fill-in. We obtain that the treewidth and the fill-in problems are also orthogonal' in the sense that for some graphs, a solution that minimizes one of those parameters cannot minimize the other. As a motivating example, we recall graph searching games which illustrates a need of simultaneous minimization of these pairs of graph parameters.
The treewidth counterpart of the length' minimization problem can be defined two-fold. It can be seen as minimizing, besides of the width of a tree decomposition, the number of its bags @cite_2 . On the other hand, minimization of width and the diameter of the underlying tree-structure of the decomposition has been studied in @cite_13 @cite_27 .
@cite_27 @cite_13 @cite_2
1574061374 1518294493 1573106092 1511821218
Tree-decompositions are the cornerstone of many dynamic programming algorithms for solving graph problems. Since the complexity of such algorithms generally depends exponentially on the width (size of the bags) of the decomposition, much work has been devoted to compute tree-decompositions with small width. However, practical algorithms computing tree-decompositions only exist for graphs with treewidth less than 4. In such graphs, the time-complexity of dynamic programming algorithms is dominated by the size (number of bags) of the tree-decompositions. It is then interesting to minimize the size of the tree-decompositions. In this extended abstract, we consider the problem of computing a tree-decomposition of a graph with width at most k and minimum size. We prove that the problem is NP-complete for any fixed k ≥ 4 and polynomial for k ≤ 2; for k = 3, we show that it is polynomial in the class of trees and 2-connected outerplanar graphs. Motivated by applications in parallel and dynamic graph algorithms, we investigate the tradeoff between width and diameter of tree decompositions. For all integers n, k and K with 1 ≤ k ≤ K ≤ n− 1, denote by D(n, k, K) the maximum, over all n-vertex graphs G of treewidth k, of the smallest diameter of a tree decomposition of G of width K. We determine D(n, k, K), up to a constant factor, for all values of n, k and K. When K is bounded by a constant (the case of greatest practical relevance), D(n, k, K) is θ(n) for K ≤ 2k-1, θ(√n) for 2k ≤ K ≤ 3k−2, and θ(log n) for K ≥ 3k−1. We provide much more accurate bounds for the case K ≤ 2k−1. This paper reviews structural problem decomposition methods, such as tree and path decompositions. It is argued that these notions can be applied in two distinct ways: Either to show that a problem is efficiently solvable when a width parameter is fixed, or to prove that the unrestricted (or some width-parameter free) version of a problem is tractable by using a width-notion as a mathematical tool for directly solving the problem at hand. Examples are given for both cases. As a new showcase for the latter usage, we report some recent results on the Partner Units Problem, a form of configuration problem arising in an industrial context. We use the notion of a path decomposition to identify and solve a tractable class of instances of this problem with practical relevance. In this paper we discuss algorithms that approximate the treewidth and pathwidth of cotriangulated graphs, permutation graphs and of cocomparability graphs. For a cotriangulated graph, of which the treewidth is at most k we show there exists an O(n2) algorithm finding a path-decomposition with width at most 3k+4. If G[π] is a permutation graph with treewidth k, then we show that the pathwidth of G[π] is at most 2k, and we give an algorithm which constructs a path-decomposition with width at most 2k in time O(nk). We assume that the permutation π is given. In this paper we also discuss the problem of finding an approximation for the treewidth and pathwidth of cocomparability graphs. We show that, if the treewidth of a cocomparability graph is at most k, then the pathwidth is at most O(k2), and we give a simple algorithm finding a path-decomposition with this width. The running time of the algorithm is dominated by a coloring algorithm of the graph. Such a coloring can be found in time O(n3).