Solving frequency assignment problems via tree-decomposition

In this paper we describe a computational study to solve hard frequency assignment problems (FAPs) to optimality using a tree decomposition of the graph that models interference constraints. We present a dynamic programming algorithm which solves FAPs based on this tree decomposition. We show that with the use of several dominance and bounding techniques it is possible to solve small and medium-size real-life instances of the frequency assignment problem to optimality. Moreover, with an iterative version of the algorithm we obtain good lower bounds for large-size instances within reasonable time and memory limitsEconomics ;

A Dynamic Programming Algorithm for the ATM Network Installation Problem on a Tree

This paper considers the ATM Network Installation Problem on a tree. To install such a communication network, decisions concerning the location of hardware devices, the capacity installation on links, and the routing of demands have to be made simultaneously. The problem is shown to be NP-hard. By exploiting the tree structure we show that the problem can be solved to optimality using a pseudo-polynomial time dynamic programming algorithm. Computational experiments on real-life problem instances indicate that the algorithm is highly efficient.Economics ;

A Branch and Bound Algorithm for Exact, Upper, and Lower Bounds on Treewidth

In this paper, a branch and bound algorithm for computing the treewidth of a graph is presented. The method incorporates extensions of existing results, and uses new pruning and reduction rules, based upon roperties of the adopted branching strategy. We discuss how the algorithm can not only be used to obtain exact bounds for the treewidth, but also to obtain upper and/or lower bounds. Computational results of the algorithm are presented

A SAT Approach to Clique-Width

Clique-width is a graph invariant that has been widely studied in combinatorics and computer science. However, computing the clique-width of a graph is an intricate problem, the exact clique-width is not known even for very small graphs. We present a new method for computing the clique-width of graphs based on an encoding to propositional satisfiability (SAT) which is then evaluated by a SAT solver. Our encoding is based on a reformulation of clique-width in terms of partitions that utilizes an efficient encoding of cardinality constraints. Our SAT-based method is the first to discover the exact clique-width of various small graphs, including famous graphs from the literature as well as random graphs of various density. With our method we determined the smallest graphs that require a small pre-described clique-width.Comment: proofs in section 3 updated, results remain unchange

Treewidth Computations I. Upper Bounds.

For more and more applications, it is important to be able to compute the treewidth of a given graph and to find tree decompositions of small width reasonably fast. This paper gives an overview of several upper bound heuristics that have been proposed and tested for the problem of determining the treewidth of a graph and finding tree decompositions. Each of the heuristics produces tree decompositions whose width is not necessarily optimal, but experiments show that many of these come often close to the exact treewidth

Treewidth computations II. Lower bounds

For several applications, it is important to be able to compute the treewidth of a given graph and to find tree decompositions of small width reasonably fast. Good lower bounds on the treewidth of a graph can, amongst others, help to speed up branch and bound algorithms that compute the treewidth of a graph exactly. A high lower bound for a specific graph instance can tell that a dynamic programming approach for solving a problem is infeasible for this instance. This paper gives an overview of several recent methods that give lower bounds on the treewidth of graphs

Treewidth Lower Bounds with Brambles

In this paper we present a new technique for computing lower bounds for graph treewidth. Our technique is based on the fact that the treewidth of a graph G is the maximum order of a bramble of G minus one. We give two algorithms: one for general graphs, and one for planar graphs. The algorithm for planar graphs is shown to give a lower bound for both the treewidth and branchwidth that is at most a constant factor away from the optimum. For both algorithms, we report on extensive computational experiments that show that the algorithms give often excellent lower bounds, in particular when applied to (close to) planar graphs

Safe reduction rules for weighted treewidth

Several sets of reductions rules are known for preprocessing a graph when computing its treewidth. In this paper, we give reduction rules for a weighted variant of treewidth, motivated by the analysis of algorithms for probabilistic networks. We present two general reduction rules that are safe for weighted treewidth. They generalise many of the existing reduction rules for treewidth. Experimental results show that these reduction rules can significantly reduce the problem size for several instances of real-life probabilistic networks