Experimental Evaluation of Approximation and Heuristic Algorithms for the Dominating Paths Problem

Monitoring flows on networks is a research area for which a number of applications are waiting for models and algorithms to face new problems emerging with a very high pace. In this paper we analyze a particular optimization problem, namely the Dominating Paths Problem (DPP), that has application in this filed specially for urban transportation networks. Given an undirected graph G = (V; E) and a subset B ⊆ V of bound vertices, we look for a set of vertices M of minimum size such that each element of M is the origin of one or more paths, and, the set of all these paths dominates B. For this NP-hard problem, we present an approximation algorithm and new heuristic procedures extensively evaluated on a set of test instances. We defined two different sets of benchmarks: grid graphs and random graphs. Moreover, we included two test cases taken from real traffic networks. Computational results, discussed in the paper, give insights both on the problem and on algorithms’ performance

Orienting dart-free clique-Helly chordal graphs

In this paper we acyclic orient dart-free clique-Helly chordal graphs in which each directed path is contained in at most two maximal cliques. As shown by the authors in previous works, this allows to give performance guarantee approximation results on a wide class of optimization problems

Partitioning Cliques of Claw-free Strongly Chordal Graphs

In this paper we find a particular partition of the vertex set of claw-free strongly chordal graphs in which each element is a clique, and we show that the adjacency graph of these cliques is a tree. In particular, the presented results imply the existence of an ordering of the vertices, and a corresponding edge orientation, such that each directed path is contained in at most two maximal cliques. As shown by the authors in previous works, this allows to give performance guarantee approximation results on a wide class of optimization problems

Vertex Partitioning of Crown-Free Interval Graphs

We study the problem of finding an acyclic orientation of an undirected graph G such that each path is contained in a limited number of maximal cliques of G. In general, in an acyclic oriented graph, each path is contained in more than one maximal cliques. We focus our attention on crown-free interval graphs, and show how to find an acyclic orientation of such a graph, which guarantees that each path is contained in at most four maximal cliques. The proposed technique is used to find approximated solutions for a class of related optimization problems where a solution corresponds to an acyclic orientation of graphs

An approximation result for the interval coloring problem on claw-free chordal graphs

We study the problem of finding an acyclic orientation of an undirected graph, such that each (oriented) path is covered by a limited number k of maximal cliques. This is equivalent to finding a k-approximate solution for the interval coloring problem on a graph. We focus our attention on claw-free chordal graphs, and show how to find an orientation of such a graph in linear time, which guarantees that each path is covered by at most two maximal cliques. This extends previous published results on other graph classes where stronger assumptions were made. (C) 2002 Elsevier Science B.V. All rights reserved

A Linear Time Approximation Algorithm for Interval Coloring on Proper Interval Graphs

Given a set of intervals on the real line, an interval graph is de®ned by a set of vertices associated to the intervals with edges between two vertices when the corresponding intervals overlap. If no interval properly contains another, we have a proper interval graph. When weights are associated to vertices, the interval coloring problem on a graph consists in assigning to each vertex a number of consecutive colors equal to the weight, such that adjacent vertices do not share any color and the total number of used colors is minimized. In this paper, we prove that this optimization problem on proper interval graphs is NP-hard. We give a linear time 2-approximation algorithm for it, and show that the bound is tight. Moreover, by exploiting the particular problem representation, the structure of the provided solutions are guaranteed to remain 2-approximated for any vertex weight function