Approximation Algorithms for the Capacitated Domination Problem

We consider the {\em Capacitated Domination} problem, which models a service-requirement assignment scenario and is also a generalization of the well-known {\em Dominating Set} problem. In this problem, given a graph with three parameters defined on each vertex, namely cost, capacity, and demand, we want to find an assignment of demands to vertices of least cost such that the demand of each vertex is satisfied subject to the capacity constraint of each vertex providing the service. In terms of polynomial time approximations, we present logarithmic approximation algorithms with respect to different demand assignment models for this problem on general graphs, which also establishes the corresponding approximation results to the well-known approximations of the traditional {\em Dominating Set} problem. Together with our previous work, this closes the problem of generally approximating the optimal solution. On the other hand, from the perspective of parameterization, we prove that this problem is {\it W[1]}-hard when parameterized by a structure of the graph called treewidth. Based on this hardness result, we present exact fixed-parameter tractable algorithms when parameterized by treewidth and maximum capacity of the vertices. This algorithm is further extended to obtain pseudo-polynomial time approximation schemes for planar graphs

Fixed-parameter approximation: Conceptual framework and approximability results

Abstract. The notion of fixed-parameter approximation is introduced to investigate the approximability of optimization problems within the framework of fixed-parameter computation. This work partially aims at enhancing the world of fixed-parameter computation in parallel with the conventional theory of computation that includes both exact and approximate computations. In particular, it is proved that fixed-parameter approximability is closely related to the approximation of small-cost solutions in polynomial time. It is also demonstrated that many fixedparameter intractable problems are not fixed-parameter approximable. On the other hand, fixed-parameter approximation appears to be a viable approach to solving some inapproximable yet important optimization problems. For instance, all problems in the class MAX SNP admit fixed-parameter approximation schemes in time O(2 O((1−ɛ/O(1))k) p(n)) for any small ɛ>0.

Geosphere

It is known that the treewidth of a planar graph with a dominating set of size d is O( # d) and this fact is used as the basis for several fixed parameter algorithms on planar graphs. An interesting question motivating our study is if similar bounds can be obtained for larger minor closed graph families. We say that a graph family has the domination-treewidth property if there is some function f(d) such that every graph G f(d). We show that a minor-closed graph family has bounded local treewidth. This result has important algorithmic consequences

Approximation algorithms for unit disk graphs

Mobile ad hoc networks are frequently modeled by unit disk graphs. We consider several classical graph theoretic problems on unit disk graphs (Maximum Independent Set, Minimum Vertex Cover, and Minimum (Connected) Dominating Set), which are relevant to such networks. We propose two new notions for unit disk graphs: thickness and density. The thickness of a graph is the number of disk centers in any width 1 slab. If the thickness of a graph is bounded, then the considered problems can be solved in polynomial time. We prove this both indirectly by presenting a relation between unit disk graphs of bounded thickness and the pathwidth of such graphs, and directly by giving dynamic programming algorithms. This result implies that the problems are fixed-parameter tractable in the thickness. We then consider unit disk graphs of bounded density. The density of a graph is the number of disk centers in any 1-by-1 box. We present a new approximation scheme for the considered problems, which uses the bounded thickness results mentioned above and the so called shifting technique. We show that the scheme is an asymptotic FPTAS and that this result is optimal, in the sense that no FPTAS can exist (unless P=NP). The scheme for Minimum Connected Dominating Set is the first FPTAS∞ for this problem. The analysis that is applied can also be used to improve existing results, which among others implies the existence of an FPTAS∞ for MCDS on planar graphs

Faster Approximation Schemes and Parameterized Algorithms on H-Minor-Free and Odd-Minor-Free Graphs

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Guard Games on Graphs: Keep the Intruder Out!

A team of mobile agents, called guards, tries to keep an intruder out of an assigned area by blocking all possible attacks. In a graph model for this setting, the guards and the intruder are located on the vertices of a graph, and they move from node to node via connecting edges. The area protected by the guards is an induced subgraph of the given graph. We investigate the algorithmic aspects the guarding problem, which is to find the minimum number of guards sufficient to patrol the area. We show that the guarding problem is PSPACE-hard and provide a set of approximation algorithms. All approximation algorithms are based on the study of a variant of the game where the intruder must reach the guarded area in a single step in order to win. This variant of the game appears to be a 2-approximation for the guarding problem, and for graphs without cycles of length 5 the minimum number of required guards in both games coincides. We give a polynomial time algorithm for solving the one-step guarding problem in graphs of bounded treewidth, and complement this result by showing that the problem is W[1]-hard parameterized by the treewidth of the input graph. We also show that the problem is fixed parameter tractable (FPT) parameterized by the treewidth and maximum degree of the input graph. Finally, we turn our attention to a large class of sparse graphs, including plana

Bidimensional parameters and local treewidth

Abstract. For several graph-theoretic parameters such as vertex cover and dominating set, it is known that if their sizes are bounded by k then the treewidth of the graph is bounded by some function of k. This fact is used as the main tool for the design of several fixed-parameter algorithms on minor-closed graph classes such as planar graphs, single-crossing-minor-free graphs, and graphs of bounded genus. In this paper we examine the question whether similar bounds can be obtained for larger minor-closed graph classes, and for general families of graph parameters including all those for which such behavior has been reported so far. Given a graph parameter P, we say that a graph family F has the parameter-treewidth property for P if there is an increasing function t such that every graph G ∈ F has treewidth at most t(P (G)). We prove as our main result that, for a large family of graph parameters called contraction-bidimensional, a minor-closed graph family F has the parameter-treewidth property if F has bounded local treewidth. We also show “if and only if ” for some graph parameters, and thus this result is in some sense tight. In addition we show that, for a slightly smaller family of graph parameters called minor-bidimensional, all minor-closed graph families F excluding some fixed graphs have the parameter-treewidth property. The contractionbidimensional parameters include many domination and covering graph parameters such as vertex cover, feedback vertex set, dominating set, edge-dominating set, q-dominating set (for fixed q). We use our theorems to develop new fixed-parameter algorithms in these contexts. Key words. Treewidth, local treewidth, graph minors, dominating set AMS subject classifications. 05C85, 68Q25, 68R10 1. Introduction. Th