Approximation Algorithms for the Capacitated Domination Problem

TL;DR

This paper introduces logarithmic approximation algorithms for the Capacitated Domination problem, a generalization of Dominating Set, and explores its computational complexity and fixed-parameter tractability.

Contribution

It provides the first logarithmic approximation algorithms for the problem on general graphs and establishes its W[1]-hardness with respect to treewidth, along with fixed-parameter algorithms.

Findings

Logarithmic approximation algorithms for general graphs.

W[1]-hardness when parameterized by treewidth.

Fixed-parameter tractable algorithms for bounded treewidth and capacity.

Abstract

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.