A unified approach for domination problems on different network topologies

Abstract

This chapter studies approximability and inapproximability for a class of important domination problems in different network topologies. In addition to the well-known connected dominating set, total dominating set, more general forms of the problems, in which each node is required to be dominated by more than one of its neighbors, are also considered. An example is the positive influence dominating set (PIDS) problem, originated from the context of influence propagation in social networks. The PIDS problem seeks for a minimal set of nodes P such that all other nodes in the network have at least a fraction p> 0 of their neighbors in P. Furthermore, domination problems can be hybridized to form new problems such as T-PIDS and C-PIDS, the total version and the connected version of PIDS. The goal of the chapter is to narrow the gaps between the approximability and inapproximability of those domination problems. Going beyond the classic O(log n) results, the chapter presents the explicit constants in the approximation factors to obtain tighter approximation bounds. While the first part of the chapter focuses on the general topology, the second part presents improved approximation results in different network topologies including power- law networks, social networks, and treelike networks.