A Unified Approach for Domination Problems on Different Network Topologies

We provide tight hardness results and approxiamation algorithms for many existing domination problems. We start with 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 ρ > 0 of their neighbors in P; in the total version (T-PIDS), nodes in P are required to have a fraction ρ of neighbors inside P; and in the connected version (C-PIDS) the dominating set have to induce a connected subgraph. Then, we unify a large variations of dominating set problem under a single generalized dominating set problem. We show a tight hardness results 1 2(1− o(1)) lnn inapproximability and a ln∆ + O(1) approximation algorithms for our generalized dominating set problem where n is the newtork size and ∆ is the maximum degree. The results apply directly to PIDS, k-tuple dominating set, m-connected k-dominating set, Fixed Threshold Dominating Set and many existing domination problems plus all connected or/and total versions of those problems. As most previous hardness results are NP-completeness or APXhardness, we effectively close many long-standing approximation gaps of domination problems, under the reasonable assumption that NP 6⊂ DTIME(nO(loglogn)). In networks with degrees bounded by a constant B, we show that all problems cannot be approximated within lnB−O(ln lnB), unless P=NP. In dense networks and scale-free networks such as Internet, WWW, social networks, etc. in which degree sequences follows a power-law distribution, we reveal trivial constant factor approximation algorithms for the class of PIDS-like domination problems. Finally, we prove that optimal solution of any domination problems can be found in linear time for networks with tree topology. Thang N. Dinh University of Florida, Gainesville, FL 32611, US; e-mail: [email protected] Dung T. Nguyen University of Florida, Gainesville, FL 32611, US; e-mail: [email protected] My T. Thai University of Florida, Gainesville, FL 32611, US; e-mail: [email protected]