# Analyzing the optimal neighborhood: Algorithms for partial and budgeted connected dominating set problems

## Abstract

We study partial and budgeted versions of the well-studied connected dominating set problem. In the partial connected dominating set (PCDS) problem, we are given an undirected graph G = (V;E) and an integer n0, and the goal is to find a minimum subset of vertices that induces a connected subgraph of G and dominates at least n0 vertices. We obtain the first polynomial time algorithm with an O(ln Δ) approximation guarantee for this problem, thereby significantly extending the results of Guha and Khuller [Algorithmica, 20(1998), pp. 374-387] for the connected dominating set problem. We note that none of the methods developed earlier can be applied directly to solve this problem. In the budgeted connected dominating set problem, there is a budget on the number of vertices we can select, and the goal is to dominate as many vertices as possible. We obtain a 1 12 (1-1 e) approximation algorithm for this problem. Finally, we show that our techniques extend to a more general setting where the profit function associated with a subset of vertices is a "special" submodular function. This generalization captures the connected dominating set problem with capacities and/or weighted profits as special cases. This implies an O(ln q) approximation (where q denotes the quota) and O(1) approximation algorithms for the partial and budgeted versions of these problems. While the algorithms are simple, the results make a surprising use of the greedy set cover framework in defining a useful profit function. Finally, we prove that (both edge and node) weighted versions of the PCDS problem are as hard as the more general group Steiner tree problem.