An approximation algorithm for maximum weight budgeted connected set cover

Abstract

This paper studies approximation algorithm for the maximum weight budgeted connected set cover (MWBCSC) problem. Given an element set $$X$$X, a collection of sets $${\mathcal {S}}\subseteq 2^X$$S⊆2X, a weight function $$w$$w on $$X$$X, a cost function $$c$$c on $${\mathcal {S}}$$S, a connected graph $$G_{\mathcal {S}}$$GS (called communication graph) on vertex set $${\mathcal {S}}$$S, and a budget $$L$$L, the MWBCSC problem is to select a subcollection $${\mathcal {S'}}\subseteq {\mathcal {S}}$$S?⊆S such that the cost $$c({\mathcal {S'}})=\sum _{S\in {\mathcal {S'}}}c(S)\le L$$c(S?)=?S?S?c(S)≤L, the subgraph of $$G_{\mathcal {S}}$$GS induced by $${\mathcal {S'}}$$S? is connected, and the total weight of elements covered by $${\mathcal {S'}}$$S? (that is $$\sum _{x\in \bigcup _{S\in {\mathcal {S'}}}S}w(x)$$?x??S?S?Sw(x)) is maximized. We present a polynomial time algorithm for this problem with a natural communication graph that has performance ratio $$O((\delta +1)\log n)$$O((?+1)logn), where $$\delta $$? is the maximum degree of graph $$G_{\mathcal {S}}$$GS and $$n$$n is the number of sets in $${\mathcal {S}}$$S. In particular, if every set has cost at most $$L/2$$L/2, the performance ratio can be improved to $$O(\log n)$$O(logn).