The B-Prize-Collecting Multicut Problem in Paths, Spider Graphs and Rings

Given a graph [Formula: see text], a set of [Formula: see text] source-sink pairs [Formula: see text] [Formula: see text] and a profit bound [Formula: see text], every edge [Formula: see text] has a cost [Formula: see text], and every source-sink pair [Formula: see text] has a profit [Formula: see text] and a penalty [Formula: see text]. The [Formula: see text]-prize-collecting multicut problem ([Formula: see text]-PCMP) is to find a multicut [Formula: see text] such that the objective cost, which consists of the total cost of the edges in [Formula: see text] and the total penalty of the pairs still connected after removing [Formula: see text], is minimized and the total profit of the disconnected pairs by removing [Formula: see text] is at least [Formula: see text]. In this paper, we firstly consider the [Formula: see text]-PCMP in paths, and prove that it is [Formula: see text]-hard even when [Formula: see text] for any [Formula: see text]. Then, we present a fully polynomial time approximation scheme (FPTAS) whose running time is [Formula: see text] for the [Formula: see text]-PCMP in paths. Based on this algorithm, we present an FPTAS whose running time is [Formula: see text] for the [Formula: see text]-PCMP in spider graphs, and an FPTAS whose running time is [Formula: see text] for the [Formula: see text]-PCMP in rings, respectively, where [Formula: see text] is the number of leaves of spider graph.