A Primal-Dual based Distributed Approximation Algorithm for Prize Collecting Steiner Tree
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Constructing a steiner tree of a graph is a fundamental problem in many applications. Prize collecting steiner tree (PCST) is a special variant of the steiner tree problem and has applications in network design, content distribution etc. There are a few centralized approximation algorithms DB_MG_DS_DW_1993, GW_1995, AA_MB_MH_2011 for solving the PCST problem. However no distributed algorithm is known that solves the PCST problem with non-trivial approximation factor. In this work we present a distributed algorithm that constructs a prize collecting steiner tree for a given connected undirected graph with non-negative weight for each edge and non-negative prize value for each node. Initially each node knows its own prize value and weight of each incident edge. Our algorithm is based on primal-dual method and it achieves an approximation factor of (2 - 1/n - 1) of the optimal. The total number of messages required by our distributed algorithm to construct the PCST for a graph with |V| nodes and |E| edges is O(|V|^2 + |E||V|). The algorithm is spontaneously initiated at a special node called the root node and when the algorithm terminates each node knows whether it is in the prize part or in the steiner tree of the PCST.
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