A Primal-Dual based Distributed Approximation Algorithm for Prize Collecting Steiner Tree
share
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.
READ FULL TEXT