Prize-collecting Steiner Tree Problem Research Articles

An Algorithmic Framework for the Exact Solution of the Prize-Collecting Steiner Tree Problem

The Prize-Collecting Steiner Tree Problem (PCST) on a graph with edge costs and vertex profits asks for a subtree minimizing the sum of the total cost of all edges in the subtree plus the total profit of all vertices not contained in the subtree. PCST appears frequently in the design of utility networks where profit generating customers and the network connecting them have to be chosen in the most profitable way.

A hybrid Lagrangian genetic algorithm for the prize collecting Steiner tree problem

We consider the version of prize collecting Steiner tree problem (PCSTP) where each node of a given weighted graph is associated with a prize and where the objective is to find a minimum weight tree spanning a subset of nodes and collecting a total prize not less that a given quota Q . We present a lower bound and a genetic algorithm for the PCSTP. The lower bound is based on a Lagrangian decomposition of a minimum spanning tree formulation of the problem. The volume algorithm is used to solve the Lagrangian dual. The genetic algorithm incorporates several enhancements. In particular, it fully exploits both primal and dual information produced by Lagrangian decomposition. The proposed lower and upper bounds are assessed through computational experiments on randomly generated instances with up to 500 nodes and 5000 edges. For these instances, the proposed lower and upper bounds exhibit consistently a tight gap: in 76% of the cases the gap is strictly less than 2%.

A Constant-Factor Approximation Algorithm for thek-MST Problem

Given an undirected graph with nonnegative edge costs and an integerk, thek-MST problem is that of finding a tree of minimum cost onknodes. This problem is known to be NP-hard. We present a simple approximation algorithm that finds a solution whose cost is less than 17 times the cost of the optimum. This improves upon previous performance ratios for this problem −O(k) due to Raviet al.,O(log2k) due to Awerbuchet al., and the previous best bound ofO(logk) due to Rajagopalan and Vazirani. Given any 0<α<1, we first present a bicriteria approximation algorithm that outputs a tree onp⩾αkvertices of total cost at most 2pL/(1−α)k, whereLis the cost of the optimalk-MST. The running time of the algorithm isO(n2log2n) on ann-node graph. We then show how to use this algorithm to derive a constant factor approximation algorithm for thek-MST problem. The main subroutine in our algorithm is an approximation algorithm of Goemans and Williamson for the prize-collecting Steiner tree problem.