Improved Approximation Algorithms for PRIZE-COLLECTING STEINER TREE and TSP

Abstract — We study the prize-collecting versions of the Steiner tree, traveling salesman, and stroll (a.k.a. PATH-TSP) problems (PCST, PCTSP, and PCS, respectively): given a graph (V, E) with costs on each edge and a penalty (a.k.a. prize) on each node, the goal is to find a tree (for PCST), cycle (for PCTSP), or stroll (for PCS) that minimizes the sum of the edge costs in the tree/cycle/stroll and the penalties of the nodes not spanned by it. In addition to being a useful theoretical tool for helping to solve other optimization problems, PCST has been applied fruitfully by AT&T to the optimization of real-world telecommunications networks. The most recent improvements for the first two problems, giving a 2-approximation algorithm for each, appeared first in 1992. (A 2-approximation for PCS appeared in 2003.) The natural linear programming (LP) relaxation of PCST has an integrality gap of 2, which has been a barrier to further improvements for this problem. We present (2 − ɛ)-approximation algorithms for all three problems, connected by a unified technique for improving prizecollecting algorithms that allows us to circumvent the integrality gap barrier. 1

Approximation algorithms for node-weighted prize-collecting Steiner tree problems on planar graphs

We study the prize-collecting version of the Node-weighted Steiner Tree problem (NWPCST) restricted to planar graphs. We give a new primal-dual Lagrangian-multiplier-preserving (LMP) 3-approximation algorithm for planar NWPCST. We then show a ($2.88 + \epsilon$)-approximation which establishes a new best approximation guarantee for planar NWPCST. This is done by combining our LMP algorithm with a threshold rounding technique and utilizing the 2.4-approximation of Berman and Yaroslavtsev for the version without penalties. We also give a primal-dual 4-approximation algorithm for the more general forest version using techniques introduced by Hajiaghay and Jain

A 3/2-approximation algorithm for some minimum-cost graph problems

International audienceWe consider a class of graph problems introduced in a paper of Goemans and Williamson that involve finding forests of minimum edge cost. This class includes a number of location/routing problems; it also includes a problem in which we are given as input a parameter k, and want to find a forest such that each component has at least k vertices. Goemans and Williamson gave a 2-approximation algorithm for this class of problems. We give an improved 3/2-approximation algorithm

Prize-Collecting TSP with a Budget Constraint

We consider constrained versions of the prize-collecting traveling salesman and the minimum spanning tree problems. The goal is to maximize the number of vertices in the returned tour/tree subject to a bound on the tour/tree cost. We present a 2-approximation algorithm for these problems based on a primal-dual approach. The algorithm relies on finding a threshold value for the dual variable corresponding to the budget constraint in the primal and then carefully constructing a tour/tree that is just within budget. Thereby, we improve the best-known guarantees from 3+epsilon and 2+epsilon for the tree and the tour version, respectively. Our analysis extends to the setting with weighted vertices, in which we want to maximize the total weight of vertices in the tour/tree subject to the same budget constraint

Steiner Tree Games

Prize-collecting Steiner tree is a network design problem in which a utility provider located at some position in a graph attempts to construct a network (subtree) of maximum profit based on the value of the vertices in the graph and the costs of the edges. I consider three network formation games where the players represent competing providers attempting to build networks in the same market. These games seek to preserve the key feature of Prize-Collecting Steiner tree, namely that players must each build a subtree that attempts to include customers who are of high value or are easy to reach. I analyze the price of anarchy and price of stability of each of these games

Steiner Tree Games

Prize-collecting Steiner tree is a network design problem in which a utility provider located at some position in a graph attempts to construct a network (subtree) of maximum profit based on the value of the vertices in the graph and the costs of the edges. I consider three network formation games where the players represent competing providers attempting to build networks in the same market. These games seek to preserve the key feature of Prize-Collecting Steiner tree, namely that players must each build a subtree that attempts to include customers who are of high value or are easy to reach. I analyze the price of anarchy and price of stability of each of these games