TY - JOUR

T1 - New approximation guarantees for minimum-weight κ-trees and prize-collecting salesmen

AU - Awerbuch, Baruch

AU - Azar, Yossi

AU - Blum, Avrim

AU - Vempala, Santosh

PY - 1998

Y1 - 1998

N2 - We consider a formalization of the following problem. A salesperson must sell some quota of brushes in order to win a trip to Hawaii. This salesperson has a map (a weighted graph) in which each city has an attached demand specifying the number of brushes that can be sold in that city. What is the best route to take to sell the quota while traveling the least distance possible? Notice that unlike the standard traveling salesman problem, not only do we need to figure out the order in which to visit the cities, but we must decide the more fundamental question: which cities do we want to visit? In this paper we give the first approximation algorithm having a polylogarithmic performance guarantee for this problem, as well as for the slightly more general "prize-collecting traveling salesman problem" (PCTSP) of Balas, and a variation we call the "bank robber problem" (also called the "orienteering problem" by Golden. Levi, and Vohra). We do this by providing an O(log2 κ) approximation to the somewhat cleaner κ-MST problem which is defined as follows. Given an undirected graph on n nodes with nonnegative edge weights and an integer κ ≤ n, find the tree of least weight that spans κ vertices. (If desired, one may specify in the problem a "root vertex" that must be in the tree as well.) Our result improves on the previous best bound of O(√κ) of Ravi et al.

AB - We consider a formalization of the following problem. A salesperson must sell some quota of brushes in order to win a trip to Hawaii. This salesperson has a map (a weighted graph) in which each city has an attached demand specifying the number of brushes that can be sold in that city. What is the best route to take to sell the quota while traveling the least distance possible? Notice that unlike the standard traveling salesman problem, not only do we need to figure out the order in which to visit the cities, but we must decide the more fundamental question: which cities do we want to visit? In this paper we give the first approximation algorithm having a polylogarithmic performance guarantee for this problem, as well as for the slightly more general "prize-collecting traveling salesman problem" (PCTSP) of Balas, and a variation we call the "bank robber problem" (also called the "orienteering problem" by Golden. Levi, and Vohra). We do this by providing an O(log2 κ) approximation to the somewhat cleaner κ-MST problem which is defined as follows. Given an undirected graph on n nodes with nonnegative edge weights and an integer κ ≤ n, find the tree of least weight that spans κ vertices. (If desired, one may specify in the problem a "root vertex" that must be in the tree as well.) Our result improves on the previous best bound of O(√κ) of Ravi et al.

KW - Approximation algorithm

KW - Prize-collecting traveling salesman problem

KW - κ-MST

UR - http://www.scopus.com/inward/record.url?scp=0032057918&partnerID=8YFLogxK

U2 - 10.1137/S009753979528826X

DO - 10.1137/S009753979528826X

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AN - SCOPUS:0032057918

VL - 28

SP - 254

EP - 262

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 1

ER -