Improved algorithms for orienteering and related problems

Abstract

In this article, we consider the orienteering problem in undirected and directed graphs and obtain improved approximation algorithms. The point to point-orienteering problem is the following: Given an edge-weighted graphG=(V, E) (directed or undirected), two nodess, t∈Vand a time limitB, find ans-twalkinGof total length at mostBthat maximizes the number of distinct nodes visited by the walk. This problem is closely related to tour problems such as TSP as well as network design problems such ask-MST. Orienteering withtime-windowsis the more general problem in which each nodevhas a specified time-window [R(v),D(v)] and a nodevis counted as visited by the walk only ifvis visited during its time-window. We design new and improved algorithms for the orienteering problem and orienteering with time-windows. Our main results are the following:— A (2+ϵ) approximation for orienteering in undirected graphs, improving upon the 3-approximation of Bansal et al. [2004].— AnO(log2OPT) approximation for orienteering in directed graphs, where OPT ≤nis the number of vertices visited by an optimal solution. Previously, only a quasipolynomial-time algorithm due to Chekuri and Pál [2005] achieved a polylogarithmic approximation (a ratio ofO(log OPT)).— Given an α approximation for orienteering, we show anO(α ċ max{log OPT, loglmax/lmin}) approximation for orienteering with time-windows, wherelmaxandlminare the lengths of the longest and shortest time-windows respectively.