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 graph G =( V, E ) (directed or undirected), two nodes s, t ∈ V and a time limit B , find an s - t walk in G of total length at most B that 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 as k -MST. Orienteering with time-windows is the more general problem in which each node v has a specified time-window [ R ( v ), D ( v )] and a node v is counted as visited by the walk only if v is 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]. — An O (log 2 OPT) approximation for orienteering in directed graphs, where OPT ≤ n is 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 of O (log OPT)). — Given an α approximation for orienteering, we show an O (α ċ max{log OPT, log l max / l min }) approximation for orienteering with time-windows, where l max and l min are the lengths of the longest and shortest time-windows respectively.
Published Version
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