Approximation Algorithms for Orienteering with Time Windows

Chandra Chekuri; Nitish Korula

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arXiv · 0711.4825

Approximation Algorithms for Orienteering with Time Windows

Chandra Chekuri·Nitish Korula

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Abstract

Orienteering is the following optimization problem: given an edge-weighted graph (directed or undirected), two nodes s,t and a time limit T, find an s-t walk of total length at most T that maximizes the number of distinct nodes visited by the walk. One obtains a generalization, namely orienteering with time-windows (also referred to as TSP with time-windows), if 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. For the time-window problem, an O(\log \opt) approximation can be achieved even for directed graphs if the algorithm is allowed quasi-polynomial time. However, the best known polynomial time approximation ratios are O(\log^2 \opt) for undirected graphs and O(\log^4 \opt) in directed graphs. In this paper we make some progress towards closing this discrepancy, and in the process obtain improved approximation ratios in several natural settings. Let L(v) = D(v) - R(v) denote the length of the time-window for v and let \lmax = \max_v L(v) and \lmin = \min_v L(v). Our results are given below with αdenoting the known approximation ratio for orienteering (without time-windows). Currently α= (2+\eps) for undirected graphs and α= O(\log^2 \opt) in directed graphs. 1. An O(α\log \lmax) approximation when R(v) and D(v) are integer valued for each v. 2. An O(α\max{\log \opt, \log \frac{\lmax}{\lmin}}) approximation. 3. An O(α\log \frac{\lmax}{\lmin}) approximation when no start and end points are specified. In particular, if \frac{\lmax}{\lmin} is poly-bounded, we obtain an O(\log n) approximation for the time-window problem in undirected graphs.

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BibTeX RIS

Chandra Chekuri, Nitish Korula. 2007-11-29. Approximation Algorithms for Orienteering with Time Windows. https://arxiv.org/abs/0711.4825

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