The input to the stochastic orienteering problem consists of a budget @math and metric @math where each vertex @math has a job with deterministic reward and random processing time (drawn from a known distribution). The processing times are independent across vertices. The goal is to obtain a non-anticipatory policy to run jobs at different vertices, that maximizes expected reward, subject to the total distance traveled plus processing times being at most @math . An adaptive policy is one that can choose the next vertex to visit based on observed random instantiations. Whereas, a non-adaptive policy is just given by a fixed ordering of vertices. The adaptivity gap is the worst-case ratio of the expected rewards of the optimal adaptive and non-adaptive policies. We prove an @math lower bound on the adaptivity gap of stochastic orienteering. This provides a negative answer to the @math adaptivity gap conjectured earlier, and comes close to the @math upper bound. This result holds even on a line metric. We also show an @math upper bound on the adaptivity gap for the correlated stochastic orienteering problem, where the reward of each job is random and possibly correlated to its processing time. Using this, we obtain an improved quasi-polynomial time approximation algorithm for correlated stochastic orienteering.
The deterministic orienteering problem was introduced by @cite_2 . It has several applications, and many exact approaches and heuristics have been applied to this problem, see eg. the survey @cite_17 . The first constant-factor approximation algorithm was due to @cite_12 . The approximation ratio has been improved @cite_15 @cite_5 to the current best @math .
@cite_2 @cite_5 @cite_15 @cite_12 @cite_17
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In this paper, we give the first constant-factor approximation algorithm for the rooted Orienteering problem, as well as a new problem that we call the Discounted-Reward traveling salesman problem (TSP), motivated by robot navigation. In both problems, we are given a graph with lengths on edges and rewards on nodes, and a start node @math . In the Orienteering problem, the goal is to find a path starting at @math that maximizes the reward collected, subject to a hard limit on the total length of the path. In the Discounted-Reward TSP, instead of a length limit we are given a discount factor @math , and the goal is to maximize the total discounted reward collected, where the reward for a node reached at time @math is discounted by @math . This problem is motivated by an approximation to a planning problem in the Markov decision process (MDP) framework under the commonly employed infinite horizon discounted reward optimality criterion. The approximation arises from a need to deal with exponentially large state spaces that emerge when trying to model one-time events and nonrepeatable rewards (such as for package deliveries). We also consider tree and multiple-path variants of these problems and provide approximations for those as well. Although the unrooted Orienteering problem, where there is no fixed start node @math , has been known to be approximable using algorithms for related problems such as @math -TSP (in which the amount of reward to be collected is fixed and the total length is approximately minimized), ours is the first to approximate the rooted question, solving an open problem in [E. M. Arkin, J. S. B. Mitchell, and G. Narasimhan, Proceedings of the @math th ACM Symposium on Computational Geometry, 1998, pp. 307-316] and [B. Awerbuch, Y. Azar, A. Blum, and S. Vempala, SIAM J. Comput., 28 (1998), pp. 254-262]. We complement our approximation result for Orienteering by showing that the problem is APX-hard. The time-dependent orienteering problem is dual to the time-dependent traveling salesman problem. It consists of visiting a maximum number of sites within a given deadline. The traveling time between two sites is in general dependent on the starting time.For any ɛ > 0, we provide a (2 + ɛ)-approximation algorithm for the time-dependent orienteering problem which runs in polynomial time if the ratio between the maximum and minimum traveling time between any two sites is constant. No prior upper approximation bounds were known for this time-dependent problem. 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+e) approximation for orienteering in undirected graphs, improving upon the 3-approximation of [2004]. — An O(log2 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 Pal [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 lmax lmin ) approximation for orienteering with time-windows, where lmax and lmin are the lengths of the longest and shortest time-windows respectively. In the team orienteering problem, start and end points are specified along with other locations which have associated scores. Given a fixed amount of time for each of the M members of the team, the goal is to determine M paths from the start point to the end point through a subset of locations in order to maximize the total score. In this paper, a fast and effective heuristic is presented and tested on 353 problems ranging in size from 21 to 102 points. The computational results are presented in detail.