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.
@cite_15: Given a metric space G on n nodes, with a start node r and deadlines D(v) for each vertex v, we consider the Deadline-TSP problem of finding a path starting at r that visits as many nodes as possible by their deadlines. We also consider the more general Vehicle Routing with Time-Windows problem, in which each node v also has a release-time R(v) and the goal is to visit as many nodes as possible within their "time-windows" [R(v),D(v)]. No good approximations were known previously for these problems on general metric spaces. We give an O(logn) approximation algorithm for Deadline-TSP, and extend this algorithm to an O(log2n) approximation for the Time-Window problem. We also give a bicriteria approximation algorithm for both problems: Given an e>0, our algorithm produces a (1 e) approximation, while exceeding the deadlines by a factor of 1+e. We use as a subroutine for these results a constant-factor approximation that we develop for a generalization of the orienteering problem in which both the start and the end nodes of the path are fixed. In the process, we give a 3-approximation to the orienteering problem, improving on the previously best known 4-approximation of [6].
@cite_14: In the Stochastic Orienteering problem, we are given a metric, where each node also has a job located there with some deterministic reward and a random size. (Think of the jobs as being chores one needs to run, and the sizes as the amount of time it takes to do the chore.) The goal is to adaptively decide which nodes to visit to maximize total expected reward, subject to the constraint that the total distance traveled plus the total size of jobs processed is at most a given budget of B. (I.e., we get reward for all those chores we finish by the end of the day). The (random) size of a job is not known until it is completely processed. Hence the problem combines aspects of both the stochastic knapsack problem with uncertain item sizes and the deterministic orienteering problem of using a limited travel time to maximize gathered rewards located at nodes. In this paper, we present a constant-factor approximation algorithm for the best non-adaptive policy for the Stochastic Orienteering problem. We also show a small adaptivity gap---i.e., the existence of a non-adaptive policy whose reward is at least an Ω(1 log log B) fraction of the optimal expected reward---and hence we also get an O(log log B)-approximation algorithm for the adaptive problem. Finally we address the case when the node rewards are also random and could be correlated with the waiting time, and give a non-adaptive policy which is an O(log n logB)-approximation to the best adaptive policy on n-node metrics with budget B.
The correlated stochastic orienteering problem was studied in @cite_14 , where the authors obtained an @math -approximation algorithm and an @math adaptivity gap. They also showed the problem to be at least as hard to approximate as the deadline orienteering problem, for which the best approximation ratio known is @math @cite_15 .