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On the Adaptivity Gap of Stochastic Orienteering

  • Nikhil Bansal
  • Viswanath Nagarajan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8494)

Abstract

The input to the stochastic orienteering problem [14] consists of a budget B and metric (V,d) where each vertex v ∈ V has a job with a deterministic reward and a random processing time (drawn from a known distribution). The processing times are independent across vertices. The goal is to obtain a non-anticipatory policy (originating from a given root vertex) to run jobs at different vertices, that maximizes expected reward, subject to the total distance traveled plus processing times being at most B. 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 \(\Omega\left((\log\log B)^{1/2}\right)\) lower bound on the adaptivity gap of stochastic orienteering. This provides a negative answer to the O(1)-adaptivity gap conjectured in [14] and comes close to the O(loglogB) upper bound proved there. This result holds even on a line metric.

We also show an O(loglogB) 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 \( \min\{\log n,\log B\}\cdot \tilde{O}(\log^2\log B)\)-approximation algorithm for correlated stochastic orienteering.

Keywords

Vehicle Rout Problem Left Child Orienteering Problem Adaptive Policy Good Approximation Ratio 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Nikhil Bansal
    • 1
  • Viswanath Nagarajan
    • 2
  1. 1.Eindhoven University of TechnologyThe Netherlands
  2. 2.IBM T.J. Watson Research CenterUSA

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