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Average-Case Performance of Rollout Algorithms for Knapsack Problems

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Abstract

Rollout algorithms have demonstrated excellent performance on a variety of dynamic and discrete optimization problems. Interpreted as an approximate dynamic programming algorithm, a rollout algorithm estimates the value-to-go at each decision stage by simulating future events while following a heuristic policy, referred to as the base policy. While in many cases rollout algorithms are guaranteed to perform as well as their base policies, there have been few theoretical results showing additional improvement in performance. In this paper, we perform a probabilistic analysis of the subset sum problem and 0–1 knapsack problem, giving theoretical evidence that rollout algorithms perform strictly better than their base policies. Using a stochastic model from the existing literature, we analyze two rollout methods that we refer to as the exhaustive rollout and consecutive rollout, both of which employ a simple greedy base policy. We prove that both methods yield a significant improvement in expected performance after a single iteration of the rollout algorithm, relative to the base policy.

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Acknowledgments

The authors would like to thank the two referees for the helpful comments. Research supported by NSF Grant 1029603 and ONR Grant N00014-12-1-0033; the first author is supported in part by a NSF graduate research fellowship.

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Authors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Andrew Mastin

  2. Department of Electrical Engineering and Computer Science, Laboratory for Information and Decision Systems, Operations Research Center, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Patrick Jaillet

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  1. Andrew Mastin
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  2. Patrick Jaillet
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Correspondence to Andrew Mastin.

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Communicated by Anita Schöbel.

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Mastin, A., Jaillet, P. Average-Case Performance of Rollout Algorithms for Knapsack Problems. J Optim Theory Appl 165, 964–984 (2015). https://doi.org/10.1007/s10957-014-0603-x

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