Average-Case Performance of Rollout Algorithms for Knapsack Problems

TL;DR

This paper provides a probabilistic analysis demonstrating that rollout algorithms outperform their base policies in subset sum and knapsack problems, with significant expected improvements after just one iteration.

Contribution

It offers the first theoretical evidence that rollout algorithms can strictly improve performance over base policies in these combinatorial problems.

Findings

Single iteration of rollout reduces expected gap by at least 30%.

Both rollout methods outperform base policies in subset sum and knapsack problems.

Theoretical analysis supports empirical effectiveness of rollout algorithms.

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 greedy 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 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 consecutive rollout and exhaustive rollout, both of which employ a simple greedy base policy. For the subset sum problem, we prove that after only a single iteration of the rollout algorithm, both methods yield at least a 30% reduction in the expected gap between the solution value and capacity, relative to the base policy. Analogous results are shown for the knapsack problem.