求解背包问题的Rollout算法的平均性能
Average-Case Performance of Rollout Algorithms for Knapsack Problems
摘要：
推出算法在各种动态和离散优化问题上表现出优异的性能。作为一种近似的动态规划算法，推出算法通过模拟未来事件同时遵循贪婪策略（称为基本策略）来估计每个决策阶段的价值。虽然在许多情况下，推出算法保证执行以及他们的基本政策，有很少的理论结果表明，在性能上的额外改进。在本文中，我们对子集和问题和背包问题进行了概率分析，给出了理论证据，证明推出算法的性能严格优于其基本策略。使用现有文献中的随机模型，我们分析了两种推出方法，我们称之为连续推出和穷举推出，这两种方法都采用了一个简单的贪婪的基础政策。对于子集和问题，我们证明，只有一个迭代的推出算法后，这两种方法产生至少30%的减少，相对于基本政策的解决方案的价值和容量之间的预期差距。类似的结果显示背包问题。
英文摘要：
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.
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