Adaptive sample size and importance sampling in estimation-based local search for the probabilistic traveling salesman problem

Abstract

The probabilistic traveling salesman problem is a paradigmatic example of a stochastic combinatorial optimization problem. For this problem, recently an estimation-based local search algorithm using delta evaluation has been proposed. In this paper, we adopt two well-known variance reduction procedures in the estimation-based local search algorithm: the first is an adaptive sampling procedure that selects the appropriate size of the sample to be used in Monte Carlo evaluation; the second is a procedure that adopts importance sampling to reduce the variance involved in the cost estimation. We investigate several possible strategies for applying these procedures to the given problem and we identify the most effective one. Experimental results show that a particular heuristic customization of the two procedures increases significantly the effectiveness of the estimation-based local search.

Introduction

The probabilistic traveling salesman problem (PTSP) (Jaillet, 1985) is a paradigmatic example of a stochastic combinatorial optimization problem. The PTSP models a number of practical problems in the areas of strategic planning, routing, transportation, and scheduling (Bertsimas, 1988). The PTSP is similar to the TSP with the difference that each node has a probability of requiring a visit. The a priori optimization approach (Jaillet, 1985, Bertsimas et al., 1990) for the PTSP consists in finding an a priori solution that visits all the nodes such that the expected cost of a posteriori solutions is minimized: the a priori solution must be found prior to knowing which nodes are to be visited; the associated a posteriori solution, which is computed after knowing which nodes need to be visited, is obtained by visiting the nodes that require being visited in the order prescribed by the a priori solution, while skipping the nodes that do not require being visited.

Two classes of techniques for tackling the PTSP by a priori optimization have been proposed in the literature: analytical computation and empirical estimation. The former exactly computes the expected cost of the a posteriori solutions using a complex analytical development. The latter estimates the expected cost through Monte Carlo simulation.

2.5-opt-EEs (Birattari et al., 2008) is an estimation-based local search algorithm for tackling the PTSP. It is an iterative improvement algorithm that starts from some initial solution and then iteratively moves to improved neighbor solutions until a local optimum is found. A particularity of 2.5-opt-EEs is that the cost of the neighbor solutions are estimated using delta evaluation, a technique that considers only the cost contribution of solution components that are not common between two neighbor solutions. The results from Birattari et al. (2008) show that the performance of 2.5-opt-EEs depends on the probability associated with the nodes of the given PTSP instance. In particular, for low probabilities, where the coefficient of variation of the PTSP solution cost is high, 2.5-opt-EEs is less effective.

The goal of this paper is to increase the effectiveness of 2.5-opt-EEs for PTSP instances with low probabilities by using two procedures that reduce the variance of the cost estimator. The first is an adaptive sampling procedure that selects the appropriate size of the sample with respect to the variance of the cost estimator; the second is a procedure that adopts the importance sampling technique in order to reduce the variance of the cost estimator.

There exists a number of prior publications where adaptive sampling and importance sampling have been studied in the context of stochastic combinatorial optimization. Alkhamis et al., 1999, Gutjahr, 2004, Homem-de-Mello, 2003, Pichitlamken and Nelson, 2003, Birattari et al., 2006 investigated adaptive sample size procedures that make use of statistical tests to determine the number of samples to be chosen. The adoption of importance sampling to reduce the variance of the cost estimator has been investigated in Gutjahr et al., 2000a, Gutjahr et al., 2000b. In all these works, the adaptive sample size and the importance sampling techniques have been used in the context of full evaluation, where the cost of each solution is estimated from scratch. This is mainly due to the fact that the usage of delta evaluation is either ignored or not feasible for the given stochastic combinatorial optimization problem. The adoption of adaptive sample size and of importance sampling procedures in delta evaluation has never been investigated. For the PTSP, where delta evaluation is feasible, we expect that the adoption of the two particular techniques will increase the effectiveness of 2.5-opt-EEs. However, as we show in this paper, the adoption is not trivial and a main contribution of the paper consists in customizing the adaptive sample size and the importance sampling procedures for the delta evaluation applied to the PTSP. In particular, we investigate several ways of applying these procedures in the PTSP delta evaluation and we use a design of experiments approach to identify the most effective one.

The paper is organized as follows: in Section 2, we introduce the proposed approach; in Section 3, we study its performance; and in Section 4, we conclude the paper.