Exact Counting and Sampling of Optima for the Knapsack Problem

TL;DR

This paper introduces an exact algorithm for counting and uniformly sampling all optimal solutions of the zero-one knapsack problem, revealing exponential growth in the number of optima for certain instance classes.

Contribution

It presents the first exact counting and sampling algorithms for knapsack optima, and analyzes how the number of solutions varies with instance parameters.

Findings

Number of optima can grow exponentially in correlated instances

Exact counting of solutions is computationally justified for certain classes

Sampling uniformly from all optima is efficiently achievable

Abstract

Computing sets of high quality solutions has gained increasing interest in recent years. In this paper, we investigate how to obtain sets of optimal solutions for the classical knapsack problem. We present an algorithm to count exactly the number of optima to a zero-one knapsack problem instance. In addition, we show how to efficiently sample uniformly at random from the set of all global optima. In our experimental study, we investigate how the number of optima develops for classical random benchmark instances dependent on their generator parameters. We find that the number of global optima can increase exponentially for practically relevant classes of instances with correlated weights and profits which poses a justification for the considered exact counting problem.