Large W limit of the knapsack problem

TL;DR

This paper applies statistical physics methods to the knapsack problem, deriving new algorithms based on the partition function and analyzing their relation to classical solutions like dynamic programming and greedy algorithms.

Contribution

It introduces a novel formalism linking the large weight limit of the partition function to greedy solutions, providing new insights into the problem's structure.

Findings

The exact partition function reproduces dynamic programming solutions.

The zero-temperature limit yields greedy solutions.

The new algorithms do not outperform traditional methods in runtime or accuracy.

Abstract

We formulate the knapsack problem (KP) as a statistical physics system and compute the corresponding partition function as an integral in the complex plane. The introduced formalism allows us to derive three statistical-physics-based algorithms for the KP: one based on the recursive definition of the exact partition function; another based on the large weight limit of that partition function; and a final one based on the zero-temperature limit of the second. Comparing the performances of the algorithms, we find that they do not consistently outperform (in terms of runtime and accuracy) dynamic programming, annealing, or standard greedy algorithms. However, the exact partition function is shown to reproduce the dynamic programming solution to the KP, and the zero-temperature algorithm is shown to produce a greedy solution. Therefore, although dynamic programming and greedy solutions to the KP are conceptually distinct, the statistical physics formalism introduced in this work reveals that the large weight-constraint limit of the former leads to the latter. We conclude by discussing how to extend this formalism in order to obtain more accurate versions of the introduced algorithms and to other similar combinatorial optimization problems.