Statistical mechanics of the multi-constraint continuous knapsack problem

TL;DR

This paper uses replica analysis to study the solution space and optimal item selection in the multi-constraint continuous knapsack problem, revealing limitations of the replica symmetric solution and applying replica symmetry breaking methods.

Contribution

It applies advanced statistical mechanics techniques to analyze the multi-constraint continuous knapsack problem, providing new insights into its solution space and stability conditions.

Findings

Replica symmetric solution fails with many constraints

Zero entropy approximation yields stable solutions

One step RSB solution relates to perceptron capacity

Abstract

We apply the replica analysis established by Gardner to the multi-constraint continuous knapsack problem,which is one of the linear programming problems and a most fundamental problem in the field of operations research (OR). For a large problem size, we analyse the space of solution and its volume, and estimate the optimal number of items to go into the knapsack as a function of the number of constraints. We study the stability of the replica symmetric (RS) solution and find that the RS calculation cannot estimate the optimal number of items in knapsack correctly if many constraints are required.In order to obtain a consistent solution in the RS region,we try the zero entropy approximation for this continuous solution space and get a stable solution within the RS ansatz.On the other hand, in replica symmetry breaking (RSB) region, the one step RSB solution is found by Parisi's scheme. It turns out that this problem is closely related to the problem of optimal storage capacity and of generalization by maximum stability rule of a spherical perceptron.