Large N limit of the knapsack problem

In the simplest formulation of the knapsack problem, one seeks to maximize the total value of a collection of objects such that the total weight remains below a certain limit. In this work, we move from computer science to physics and formulate the knapsack problem as a statistical physics system and compute the corresponding partition function. We approximate the result in the large number limit and from this approximation develop a new algorithm for the problem. We compare the performance of this algorithm to that of other approximation algorithms, finding that the new algorithm is faster than most of these approaches while still retaining high accuracy. From its speed and accuracy relationship, we argue that the algorithm is a manifestation of a greedy algorithm. We conclude by discussing ways to extend the formalism to make its underlying heuristics more rigorous or to apply the approach to other combinatorial optimization problems. In all, this work exists at the intersection between computer science and statistical physics and represents a new analytical approach to solving the problems in the former using methods of the latter.