Random Costs in Combinatorial Optimization

TL;DR

This paper explores the random cost problem, demonstrating its equivalence to the NP-hard number partitioning problem, and provides probabilistic insights into optimal and sub-optimal solutions, explaining heuristic limitations.

Contribution

It establishes the equivalence between the random cost problem and number partitioning, enabling analysis of solution distributions and explaining heuristic inefficacies.

Findings

Number partitioning is essentially equivalent to the random cost problem.

Heuristic approaches perform poorly due to the problem's complexity.

Probabilistic distributions of optimal and sub-optimal costs are derived.

Abstract

The random cost problem is the problem of finding the minimum in an exponentially long list of random numbers. By definition, this problem cannot be solved faster than by exhaustive search. It is shown that a classical NP-hard optimization problem, number partitioning, is essentially equivalent to the random cost problem. This explains the bad performance of heuristic approaches to the number partitioning problem and allows us to calculate the probability distributions of the optimum and sub-optimum costs.