Social interaction as a heuristic for combinatorial optimization problems

TL;DR

This paper explores a social interaction-inspired heuristic, based on Axelrod's model, for solving NP-Complete optimization problems, demonstrating how agent-based dynamics can efficiently find optimal solutions with specific scaling properties.

Contribution

Introduces the Adaptive Culture Heuristic (ACH), a novel social interaction-based method for combinatorial optimization, analyzing its performance and scaling behavior on a Boolean Binary Perceptron problem.

Findings

Probability of success scales with F/N^{1/4}

Number of agents must scale as F^4 for fixed success probability

Relaxation time scales as F^6, indicating computational cost

Abstract

We investigate the performance of a variant of Axelrod's model for dissemination of culture - the Adaptive Culture Heuristic (ACH) - on solving an NP-Complete optimization problem, namely, the classification of binary input patterns of size $F$ by a Boolean Binary Perceptron. In this heuristic, $N$ agents, characterized by binary strings of length $F$ which represent possible solutions to the optimization problem, are fixed at the sites of a square lattice and interact with their nearest neighbors only. The interactions are such that the agents' strings (or cultures) become more similar to the low-cost strings of their neighbors resulting in the dissemination of these strings across the lattice. Eventually the dynamics freezes into a homogeneous absorbing configuration in which all agents exhibit identical solutions to the optimization problem. We find through extensive simulations that the probability of finding the optimal solution is a function of the reduced variable $F/N^{1/4}$ so that the number of agents must increase with the fourth power of the problem size, $N \propto F^ 4$, to guarantee a fixed probability of success. In this case, we find that the relaxation time to reach an absorbing configuration scales with $F^ 6$ which can be interpreted as the overall computational cost of the ACH to find an optimal set of weights for a Boolean Binary Perceptron, given a fixed probability of success.