Social interaction as a heuristic for combinatorial optimization problems

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.

💡 Research Summary

The paper introduces the Adaptive Culture Heuristic (ACH), a novel meta‑heuristic derived from Axelrod’s cultural dissemination model, and evaluates its ability to solve a classic NP‑Complete combinatorial optimization task: finding the optimal binary weight vector of a Boolean Binary Perceptron that correctly classifies a set of binary input patterns of size F. In ACH, N agents are placed on the sites of a two‑dimensional square lattice. Each agent carries a binary string of length F, which directly encodes a candidate solution (the perceptron’s weight vector). At each discrete time step an agent randomly selects one of its four nearest neighbours. The two strings are compared by their Hamming distance; if they differ, the agent copies a randomly chosen subset of bits from the neighbour that has the lower classification error (i.e., the lower cost). This “cultural assimilation” rule is repeated until the system reaches an absorbing configuration in which all agents share an identical string.

Extensive Monte‑Carlo simulations were performed for a wide range of problem sizes (F) and population sizes (N). The key empirical findings are two scaling relationships. First, the probability P_opt of ending in the global optimum depends only on the reduced variable x = F/N^{1/4}. Consequently, to keep a fixed success probability as the problem size grows, the number of agents must scale as N ∝ F^{4}. Second, the average relaxation time τ (the number of update steps required to reach the homogeneous absorbing state) grows as τ ∝ F^{6}. This τ can be interpreted as the overall computational cost of ACH for a given success probability.

The dynamics exhibit strong robustness: the same scaling laws hold regardless of whether the initial strings are completely random or whether a fraction of agents are seeded with near‑optimal solutions. The local nature of interactions (only nearest neighbours) nevertheless yields global convergence through the formation and expansion of cultural clusters that eventually dominate the lattice. The authors note that while the memory footprint of ACH is modest (N·F bits), the N ∝ F^{4} requirement leads to rapidly increasing resource demands for large F. They suggest possible mitigations such as parallel implementation, use of small‑world or scale‑free interaction topologies to reduce average path length, or hybridization with other search techniques.

In summary, ACH demonstrates that simple social‑interaction rules can be harnessed to perform combinatorial optimization, achieving a polynomial‑time scaling of success probability and relaxation time with respect to problem size. The work opens avenues for extending the approach to other NP‑Complete problems, to non‑binary decision variables, and to more complex network structures, thereby contributing a fresh perspective to the design of nature‑inspired optimization algorithms.