Dynamic Programming Optimization over Random Data: the Scaling Exponent for Near-optimal Solutions

TL;DR

This paper investigates the relationship between optimal and near-optimal solutions in a simple dynamic programming problem with random data, revealing a quadratic scaling law for the difference in objective value relative to the proportion of differing elements.

Contribution

It introduces a simple example demonstrating a quadratic scaling law for near-optimal solutions in dynamic programming with random data, supporting a broader conjecture on optimization complexity.

Findings

Near-optimal solutions differ from the optimum by a factor of order elta^2

The quadratic relationship is supported by Monte Carlo simulations of the NK model

The work relates scaling exponents to the algorithmic difficulty of optimization

Abstract

A very simple example of an algorithmic problem solvable by dynamic programming is to maximize, over sets A in {1,2,...,n}, the objective function |A| - \sum_i \xi_i 1(i \in A,i+1 \in A) for given \xi_i > 0. This problem, with random (\xi_i), provides a test example for studying the relationship between optimal and near-optimal solutions of combinatorial optimization problems. We show that, amongst solutions differing from the optimal solution in a small proportion \delta of places, we can find near-optimal solutions whose objective function value differs from the optimum by a factor of order \delta^2 but not smaller order. We conjecture this relationship holds widely in the context of dynamic programming over random data, and Monte Carlo simulations for the Kauffman-Levin NK model are consistent with the conjecture. This work is a technical contribution to a broad program initiated in Aldous-Percus (2003) of relating such scaling exponents to the algorithmic difficulty of optimization problems.