Dynamic Programming Optimization over Random Data: The Scaling Exponent for Near-Optimal Solutions

Abstract

A very simple example of an algorithmic problem solvable by dynamic programming is to maximize, over $A\subseteq\{1,2,\ldots,n\}$, the objective function $|A|-\sum_i\xi_i{\rm1\hspace{-0.90ex}1}(i\in A,i+1\in A)$ 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 of 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 [D. J. Aldous and A. G. Percus, Proc. Natl. Acad. Sci. USA, 100 (2003), pp. 11211–11215] of relating such scaling exponents to the algorithmic difficulty of optimization problems.