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A very simple example of an algorithmic problem solvable by dynamicprogramming 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, withrandom (\xi_i), provides a test example for studying the relationship betweenoptimal and near-optimal solutions of combinatorial optimization problems. Weshow that, amongst solutions differing from the optimal solution in a smallproportion \delta of places, we can find near-optimal solutions whose objectivefunction value differs from the optimum by a factor of order \delta^2 but notsmaller order. We conjecture this relationship holds widely in the context ofdynamic programming over random data, and Monte Carlo simulations for theKauffman-Levin NK model are consistent with the conjecture. This work is atechnical contribution to a broad program initiated in Aldous-Percus (2003) ofrelating such scaling exponents to the algorithmic difficulty of optimizationproblems.

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