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SIAM J. COMPUT. c 2009 Society for Industrial and Applied Mathematics
Vol. 38, No. 6, pp. 23822410
DYNAMIC PROGRAMMING OPTIMIZATION OVER RANDOM
DATA: THE SCALING EXPONENT FOR
NEAROPTIMAL SOLUTIONS
DAVID J. ALDOUS, CHARLES BORDENAVE, AND MARC LELARGE§
Abstract. A very simple example of an algorithmic problem solvable by dynamic programming
is to maximize, over A {1, 2, . . . , n}, the objective function A  i i11(i A, i + 1 A) for
given i > 0. This problem, with random (i), provides a test example for studying the relationship
between optimal and nearoptimal solutions of combinatorial optimization problems. We show that,
amongst solutions differing from the optimal solution in a small proportion of places, we can find
nearoptimal solutions whose objective function value differs from the optimum by a factor of order
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 KauffmanLevin 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. 1121111215] of
relating such scaling exponents to the algorithmic difficulty of optimization problems.
Key words. dynamic programming, local weak convergence, Markov chain, nearoptimal solu
tions, optimization, probabilistic analysis of algorithms, scaling exponent
