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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
NEAR-OPTIMAL 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 near-optimal solutions of combinatorial optimization problems. We show that,
amongst solutions differing from the optimal solution in a small proportion of places, we can find
near-optimal 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, near-optimal solu-
tions, optimization, probabilistic analysis of algorithms, scaling exponent

Source: Aldous, David J. - Department of Statistics, University of California at Berkeley
Lelarge, Marc - Département d'Informatique, École Normale Supérieure

Collections: Computer Technologies and Information Sciences; Engineering; Mathematics