Approximation of some NP-hard optimization problems by finite machines, in probability

We introduce a subclass of NP optimization problems which contains some NP-hard problems, e.g., bin covering and bin packing. For each problem in this subclass we prove that with probability tending to 1 (exponentially fast as the number of input items tends to in)nity), the problem is approximable up to any chosen relative error bound ¿ 0 by a deterministic )nite-state machine. More precisely, let be a problem in our subclass of NP optimization problems, let ¿ 0 be any chosen bound, and assume there is a )xed (but arbitrary) probability distribution for the inputs. Then there exists a )nite-state machine which does the following: On an input I (random according to this probability distribution), the )nite-state machine produces a feasible solution whose objective value M (I) satis)es P ( |Opt(I)−M (I)| max{Opt(I); M (I)} ) 6Ke−hn; when n is large enough. Here K and h are positive constants. c © 2001 Elsevier Science B.V. All rights reserved.