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 infinity), the problem is approximable up to any chosen relative error bound ε>0 by a deterministic finite-state machine. More precisely, let Π be a problem in our subclass of NP optimization problems, let ε>0 be any chosen bound, and assume there is a fixed (but arbitrary) probability distribution for the inputs. Then there exists a finite-state machine which does the following: On an input I (random according to this probability distribution), the finite-state machine produces a feasible solution whose objective value M(I) satisfiesP|Opt(I)−M(I)|max{Opt(I),M(I)}⩾ε⩽Ke−hn,when n is large enough. Here K and h are positive constants.