Polynomial-time approximation for an arbitrary number of machines

Develop a polynomial-time approximation scheme, or at least a polynomial-time constant-factor approximation, for stochastic scheduling of Bernoulli jobs when the number of identical machines is part of the input.

Background

The paper establishes a quasi-polynomial-time approximation scheme for Bernoulli jobs on any number of identical machines, but the resulting dynamic program remains quasi-polynomial after the size-reduction step. Its state description records both the remaining number of jobs in each of the O(ε^{-1} log(N/ε)) size classes and the release times as a multiset over O(ε^{-2} log(N/ε)) candidate values. A polynomial-time approximation scheme would therefore require substantially more compact representations of both components. The paper also proves that non-adaptive list policies can lose an Ω(log N) factor even on two machines, so any constant-factor approximation for arbitrary machine counts must use adaptivity.