The guess approximation technique and its application to the Discrete Resource Sharing Scheduling Problem

We present in this work a technique for designing approximation schemes and its application to the discrete resource sharing problem (dRSSP ). We use the (classical) oracle formalism, in which a reliable oracle provides some information (usually called “a guess”) that helps to construct a good approximated solution. This oracle technique is sometimes called “partial enumeration” [ST07] or “output structuring” [SW00]. Generally, this oracle design technique consists of two steps: construct an algorithm that, given the oracle guess, provides the approximated solution, and then simply enumerates the set of all the possible guesses. The bigger the guess length (number of bits), the better the approximated solution. However the set of all possible guesses is also larger. The goal is to find an “efficient” guess, that is a small guess containing a lot of information. Thus, we are interested in finding a compact way to encode the guess. This (natural) idea has already been addressed [HS89] by mixing two techniques (structuring the input, and apply partial enumeration on this simplified input). We propose in this paper a slightly different solution by approximating the guess itself through a contraction function. We believe that this refinement may lead to non trivial guesses, and enable to use guesses which were too long before the contraction. Moreover, approximation schemes derived from this technique have a complexity depending on two parameters (the complexity indeed depends on the length of the guess, and the roughness of the guess approximation), which allows a more flexible tradeoff between quality of the approximation and computational complexity. In the second section, we present the targeted problem (dRSSP ) and some associated results (NP completeness, approximations algorithms and a worst case); in section three, we define the guess approximation technique and apply it to the dRSSP .