Bridging gap between standard and differential polynomial approximation: the case of bin-packing

The purpose of this paper is to mainly prove the following theorem: for every polynomial time algorithm running in time T (n) and guaranteeing standard-approximation ratio ρ for bin-packing, there exists an algorithm running in time O(nT (n)) and achieving differentialapproximation ratio 2 − ρ for BP. This theorem has two main impacts. The first one is “operational”, deriving a polynomial time differential-approximation schema for bin-packing. The second one is structural, establishing a kind of reduction (to our knowledge not existing until now) between standard approximation and differential one. 1 Standard and differential approximation A current and very active research area coping with NP-completeness is polynomial approximation theory. In this domain, the main objective is either finding a good approximation algorithm for a given NP-complete problem, or establishing proofs that such algorithms cannot exist unless an unlikely complexity-theory condition (for example, P=NP) holds. The “goodness” of an approximation algorithm is commonly measured by its approximation ratio. Given an instance I of a combinatorial optimization problem Π and an approximation algorithm A supposed to feasibly solve Π, we will denote by ω(I), λA(I) and β(I) the values of the worst case solution, the approximated one (provided by A), and the optimal one, respectively. There exist mainly two thought processes dealing with polynomial approximation. Traditionally ([8, 14]), the quality of an approximation algorithm for an NP-complete minimization (resp., maximization) problem Π is expressed by the ratio (called standard in what follows) ρA(I) = λ(I)/β(I), and the quantity ρA = inf{r : ρA(I) < r, I instance of Π} (resp., ρA = sup{r : ρA(I) > r, I instance of Π}) constitutes the approximation ratio of A for Π. Recent works ([5, 4]), strongly inspired by former ones (see, for example, [2]), bring to the fore another approximation measure, as powerful as the traditional one (concerning the type, the diversity and the quantity of the produced results), the ratio (called differential in what follows) δA(I) = [ω(I) − λ(I)]/[ω(I) − β(I)]. The quantity δA = sup{r : δA(I) > r, I instance of Π} is now the approximation ratio of A for Π. A special case of a polynomial time approximation algorithm, inducing the strongest possible positive approximation result, is the one of polynomial time approximation schema. A polynomial time standard-approximation schema for a problem Π is a sequence A2 of polynomial ∗Also, CERMSEM, Université Paris I, Maison des Sciences Economiques, 106-112 boulevard de l’Hôpital, 75647 Paris Cedex 13, France