Optima locaux garantis pour l'approximation différentielle

A survey on the structure of approximation classes

“…Under this reduction, they obtain completeness results for a class of problems admitting "good (guaranteed)" local optima. It would be interesting that such structural studies the beginnings of which are presented in [6,7,75,87] are undertaken also in differential paradigm.…”

“…Several results (positive or negative) for classical combinatorial problems have appeared (min coloring [34,42,55], min tsp, max tsp and vehicle routing problems [74,73,18,54], bin packing [35,36], min set cover [19], optimal satisfiability problems [20,45], etc.). Several structural and computational aspects are also investigated in [33,75,72,87].…”

Differential approximation of min sat, max sat and related problems

“…. , x p−1 i we will build one new variable and we will transform any of the m equations of I into an equation as in (16). Then, for any of these new equations we add in the instance of  3(p − 1) the set of clauses as built just above.…”

“…There, among other results, it was shown that   and  , as well as   and   are equivalent for the differential approximation, that all these problems are not solvable by polynomial time differential approximation schemata, unless P = NP, and, finally, that   cannot be approximately solved within differential approximation ratio 1/m 1−ǫ , for any ǫ > 0 (where m is the number of the clauses in its instance), unless NP = co-RP. Finally, let us mention here that both   and   belong to 0-DAPX, the class of the problems for which no algorithm can guarantee differential approximation ratio strictly greater than 0, unless P = NP ( [16]). This class has been also introduced in [6].…”

Differential Approximation of min sat, max sat and Related Problems

“…we will build one new variable and we will transform any of the m equations of I into an equation as in (16). Then, for any of these new equations we add in the instance of  3(p − 1) the set of clauses as built just above.…”