Min-max and min-max regret criteria are commonly used to define robust solutions. After motivating the use of these criteria, we present general results. Then, we survey complexity results for the min-max and min-max regret versions of some combinatorial optimization problems: shortest path, spanning tree, assignment, min cut, min s-t cut, knapsack. Since most of these problems are NP -hard, we also investigate the approximability of these problems. Furthermore, we present algorithms to solve these problems to optimality.

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Several problems are known to be APX-, DAPX-, PTAS-, or Poly-APX-PB-complete under suitably defined approximation-preserving reductions. But, to our knowledge, no natural problem is known to be PTAS-complete and no problem at all is known to be Poly-APX-complete. On the other hand, DPTAS-and Poly-DAPX-completeness have not been studied until now. We first prove in this paper the existence of natural Poly-APX-and Poly-DAPX-complete problems under the well known PTASreduction and under the DPTAS-reduction (defined in "G. Ausiello, C. Bazgan, M. Demange, and V. Th. Paschos, Completeness in differential approximation classes, MFCS'03"), respectively. Next, we deal with PTAS-and DPTAS-completeness. We introduce approximation preserving reductions, called FT and DFT, respectively, and prove that, under these new reductions, natural problems are PTAS-complete, or DPTAS-complete. Then, we deal with the existence of intermediate problems under our reductions and we partially answer this question showing that the existence of NPO-intermediate problems under Turing-reduction is a sufficient condition for the existence of intermediate problems under both FTand DFT-reductions. Finally, we show that   is DAPX-complete under the DPTASreduction. This is the first DAPX-complete problem that is not simultaneously APX-complete. Cahiers du LAMSADE 217 DAPX, DPTAS and DFPTAS (see section 2 for formal definitions), are the differential counterparts of Poly-APX, APX, PTAS and FPTAS, respectively. Note that FPTAS PTAS APX Poly-APX, and DFPTAS DPTAS DAPX Poly-DAPX; these inclusions are strict unless P = NP. During last two decades, several approximation preserving reductions have been introduced and, using them, hardness results in several approximability classes have been studied. Consider two classes C 1 and C 2 with C 1 ⊆ C 2 , and assume a reduction preserving membership in C 1 (i.e., if Π reduces to Π and Π ∈ C 1 , then Π ∈ C 1). A problem C 2-complete under this reduction is in C 1 if and only if C 2 = C 1 (for example, assume C 1 = P and C 2 = NP). Consider, for instance, the P-reduction defined in [6]; this reduction, extended in [4, 7] (and renamed PTAS-reduction), preserves membership in PTAS. Natural problems, such as maximum independent set in bounded degree graphs (called   -B in what follows 1), or   , are APXcomplete under the PTAS-reduction (see, respectively, [15, 16]). This implies that such problems are not in PTAS unless P = NP (since, as we have mentioned previously, provided that P = NP, PTAS APX). In differential approximation, analogous results have been obtained in [1], where a DPTAS-reduction, preserving membership in DPTAS, is defined and natural problems such as   -B, or   -B are shown to be DAPX-complete. In the same way, the F-reduction of [6] preserves membership in FPTAS. Under this reduction, only one (not very natural) problem (derived from  - ) is known to be PTAScomplete. Despite some restrictive notions of DPT...

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This paper investigates, for the first time in the literature, the approximation of minmax (regret) versions of classical problems like shortest path, minimum spanning tree, and knapsack. For a constant number of scenarios, we establish fully polynomial-time approximation schemes for the min-max versions of these problems, using relationships between multi-objective and min-max optimization. Using dynamic programming and classical trimming techniques, we construct a fully polynomial-time approximation scheme for min-max regret shortest path. We also establish a fully polynomial-time approximation scheme for min-max regret spanning tree and prove that min-max regret knapsack is not at all approximable. For a non constant number of scenarios, in which case minmax and min-max regret versions of polynomial-time solvable problems usually become strongly NP -hard, non-approximability results are provided for min-max (regret) versions of shortest path and spanning tree.

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The SATISFACTORY PARTITION problem consists in deciding if a given graph has a partition of its vertex set into two nonempty parts such that each vertex has at least as many neighbors in its part as in the other part. This problem was introduced by Gerber and Kobler [Partitioning a graph to satisfy all vertices, Technical report, Swiss Federal Institute of Technology, Lausanne, 1998; Algorithmic approach to the satisfactory graph partitioning problem, European J. Oper. Res. 125 (2000) 283-291] and further studied by other authors but its complexity remained open until now. We prove in this paper that SATISFACTORY PARTITION, as well as a variant where the parts are required to be of the same cardinality, are NP-complete. However, for graphs with maximum degree at most 4 the problem is polynomially solvable. We also study generalizations and variants of this problem where a partition into k nonempty parts (k 3) is requested.

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