No abstract

In this paper, reoptimization versions of the traveling salesman problem (TSP) are addressed. Assume that an optimum solution of an instance is given and the goal is to determine if one can maintain a good solution when the instance is subject to minor modifications. We study the case where nodes are inserted in, or deleted from, the graph. When inserting a node, we show that the reoptimization problem for MinTSP is approximable within ratio 4/3 if the distance matrix is metric. We show that, dealing with metric MaxTSP, a simple heuristic is asymptotically optimum when a constant number of nodes are inserted. In the general case, we propose a 4/5-approximation algorithm for the reoptimization version of MaxTSP.

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...