Marc Demange

On an approximation measure founded on the links between optimization and polynomial approximation theory

In order to define a polynomial approximation theory linked to combinatorial optimization closer than the existing one, we first formally define the notion of a combinatorial optimization problem and then, based upon this notion, we introduce a notion of equivalence among optimization problems. This equivalence includes, for example, translation or affine transformation of the objective function or yet some aspects of equivalencies between maximization and minimization problems (for example, the equivalence between minimum vertex cover and maximum independent set). Next, we adress the question of the adoption of an approximation ratio respecting the defined equivalence. We prove that an approximation ratio defined as a two-variable function cannot respect this equivalence. We then adopt a three-variable function as a new approximation ratio (already used by a number of researchers), which is coherent to the equivalence and, under the choice of the variables, the new ratio is introduced by an axiomatic approach. Finally, using the new ratio, we prove approximation results for a number of combinatorial problems.

Differential approximation algorithms for some combinatorial optimization problems

We use a new approximation measure, the differential approximation ratio, to derive polynomial-time approximation algorithms for minimum set covering (for both weighted and unweighted cases), minimum graph coloring and bin-packing. We also propose differential-approximation-ratio preserving reductions linking minimum coloring, minimum vertex covering by cliques, minimum edge covering by cliques and minimum edge covering of a bipartite graph by complete bipartite graphs.

Algorithms for the on-line quota traveling salesman problem

The Quota Traveling Salesman Problem is a generalization of the well-known Traveling Salesman Problem. The goal of the traveling salesman is, in this case, to reach a given quota of sales, minimizing the amount of time. In this paper we address the on-line version of the problem, where requests are given over time. We present algorithms for various metric spaces, and analyze their performance in the usual framework of competitive analysis. In particular we present a 2-competitive algorithm that matches the lower bound for general metric spaces. In the case of the halfline metric space, we show that it is helpful not to move at full speed, and this approach is also used to derive the best on-line polynomial time algorithm known so far for the On-Line TSP (in the homing version).

Weighted Node Coloring: When Stable Sets Are Expensive

A version of weighted coloring of a graph is introduced: each node v of a graph G = (V, E) is provided with a positive integer weight w(v) and the weight of a stable set S of G is w(S) = max{w(v) : v ? V ? S}. A k-coloring S = (S1, . . . , Sk) of G is a partition of V into k stable sets S1, . . . , Sk and the weight of S is w(S1) + . . . + w(Sk). The objective then is to find a coloring S = (S1, . . . , Sk) of G such that w(S1) + . . . + w(Sk) is minimized. Weighted node coloring is NP-hard for general graphs (as generalization of the node coloring problem). We prove here that the associated decision problems are NP-complete for bipartite graphs, for line-graphs of bipartite graphs and for split graphs. We present approximation results for general graphs. For the other families of graphs dealt, properties of optimal solutions are discussed and complexity and approximability results are presented.

Time slot scheduling of compatible jobs

A version of weighted coloring of a graph is introduced which is motivated by some types of scheduling problems: each node v of a graph G corresponds to some operation to be processed (with a processing time w(v)), edges represent nonsimultaneity requirements (incompatibilities). We have to assign each operation to one time slot in such a way that in each time slot, all operations assigned to this slot are compatible; the length of a time slot will be the maximum of the processing times of its operations. The number k of time slots to be used has to be determined as well. So, we have to find a k-coloring

Partitioning cographs into cliques and stable sets

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New results on maximum induced matchings in bipartite graphs and beyond

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