Vangelis Th. Paschos

A survey of approximately optimal solutions to some covering and packing problems

We survey approximation algorithms for some well-known and very natural combinatorial optimization problems, the minimum set covering, the minimum vertex covering, the maximum set packing, and maximum independent set problems; we discuss their approximation performance and their complexity. For already known results, any time we have conceived simpler proofs than those already published, we give these proofs, and, for the rest, we cite the simpler published ones. Finally, we discuss how one can relate the approximability behavior (from both a positive and a negative point of view) of vertex covering to the approximability behavior of a restricted class of independent set problems.

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).

A simulated annealing approach for the circular cutting problem

We propose a heuristic for the constrained and the unconstrained circular cutting problem based upon simulated annealing. We define an energy function, the small values of which provide a good concentration of the circular pieces on the left bottom corner of the initial rectangle. Such values of the energy correspond to configurations where pieces are placed in the rectangle without overlapping. Appropriate software has been devised and computational results and comparisons with some other algorithms are also provided and discussed.

A new single model and derived algorithms for the satellite shot planning problem using graph theory concepts

The satellite shot sequencing problem consists in choosing the pictures to be completed by defining sequences of shots which must respect technical constraints and limits. We propose a graph-theoretic model for both the medium- and the short-term sequencing and present algorithmic solutions by using properties of the model.

The probabilistic longest path problem

We study the probabilistic longest path problem. We propose a modification strategy adapting a solution for a deterministic instance to a solution for the probabilistic one, we compute the functional associated with this strategy, and we evaluate the complexities of computing this functional and of computing the deterministic solution maximizing it.

Reoptimization of minimum and maximum traveling salesman's tours

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

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.

An improved general procedure for lexicographic bottleneck problems

In combinatorial optimization, the bottleneck (or minmax) problems are those problems where the objective is to find a feasible solution such that its largest cost coefficient elements have minimum cost. Here we consider a generalization of these problems, where under a lexicographic rule we want to minimize the cost also of the second largest cost coefficient elements, then of the third largest cost coefficients, and so on. We propose a general rule which leads, given the considered problem, to a vectorial version of the solution procedure for the underlying sum optimization (minsum) problem. This vectorial procedure increases by a factor of k (where k is the number of different cost coefficients) the complexity of the corresponding sum optimization problem solution procedure.