Compatibilité entre structures d''intervalles et relations d''orde

Compatibility between interval structures and partial orderings.

If H=(X,E) is a hypergraph, n the cardinality of X,I_n the ordered set {1..n} and < an order relation on X, we call F(X,<) the set of the one-to-one functions from X to I_n which are compatible with <. If AI_n we denote by (A) the length of the smallest interval of I_n which contains A.

We first deal with the following problem: Find ƒF(X,<) which minimise . The a_e, eR are positive coefficients.

This problem can be understood as a scheduling problem and is checked to be NP-complete. We learn how to recognize in polynomial time those hypergraphs H=(X,E) which induce an optimal value of z min equal to .

Next we work on a dual question which arises about interval graphs, when some partial orderings on the vertex set of these graphs intend to represent inclusion, overlapping or anteriority relations between closed intervals of the real line.     

Optimal preemptive scheduling on a fixed number of identical parallel machines

In this paper, we consider the preemptive scheduling problem on a fixed number of identical parallel machines. We present a polynomial-time algorithm for finding a minimal length schedule for an order class which contains properly interval orders.     

On the helly property working as a compactness criterion on graphs

The study of combinatorial topology and of the most important methods in algebraic topology (simplicial complexes, discretization) leads to the idea that it may be useful to translate some of the most classical problems in topology into a discrete context. Following this principle, several authors have already tried to study fixed point and retraction problems inside the theory of partially ordered sets. We try here to make a special study about the extension of homomorphisms and the fixed point problems on graphs. We introduce here, using the Helly property, a kind of compactness tool working on graphs, and we prove a generalization of Sperner's lemma which is used in the proof of the Brouwer fixed-point theorem by Kuratowski.     

Homogeneously non-idling schedules of unit-time jobs on identical parallel machines

In this paper, we study the basic homogeneous $m$-machine scheduling problem where weakly dependent unit-time jobs have to be scheduled within the time windows between their release dates and due dates so that, for any subset of machines, the set of the time units at which at least one machine is busy, is in interval. We first introduce the notions of pyramidal structure, $k$-hole, $m$-matching, preschedule, $k$-schedule and schedule for this problem. Then we provide a feasibility criteria for a preschedule. The key result of the paper is then to provide a structural necessary and sufficient condition for an instance of the problem to be feasible. We conclude by giving the directions of ongoing works and by bringing open questions related to different variants of the basic non-idling $m$-machine scheduling problem.     

Preemptive scheduling and antichain polyhedra

We present a theoretical framework, which is based upon notions of ordered hypergraphs and antichain polyhedra, and which is dedicated to the combinatorial analysis of preemptive scheduling problems submitted to parallelization constraints.This framework allows us to characterize specific partially ordered structures which are such that induced preemptive scheduling problems may be solved through linear programming. To prove that, in the general case, optimal preemptive schedules may be searched inside some connected subset of the vertex set of an Antichain Polyhedron.     

Circular representation problem on hypergraphs

A hypergraph J=(X,E) is said to be circular representable, if its vertices can be placed on a circle, in such way that every edge of H induces an interval. This concept is a translation into the vocabulary of hypergraphs of the circular one's property for the (0, 1) matrices [6] studied by Tucker [9, 10]. We give here a characterization of the hypergraphs which are circular representable. We study when the associated representation is unique, and we characterize the possible transformations of a representation into another, a kind of problem which has already been treated from the algorithmic point of view by Booth and Lueker [1] or Duchet [2] in the case of the interval representable hypergraphs.Finally, we establish a connection between circular graphs and circular representable hypergraphs of the type of the Fulkerson-Gross connection between interval graphs and matrices having the consecutive one's property [5], in some special cases.     

Linear programming based algorithms for preemptive and non-preemptive RCPSP

In this paper, the RCPSP (resource constrained project scheduling problem) is solved using a linear programming model. Each activity may or may not be preemptive. Each variable is associated to a subset of independent activities (antichains). The properties of the model are first investigated. In particular, conditions are given that allow a solution of the linear program to be a feasible schedule. From these properties, an algorithm based on neighbourhood search is derived. One neighbour solution is obtained through one Simplex pivoting, if this pivoting preserves feasibility. Methods to get out of local minima are provided. The solving methods are tested on the PSPLIB instances in a preemptive setting and prove efficient. They are used when preemption is forbidden with less success, and this difference is discussed.     

A Relation between Multiprocessor Scheduling and Linear Programming

The general non preemptive multiprocessor scheduling problem (NPMS) is NP-Complete, while in many specific cases, the same problem is Time-polynomial. A first connection between PMS and linear programming was established by Yannanakis, Sauer and Stone, who associated to any PMS instance some specific linear program. The main result inside this paper consists in a characterization of the partially ordered structures which allow the optimal values of any associated PMS instance to be equal to the optimal values of the corresponding linear programs.     

An application of the Helly property to the partially ordered sets

Quilliot (Discrete Math. 1982.) showed that when the bowls of a connected graph satisfy the Helly property it is possible to deduce for this graph some fixed point and homomorphism extension theorems. For a partially ordered set E a special family of subsets is defined which, when it satisfies the Helly property, permits the deductions that every homomorphism from E into E has a fixed point, that every antitone function from E has “almost” a fixed point, and that there exists a simple criterion letting us know when a function f from a subset A of a partially ordered set G can be extended into a homomorphism from G to E.