Marie-Christine Costa

An efficient algorithm for a task allocation problem

This paper presents an efficient algorithm to solve one of the task allocation problems. Task assignment in an heterogeneous multiple processors system is investigated. The cost function is formulated in order to measure the intertask communication and processing costs in an uncapacited network. A formulation of the problem in terms of the minimization of a submodular quadratic pseudo-Boolean function with assignment constraints is then presented. The use of a branch-and-bound algorithm using a Lagrangean relaxation of these constraints is proposed. The lower bound is the value of an approximate solution to the Lagrangean dual problem. A zero-duality gap, that is, a saddle point, is characterized by checking the consistency of a pseudo-Boolean equation. A solution is found for large-scale problems (e.g., 20 processors, 50 tasks, and 200 task communications or 10 processors, 100 tasks, and 300 task communications). Excellent experimental results were obtained which are due to the weak frequency of a duality gap and the efficient characterization of the zero-gap (for practical purposes, this is achieved in linear time). Moreover, from the saddle point, it is possible to derive the optimal task assignment.

Minimal multicut and maximal integer multiflow: A survey

We present a survey about the maximum integral multiflow and minimum multicut problems and their subproblems, such as the multiterminal cut and the unsplittable flow problems. We consider neither continuous multiflow nor minimum cost multiflow. Most of the results are very recent and some are new. We recall the dual relationship between both problems, give complexity results and algorithms, firstly in unrestricted graphs and secondly in several special graphs: trees, bipartite or planar graphs. A table summarizes the most important results.

Solving the dynamic facility location problem

This paper addresses with the multiperiod, or dynamic, uncapacited facility location problem (DUFLP): The demand varies between time-periods and the solution should answer the questions of where and when establish facilities. We modelize the problem as a 0–1 quadratic program and, since the DUFLP is NP-hard, we focus on methods for generating heuristic solutions (by simulated annealing) and good lower bounds (by Lagrangian relaxation). We prove that our bound is equal to the optimal solution of the continuous relaxation of a linearization of the initial program. Then we show that the method can take into account some additional costs needed to solve some practical problems arising in telecommunication and intelligent networks. Finally, we present experimental results: The small size of the duality gaps explains the good quality of the obtained solutions.

Optimizing splitter and fiber location in a multilevel optical FTTH network

Due to the emergence of bandwidth-requiring services, telecommunication operators are brought to renew their fixed access network, most of them favoring the Fiber To The Home (FTTH) technology. This paper focuses on the optimization of FTTH deployment, which is of prime importance due to the economic stakes. The key design issue here is locating splitters and routing fibers in an existing network infrastructure to which is associated a graph with given capacities on the edges. No assumption is made on the structure of the graph. First we propose a mixed integer formulation for this decision problem and we prove it is NP-hard. Then, valid inequalities and problem size reduction schemes are presented. Finally, the efficiency of solution approaches is assessed through extensive numerical tests performed on France Telecom-Orange real-life data.

Blockers and transversals

Given an undirected graph G=(V,E) with matching number @n(G), we define d-blockers as subsets of edges B such that @n((V,E@?B))@?@n(G)-d. We define d-transversals T as subsets of edges such that every maximum matching M has |M@?T|>=d. We explore connections between d-blockers and d-transversals. Special classes of graphs are examined which include complete graphs, regular bipartite graphs, chains and cycles and we construct minimum d-transversals and d-blockers in these special graphs. We also study the complexity status of finding minimum transversals and blockers in arbitrary graphs.

Blockers and transversals in some subclasses of bipartite graphs: When caterpillars are dancing on a grid

Given an undirected graph G=(V,E) with matching number @n(G), a d-blocker is a subset of edges B such that @n((V,E@?B))@?@n(G)-d and a d-transversal T is a subset of edges such that every maximum matching M has |M@?T|>=d. While the associated decision problem is NP-complete in bipartite graphs we show how to construct efficiently minimum d-transversals and minimum d-blockers in the special cases where G is a grid graph or a tree.

Using graphs for some discrete tomography problems

Given a rectangular array whose entries represent the pixels of a digitalized image, we consider the problem of reconstructing an image from the number of occurrences of each color in every column and in every row. The complexity of this problem is still open when there are just three colors in the image. We study some special cases where the number of occurrences of each color is limited to small values. Formulations in terms of edge coloring in graphs and as timetabling problems are used; complexity results are derived from the model.

A solvable case of image reconstruction in discrete tomography

A graph-theoretical model is used to show that a special case of image reconstruction problem (with 3 colors) can be solved in polynomial time. For the general case with 3 colors, the complexity status is open. Here we consider that among the three colors there is one for which the total number of multiple occurrences in a same line (row or column) is bounded by a fixed parameter. There is no assumption on the two remaining colors.

An acyclic days-off scheduling problem

Abstract.This paper studies the days off scheduling problem when the demand for staffing fluctuates from day to another and when the number of total workdays is fixed in advance for each employee. The scheduling problem is then to allocate rests to employees with different days off policies: (1) two or three consecutive days off for each employee per week and (2) at least three consecutive days off for each employee per month. For each one, we propose a polynomial time algorithm to construct a solution if it exists.

Reconstruction of Binary Matrices under Adjacency Constraints

We consider a generalization of the classical binary matrix reconstruction problem by considering adjacency constraints between the cells: if a given cell is of value 1 then all its neighbors are of value 0. This problem arises especially on statistical physics. We consider several definitions of neighborhood and for each one we give complexity results, necessary and/or sufficient conditions for the existence of a solution and in some cases, polynomial time algorithms.