Christophe Picouleau

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.

New complexity results on scheduling with small communication delays

Although most of the scheduling problems with interprocessor communication delays have been shown to be NP-complete, some important special cases were still unsolved. This paper deals with the problem where communication times are smaller than processing times and task duplication is not allowed. We prove that this problem is NP-complete and we give an efficient approximate algorithm with performance guarantee.

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.

Reconstruction of domino tiling from its two orthogonal projections

We are interested in the reconstruction of a domino tiling of a rectangle from its two orthogonal projections. We give polynomial algorithms for some subproblems when all the dominoes are of the same type and prove NP-completeness results when there are three types of dominoes. When two types of dominoes are allowed, we give a polynomial-time transformation from a well-known open problem.

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.

Reconstruction of binary matrices under fixed size neighborhood constraints

Using a dynamic programming approach, we prove that a large variety of matrix reconstruction problems from two projections can be solved in polynomial time whenever the number of rows (or columns) is fixed. We also prove some complexity results for several problems concerning the reconstruction of a binary matrix when a neighborhood constraint occurs.

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.

Minimum d-blockers and d-transversals in graphs

We consider a set V of elements and an optimization problem on V: the search for a maximum (or minimum) cardinality subset of V verifying a given property ℘. A d-transversal is a subset of V which intersects any optimum solution in at least d elements while a d-blocker is a subset of V whose removal deteriorates the value of an optimum solution by at least d. We present some general characteristics of these problems, we review some situations which have been studied (matchings, s–t paths and s–t cuts in graphs) and we study d-transversals and d-blockers of stable sets or vertex covers in bipartite and in split graphs.