Cédric Bentz

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.

A simple algorithm for multicuts in planar graphs with outer terminals

Given an edge-weighted graph G and a list of source-sink pairs of terminal vertices of G, the minimum multicut problem consists in selecting a minimum weight set of edges of G whose removal leaves no path from the ith source to the ith sink, for each i. Few tractable special cases are known for this problem. In this paper, we give a simple polynomial-time algorithm solving it in undirected planar graphs where (I) all the terminals lie on the outer face and (II) there is a bounded number of terminals.

Packing and covering with linear programming: A survey.

This paper considers the polyhedral results and the min–max results on packing and covering problems of the decade. Since the strong perfect graph theorem (published in 2006), the main such results are available for the packing problem, however there are still important polyhedral questions that remain open. For the covering problem, the main questions are still open, although there has been important progress. We survey some of the main results with emphasis on those where linear programming and graph theory come together. They mainly concern the covering of cycles or dicycles in graphs or signed graphs, either with vertices or edges; this includes the multicut and integral multiflow problems.

Degree-constrained edge partitioning in graphs arising from discrete tomography

Starting from the basic problem of reconstructing a 2-dimensional im- age given by its projections on two axes, one associates a model of edge coloring in a complete bipartite graph. The complexity of the case with k = 3 colors is open. Variations and special cases are considered for the case k = 3 colors where the graph corresponding to the union of some color classes (for instance colors 1 and 2) has a given structure (tree, vertex- disjoint chains, 2-factor, etc.). We also study special cases corresponding to the search of 2 edge-disjoint chains or cycles going through specified vertices. A variation where the graph is oriented is also presented. In addition we explore similar problems for the case where the under- lying graph is a complete graph (instead of a complete bipartite graph).

Multicuts and integral multiflows in rings

We show how to solve in polynomial time the multicut and the maximum integral multiflow problems in rings. Moreover, we give linear-time procedures to solve both problems in rings with uniform capacities.

Directed Steiner Tree with Branching Constraint

Given a directed weighted graph G, a root r and k terminals, the k-Directed Steiner Tree problem is to find a minimum cost tree rooted at r and spanning all terminals. If this problem has several applications in multicast routing in packet switching networks, the modeling is not adapted anymore in networks based upon the circuit switching principle in which some nodes, called non diffusing nodes, are not able to duplicate packets. We define a more general problem, named Directed Steiner Tree with Limited number of Diffusing nodes (DSTLD), able to model the multicast in a network containing at most d diffusing nodes. We show that DSTLD is XP with respect to d, and use this result to build a \(\lceil \frac{k-1}{d} \rceil\)-approximation XP in d for DST. Finally, we prove that, under the assumption that NP \(\not\subseteq\) DTIME[n O(loglogn)], there is no polynomial approximation algorithm for DSTLD with ratio \(1+(\frac{1}{e} - \varepsilon) \cdot \frac{k}{d-1}\) for every constant e > 0.

Maximum integer multiflow and minimum multicut problems in two-sided uniform grid graphs

In this paper, we deal with the maximum integer multiflow and the minimum multicut problems in rectilinear grid graphs with uniform capacities on the edges. The first problem is known to be NP-hard when any vertex can be a terminal, and we show that the second one is also NP-hard. Then, we study the case where the terminals are located in a two-sided way on the boundary of the outer face. We prove that, in this case, both problems are polynomial-time solvable. Furthermore, we give two efficient combinatorial algorithms using a primal-dual approach. Our work is based on previous results concerning related decision problems.

Edge disjoint paths and max integral multiflow/min multicut theorems in planar graphs

Abstract We generalize all the results obtained for integer multiflow and multicut problems in trees by Garg et al. [N. Garg, V.V. Vazirani and M. Yannakakis, Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica 18 (1997) 3–20] to planar graphs with a fixed number of faces, although other classical generalizations do not lead to such results. We also introduce the class of k-edge-outerplanar graphs and bound the integrality gap for the maximum edge-disjoint paths problem in these graphs.

Directed Steiner trees with diffusion costs

Given a directed arc-weighted graph G with n nodes, a root r and k terminals, the directed steiner tree problem (DST) consists in finding a minimum-weight tree rooted at r and spanning all the terminals. If this problem has several applications in multicast routing in packet switching networks, the modeling is not adapted anymore in networks based upon the circuit switching principle in which some nodes, called non diffusing nodes, are unable to duplicate packets. We define a more general problem, namely the directed steiner tree with a limited number of diffusing nodes (DSTLD), that enables us to model multicast in a network containing at most d diffusing nodes. We show that DSTLD is XP with respect to d, and use this result to build a