78 research outputs found

### A Branch-and-Reduce Algorithm for Finding a Minimum Independent Dominating Set

An independent dominating set D of a graph G = (V,E) is a subset of vertices
such that every vertex in V \ D has at least one neighbor in D and D is an
independent set, i.e. no two vertices of D are adjacent in G. Finding a minimum
independent dominating set in a graph is an NP-hard problem. Whereas it is hard
to cope with this problem using parameterized and approximation algorithms,
there is a simple exact O(1.4423^n)-time algorithm solving the problem by
enumerating all maximal independent sets. In this paper we improve the latter
result, providing the first non trivial algorithm computing a minimum
independent dominating set of a graph in time O(1.3569^n). Furthermore, we give
a lower bound of \Omega(1.3247^n) on the worst-case running time of this
algorithm, showing that the running time analysis is almost tight.Comment: Full version. A preliminary version appeared in the proceedings of WG
200

### TREEWIDTH and PATHWIDTH parameterized by vertex cover

After the number of vertices, Vertex Cover is the largest of the classical
graph parameters and has more and more frequently been used as a separate
parameter in parameterized problems, including problems that are not directly
related to the Vertex Cover. Here we consider the TREEWIDTH and PATHWIDTH
problems parameterized by k, the size of a minimum vertex cover of the input
graph. We show that the PATHWIDTH and TREEWIDTH can be computed in O*(3^k)
time. This complements recent polynomial kernel results for TREEWIDTH and
PATHWIDTH parameterized by the Vertex Cover

### Complexity of Splits Reconstruction for Low-Degree Trees

Given a vertex-weighted tree T, the split of an edge xy in T is min{s_x(xy),
s_y(xy)} where s_u(uv) is the sum of all weights of vertices that are closer to
u than to v in T. Given a set of weighted vertices V and a multiset of splits
S, we consider the problem of constructing a tree on V whose splits correspond
to S. The problem is known to be NP-complete, even when all vertices have unit
weight and the maximum vertex degree of T is required to be no more than 4. We
show that the problem is strongly NP-complete when T is required to be a path,
the problem is NP-complete when all vertices have unit weight and the maximum
degree of T is required to be no more than 3, and it remains NP-complete when
all vertices have unit weight and T is required to be a caterpillar with
unbounded hair length and maximum degree at most 3. We also design polynomial
time algorithms for the variant where T is required to be a path and the number
of distinct vertex weights is constant, and the variant where all vertices have
unit weight and T has a constant number of leaves. The latter algorithm is not
only polynomial when the number of leaves, k, is a constant, but also
fixed-parameter tractable when parameterized by k. Finally, we shortly discuss
the problem when the vertex weights are not given but can be freely chosen by
an algorithm.
The considered problem is related to building libraries of chemical compounds
used for drug design and discovery. In these inverse problems, the goal is to
generate chemical compounds having desired structural properties, as there is a
strong correlation between structural properties, such as the Wiener index,
which is closely connected to the considered problem, and biological activity

### Algorithmes exponentiels pour l'Ă©tiquetage, la domination et l'ordonnancement

This manuscript of Habilitation aÌ Diriger des Recherches enlights some results obtained since my PhD, I defended in 2007. The presented results have been mainly published in international conferences and journals. Exponential-time algorithms are given to solve various decision, optimization and enumeration problems. First, we are interested in solving the L(2,1)-labeling problem for which several algorithms are described (based on branching, divide-and-conquer and dynamic programming). Some combinatorial bounds are also established to analyze those algorithms. Then we solve domination-like problems. We develop algorithms to solve a generalization of the dominating set problem and we give algorithms to enumerate minimal dominating sets in some graph classes. As a consequence, the analysis of these algorithms implies combinatorial bounds. Finally, we extend our field of applications of moderately exponential-time algorithms to scheduling problems. By using dynamic programming paradigm and by extending the sort-and-search approach, we are able to solve a family of scheduling problems.Ce manuscrit dâHabilitation aÌ Diriger des Recherches met en lumieÌre quelques reÌsultats obtenus depuis ma theÌse de doctorat soutenue en 2007. Ces reÌsultats ont eÌteÌ, pour lâessentiel, publieÌs dans des confeÌrences et des journaux internationaux. Des algorithmes exponentiels sont donneÌs pour reÌsoudre des probleÌmes de deÌcision, dâoptimisation et dâeÌnumeÌration. On sâinteÌresse tout dâabord au probleÌme dâeÌtiquetage L(2,1) dâun graphe, pour lequel diffeÌrents algorithmes sont deÌcrits (baseÌs sur du branchement, le paradigme diviser-pour-reÌgner, ou la programmation dynamique). Des bornes combinatoires, neÌcessaires aÌ lâanalyse de ces algorithmes, sont eÌgalement eÌtablies. Dans un second temps, nous reÌsolvons des probleÌmes autour de la domination. Nous deÌveloppons des algorithmes pour reÌsoudre une geÌneÌralisation de la domination et nous donnons des algorithmes pour eÌnumeÌrer les ensembles dominants minimaux dans des classes de graphes. Lâanalyse de ces algorithmes implique des bornes combinatoires. Finalement, nous eÌtendons notre champ dâapplications de lâalgorithmique modeÌreÌment exponentielle aÌ des probleÌmes dâordonnancement. Par le deÌveloppement dâapproches de type programmation dynamique et la geÌneÌralisation de la meÌthode trier-et-chercher, nous proposons la reÌsolution de toute une famille de probleÌmes dâordonnancement

### Exponential Algorithms for Scheduling Problems

This report focuses on the challenging issue of designing exponential algorithms for scheduling problems. Despite a growing literature dealing with such algorithms for other combinatorial optimization problems, it is still a recent research area in scheduling theory and few results are known. An exponential algorithm solves optimaly an NP-hard optimization problem with a worst-case time, or space, complexity that can be established and, which is lower than the one of a brute-force search. By the way, an exponential algorithm provides information about the complexity in the worst-case of solving a given NP-hard problem. In this report, we provide a survey of the few results known on schduling problems as well as some techniques for deriving exponential algorithms. In a second part we focus on some basic scheduling problems for which we propose exponential algorithms

### Programmation dynamique exponentielle pour des problĂšmes d'ordonnancement de type flowshop Ă 3 machines

National audienceDans cet article, nous traitons du problĂšme F3Cmax de flowshop Ă 3 machines oĂč nous cherchons Ă minimiser le makespan. Nous proposons d'abord un algorithme de la programmation dynamique modĂ©rĂ©ment exponentiel pour le problĂšme F3Cmax. Nousmontrons ensuite que notre approche peut se gĂ©nĂ©raliser Ă d'autres problĂšmes de flowshop

### On Finding Optimal Polytrees

Inferring probabilistic networks from data is a notoriously difficult task.
Under various goodness-of-fit measures, finding an optimal network is NP-hard,
even if restricted to polytrees of bounded in-degree. Polynomial-time
algorithms are known only for rare special cases, perhaps most notably for
branchings, that is, polytrees in which the in-degree of every node is at most
one. Here, we study the complexity of finding an optimal polytree that can be
turned into a branching by deleting some number of arcs or nodes, treated as a
parameter.
We show that the problem can be solved via a matroid intersection formulation
in polynomial time if the number of deleted arcs is bounded by a constant. The
order of the polynomial time bound depends on this constant, hence the
algorithm does not establish fixed-parameter tractability when parameterized by
the number of deleted arcs. We show that a restricted version of the problem
allows fixed-parameter tractability and hence scales well with the parameter.
We contrast this positive result by showing that if we parameterize by the
number of deleted nodes, a somewhat more powerful parameter, the problem is not
fixed-parameter tractable, subject to a complexity-theoretic assumption.Comment: (author's self-archived copy

### On Finding Optimal Polytrees

Peer reviewe

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