Assigning channels via the meet-in-the-middle approach

We study the complexity of the Channel Assignment problem. By applying the meet-in-the-middle approach we get an algorithm for the $\ell$-bounded Channel Assignment (when the edge weights are bounded by $\ell$) running in time $O^*((2\sqrt{\ell+1})^n)$. This is the first algorithm which breaks the $(O(\ell))^n$ barrier. We extend this algorithm to the counting variant, at the cost of slightly higher polynomial factor. A major open problem asks whether Channel Assignment admits a $O(c^n)$-time algorithm, for a constant $c$ independent of $\ell$. We consider a similar question for Generalized T-Coloring, a CSP problem that generalizes \CA. We show that Generalized T-Coloring does not admit a $2^{2^{o\left(\sqrt{n}\right)}} {\rm poly}(r)$-time algorithm, where $r$ is the size of the instance.Comment: SWAT 2014: 282-29

Tight lower bound for the channel assignment problem

We study the complexity of the Channel Assignment problem. A major open problem asks whether Channel Assignment admits an $O(c^n)$-time algorithm, for a constant $c$ independent of the weights on the edges. We answer this question in the negative i.e. we show that there is no $2^{o(n\log n)}$-time algorithm solving Channel Assignment unless the Exponential Time Hypothesis fails. Note that the currently best known algorithm works in time $O^*(n!) = 2^{O(n\log n)}$ so our lower bound is tight

The Fine-Grained Complexity of Computing the Tutte Polynomial of a Linear Matroid

We show that computing the Tutte polynomial of a linear matroid of dimension $k$ on $k^{O(1)}$ points over a field of $k^{O(1)}$ elements requires $k^{\Omega(k)}$ time unless the \#ETH---a counting extension of the Exponential Time Hypothesis of Impagliazzo and Paturi [CCC 1999] due to Dell {\em et al.} [ACM TALG 2014]---is false. This holds also for linear matroids that admit a representation where every point is associated to a vector with at most two nonzero coordinates. We also show that the same is true for computing the Tutte polynomial of a binary matroid of dimension $k$ on $k^{O(1)}$ points with at most three nonzero coordinates in each point's vector. This is in sharp contrast to computing the Tutte polynomial of a $k$-vertex graph (that is, the Tutte polynomial of a {\em graphic} matroid of dimension $k$---which is representable in dimension $k$ over the binary field so that every vector has two nonzero coordinates), which is known to be computable in $2^k k^{O(1)}$ time [Bj\"orklund {\em et al.}, FOCS 2008]. Our lower-bound proofs proceed via (i) a connection due to Crapo and Rota [1970] between the number of tuples of codewords of full support and the Tutte polynomial of the matroid associated with the code; (ii) an earlier-established \#ETH-hardness of counting the solutions to a bipartite $(d,2)$-CSP on $n$ vertices in $d^{o(n)}$ time; and (iii) new embeddings of such CSP instances as questions about codewords of full support in a linear code. We complement these lower bounds with two algorithm designs. The first design computes the Tutte polynomial of a linear matroid of dimension~$k$ on $k^{O(1)}$ points in $k^{O(k)}$ operations. The second design generalizes the Bj\"orklund~{\em et al.} algorithm and runs in $q^{k+1}k^{O(1)}$ time for linear matroids of dimension $k$ defined over the $q$-element field by $k^{O(1)}$ points with at most two nonzero coordinates each.Comment: This version adds Theorem

Algorithmes exponentiels pour l'étiquetage, la domination et l'ordonnancement

This manuscript of Habilitation à 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 d’Habilitation à Diriger des Recherches met en lumière quelques résultats obtenus depuis ma thèse de doctorat soutenue en 2007. Ces résultats ont été, pour l’essentiel, publiés dans des conférences et des journaux internationaux. Des algorithmes exponentiels sont donnés pour résoudre des problèmes de décision, d’optimisation et d’énumération. On s’intéresse tout d’abord au problème d’étiquetage L(2,1) d’un graphe, pour lequel différents algorithmes sont décrits (basés sur du branchement, le paradigme diviser-pour-régner, ou la programmation dynamique). Des bornes combinatoires, nécessaires à l’analyse de ces algorithmes, sont également établies. Dans un second temps, nous résolvons des problèmes autour de la domination. Nous développons des algorithmes pour résoudre une généralisation de la domination et nous donnons des algorithmes pour énumérer les ensembles dominants minimaux dans des classes de graphes. L’analyse de ces algorithmes implique des bornes combinatoires. Finalement, nous étendons notre champ d’applications de l’algorithmique modérément exponentielle à des problèmes d’ordonnancement. Par le développement d’approches de type programmation dynamique et la généralisation de la méthode trier-et-chercher, nous proposons la résolution de toute une famille de problèmes d’ordonnancement