6 research outputs found
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 -bounded Channel
Assignment (when the edge weights are bounded by ) running in time
. This is the first algorithm which breaks the
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 -time
algorithm, for a constant independent of . 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
-time algorithm, where 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 -time algorithm, for
a constant independent of the weights on the edges. We answer this question
in the negative i.e. we show that there is no -time algorithm
solving Channel Assignment unless the Exponential Time Hypothesis fails. Note
that the currently best known algorithm works in time 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
on points over a field of elements requires
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 on points with at
most three nonzero coordinates in each point's vector. This is in sharp
contrast to computing the Tutte polynomial of a -vertex graph (that is, the
Tutte polynomial of a {\em graphic} matroid of dimension ---which is
representable in dimension over the binary field so that every vector has
two nonzero coordinates), which is known to be computable in
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 -CSP on vertices in 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~ on points in operations. The second design
generalizes the Bj\"orklund~{\em et al.} algorithm and runs in
time for linear matroids of dimension defined over the
-element field by points with at most two nonzero coordinates
each.Comment: This version adds Theorem
Algorithmes exponentiels pour l'Ć©tiquetage, la domination et l'ordonnancement
This manuscript of Habilitation aĢ 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 dāHabilitation aĢ Diriger des Recherches met en lumieĢre quelques reĢsultats obtenus depuis ma theĢse de doctorat soutenue en 2007. Ces reĢsultats ont eĢteĢ, pour lāessentiel, publieĢs dans des confeĢrences et des journaux internationaux. Des algorithmes exponentiels sont donneĢs pour reĢsoudre des probleĢmes de deĢcision, dāoptimisation et dāeĢnumeĢration. On sāinteĢresse tout dāabord au probleĢme dāeĢtiquetage L(2,1) dāun graphe, pour lequel diffeĢrents algorithmes sont deĢcrits (baseĢs sur du branchement, le paradigme diviser-pour-reĢgner, ou la programmation dynamique). Des bornes combinatoires, neĢcessaires aĢ lāanalyse de ces algorithmes, sont eĢgalement eĢtablies. Dans un second temps, nous reĢsolvons des probleĢmes autour de la domination. Nous deĢveloppons des algorithmes pour reĢsoudre une geĢneĢralisation de la domination et nous donnons des algorithmes pour eĢnumeĢrer les ensembles dominants minimaux dans des classes de graphes. Lāanalyse de ces algorithmes implique des bornes combinatoires. Finalement, nous eĢtendons notre champ dāapplications de lāalgorithmique modeĢreĢment exponentielle aĢ des probleĢmes dāordonnancement. Par le deĢveloppement dāapproches de type programmation dynamique et la geĢneĢralisation de la meĢthode trier-et-chercher, nous proposons la reĢsolution de toute une famille de probleĢmes dāordonnancement