NP-difficile

Résumé

vignette|300px|Mise en évidence d'un problème NP-difficile si Problème P ≟ NP. Un problème NP-difficile est, en théorie de la complexité, un problème appartenant à la classe NP-difficile, ce qui revient à dire qu'il est au moins aussi difficile que les problèmes les plus difficiles de la classe NP. Ainsi, un problème H est NP-difficile, si tout problème L de la classe NP peut être réduit en temps polynomial à H. Si un problème NP-difficile est dans NP, alors c'est un problème NP-complet. Donc tous les problèmes NP-complets sont NP-difficiles. Références Catégorie:Classe de complexité

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https://fr.wikipedia.org/wiki/NP-difficile

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The weighted stable matching problem

Linda Farczadi

We study the stable matching problem in non-bipartite graphs with incomplete but strict preference lists, where the edges have weights and the goal is to compute a stable matching of minimum or maximum weight. This problem is known to be NP-hard in general. Our contribution is two fold: a polyhedral characterization and an approximation algorithm. Previously Chen et al. have shown that the stable matching polytope is integral if and only if the subgraph obtained after running phase one of Irving's algorithm is bipartite. We improve upon this result by showing that there are instances where this subgraph might not be bipartite but one can further eliminate some edges and arrive at a bipartite subgraph. Our elimination procedure ensures that the set of stable matchings remains the same, and thus the stable matching polytope of the final subgraph contains the incidence vectors of all stable matchings of our original graph. This allows us to characterize a larger class of instances for which the weighted stable matching problem is polynomial-time solvable. We also show that our edge elimination procedure is best possible, meaning that if the subgraph we arrive at is not bipartite, then there is no bipartite subgraph that has the same set of stable matchings as the original graph. We complement these results with a 2-approximation algorithm for the minimum weight stable matching problem for instances where each agent has at most two possible partners in any stable matching. This is the first approximation result for any class of instances with general weights.

2017

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In this thesis we consider two optimal routing problems in networks. Both are NP-hard and are often used to modelize real life problems of large size. We first present the capacitated arc routing problem (CARP). In this problem, we have to serve a set of customers represented by the arcs of a network with vehicles of restricted capacity. The aim is to design a set of vehicle routes of least total cost. We describe two heuristic methods for the CARP based on local search techniques. The first one is a Tabu Search heuristic using an original post-optimization procedure. Principles appearing in this procedure are used to define a class of neighborhoods we combine in a Variable Neighborhood Descent method. Our algorithms compare favorably with some of her methods. They were initially developed to deal with the undirected CARP. We describe how to transform them so that they can be applied to the directed CARP. We also propose an adaptation to the directed case of a lower bound originally developed for the undirected CARP. This lower bound is used to evaluate the quality of the solutions produced by our algorithms for the directed CARP. Our methods obtain solutions whose value is, on average, close to known lower bounds. We use the CARP to tackle a real life problem of boat routing and scheduling. This problem can be viewed as a directed CARP with time windows. We describe in detail its modeling process and we analyze our first results. We obtain solutions that require less boats than the present schedule. Nevertheless, these solutions are not completely feasible in practice. We suggest some ways to improve them. The second problem treated in this thesis is the shortest capacitated paths problem (SCPP). We present a Tabu Search heuristic and some post-optimization procedures for the SCPP. These procedures are called in our Tabu Search algorithm. Two versions of this algorithm are considered. In the first one, post-optimization procedures are intensively used. In the second one, they are less frequently applied. This second version is faster. Compared with a greedy approach, the two versions of our Tabu Search heuristic obtain solutions which are, on average, of better quality. The SCPP was proposed to us by EDF (Electricité de France) in order to minimize the total length of the paths followed by cables in a nuclear power-station. It can also be viewed as a subproblem of the bandwidth packing problem (BWP) appearing in telecommunications.

EPFL2000

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Correlation-aware Resource Allocation in Multi-Cell Networks

Dorna Bandari, Pascal Frossard

We propose a cross-layer strategy for resource allocation between spatially correlated sources in the uplink of multi-cell FDMA networks. Our objective is to find the optimum power and channel allocation to the different sources, in order to minimize the maximum distortion achieved in decoding any source data in the network. This problem is NP-hard and finding the optimal solution is not computationally feasible. We propose a three-step algorithm to be performed separately in each cell, which finds cross-layer resource allocation in simple steps. This method separates the problem into inter-cell resource management, grouping of sources for joint decoding, and intra-cell channel assignment. For each of these steps we propose methods that satisfy different design constraints and analyze them by simulations. We show that, while using correlation in compression and joint decoding can achieve 25% distortion reduction over independent decoding, the improvement grows to 37% when correlation is also utilized in resource allocation. This significant distortion reduction motivates further work in correlation-aware resource allocation. Overall, our solution is able to achieve a 60% decrease in 5 percentile distortion compared to independent allocation methods.

Institute of Electrical and Electronics Engineers2012

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EPFL2000