The complexity of the Pk partition problem and related problems in bipartite graphs

In this paper, we continue the investigation made in [MT05] about the approximability of Pk partition problems, but focusing here on their complexity. Precisely, we aim at designing the frontier between polynomial and NP-complete versions of the Pk partition problem in bipartite graphs, according to both the constant k and the maximum degree of the input graph. We actually extend the obtained results to more general classes of problems, namely, the minimum k-path partition problem and the maximum Pk packing problem. Moreover, we propose some simple approximation algorithms for those problems

Approximation algorithms for the traveling salesman problem

We first prove that the minimum and maximum traveling salesman problems, their metric versions as well as some versions defined on parameterized triangle inequalities (called sharpened and relaxed metric traveling salesman) are all equi-approximable under an approximation measure, called differential-approximation ratio, that measures how the value of an approximate solution is placed in the interval between the worst- and the best-value solutions of an instance. We next show that the 2-OPT, one of the most-known traveling salesman algorithms, approximately solves all these problems within differential-approximation ratio bounded above by 1/2. We analyze the approximation behavior of 2-OPT when used to approximately solve traveling salesman problem in bipartite graphs and prove that it achieves differential-approximation ratio bounded above by 1/2 also in this case. We also prove that, it is NP-hard to differentially approximate metric traveling salesman within better than 649/650 and traveling salesman with distances 1 and 2 within better than 741/742. Finally, we study the standard approximation of the maximum sharpened and relaxed metric traveling salesman problems. These are versions of maximum metric traveling salesman defined on parameterized triangle inequalities and, to our knowledge, they have not been studied until now

Segregation of mtDNA Throughout Human Embryofetal Development: m.3243A > G as a Model System

Mitochondrial DNA (mtDNA) mutations cause a wide range of serious diseases with high transmission risk and maternal inheritance. Tissue heterogeneity of the heteroplasmy rate (“mutant load”) accounts for the wide phenotypic spectrum observed in carriers. Owing to the absence of therapy, couples at risk to transmit such disorders commonly ask for prenatal (PND) or preimplantation diagnosis (PGD). The lack of data regarding heteroplasmy distribution throughout intrauterine development, however, hampers the implementation of such procedures. We tracked the segregation of the m.3243A > G mutation (MT-TL1 gene) responsible for the MELAS syndrome in the developing embryo/fetus, using tissues and cells from eight carrier females, their 38 embryos and 12 fetuses. Mutant mtDNA segregation was found to be governed by random genetic drift, during oogenesis and somatic tissue development. The size of the bottleneck operating for m.3243A > G during oogenesis was shown to be individual-dependent. Comparison with data we achieved for the m.8993T > G mutation (MT-ATP6 gene), responsible for the NARP/Leigh syndrome, indicates that these mutations differentially influence mtDNA segregation during oogenesis, while their impact is similar in developing somatic tissues. These data have major consequences for PND and PGD procedures in mtDNA inherited disorders. Hum Mutat 32:116–125, 2011. © 2010 Wiley-Liss, Inc

Le voyageur de commerce et ses variations : un tour d'horizon de ses résolution

[extrait de l'intro] La difficulté de résolution du TSP a d'autant plus porté sur lui l'attention des chercheurs. Aussi a-t-il été étudié et retourné sous tous les angles~: théorie des graphes, programmation linéaire, programmation dynamique, optima locaux, etc. Les premières formalisations de la recherche exhaustive par la stratégie d'évaluation et séparation seraient même nées de recherches sur le voyageur de commerce. Ce chapitre retrace, certainement pas de façon exhaustive, l'histoire conjointe de la recherche opérationnelle et du voyageur de commerce. Après avoir présenté le problème, nous proposons des algorithmes, exacts puis approchés, pour différentes versions du problème~: minimisation ou maximisation, instances métriques, distances binaires, etc. Certains de ces algorithmes mettent en oeuvre des modèles généraux de résolution tels que la stratégie par séparation et évaluation, la programmation dynamique ou la recherche locale. Certains encore utilisent des heuristiques, qui ne sont autres que des idées de bon sens quant à la constitution d'une solution pour le problème étudié~: nous pensons par exemple aux heuristiques du regret et du plus proche voisin. D'autres enfin, exploitant la relative facilité de sous-problèmes du TSP, partent d'une solution de ces sous-problèmes et construisent à partir de celle-ci un cycle hamiltonien~; c'est le parti pris par l'algorithme de Christofides avec l'arbre couvrant, mais de nombreux résultats sont également obtenus par le biais d'un 2-couplage optimal

Complexity and approximation results for bounded-size paths packing problems

This chapter presents some recent works given by the authors (\cite{MT05,MT07}) about the complexity and the approximation of several problems on computing collections of (vertex)-disjoint paths of bounded size.ou

Approximation results for the weighted P4 partition problem

We present several new standard and differential approximation results for the P4partition problem using the Hassin and Rubinstein algorithm (Information Processing Letters, 63: 63-67, 1997). Those results concern both minimization and maximization versions of the problem. However, the main point of this paper lies in the establishment of the robustness of this algorithm, in the sense that it provides good quality solutions for a variety of versions of the problem, under both standard and differential approximation ratio

The path partition problem and related problems in bipartite graphs

International audienceWe prove that it is NP-complete to decide whether a bipartite graph of maximum degree three on nk vertices can be partitioned into n paths of length k. Finally, we propose some approximation and inapproximation results for several related problems

Approximation results for the weighted P4 partition problem

Abstract. We present several new standard and differential approximation results for P4-partition problem by using the algorithm proposed in Hassin and Rubinstein (Information Processing Letters, 63: 63-67, 1997), for both minimization and maximization versions of the problem. However, the main point of this paper is the robustness of this algorithm, since it provides good solutions, whatever version of the problem we deal with, whatever the approximation framework within which we estimate its approximate solutions

Pk partition problem and related problems in bipartite graphs

In this paper, we continue the investigation proposed in [15] about the approximability of P k partition problems, but focusing here on their complexity. More precisely, we prove that the problem consisting of deciding if a graph of nk vertices has n vertex disjoint simple paths {P 1, ⋯ ,P n } such that each path P i has k vertices is NP-complete, even in bipartite graphs of maximum degree 3. Note that this result also holds when each path P i is chordless in G[V(P i )]. Then, we prove that the optimization version of these problems, denoted by Max P 3 Packing and MaxInduced P 3 Packing, are not in PTAS in bipartite graphs of maximum degree 3. Finally, we propose a 3/2-approximation for Min3-PathPartition in general graphs within O(nm + n 2logn) time and a 1/3 (resp., 1/2)-approximation for MaxW P 3 Packing in general (resp., bipartite) graphs of maximum degree 3 within O(α(n,3n/2)n) (resp., O(n 2logn)) time, where α is the inverse Ackerman’s function and n = |V|, m = |E|.ou