164 research outputs found
Algebra and the Complexity of Digraph CSPs: a Survey
No full textWe present a brief survey of some of the key results on the interplay between algebraic and graph-theoretic methods in the study of the complexity of digraph-based constraint satisfaction problems
Stable sets of maximal size in Kneser-type graphs
No full textWe introduce a family of vertex-transitive graphs with specified subgroups of automorphisms which generalise Kneser graphs, powers of complete graphs and Cayley graphs of permutations. We compute the stability ratio for a wide class of these. Under certain conditions we characterise their stable sets of maximal size. (C) 2003 Elsevier Ltd. All rights reserved
Rigid and strongly rigid relations on small domains
No full textLes relations fortement rigides jouent un rôle important dans l’étude de la complexité
des problèmes de satisfaction de contraintes (CSPs) (Feder et Vardi [22], Schaefer [9],
Jeavons [23], Bulatov, Jeavons et Krokhin [24], Larose et Tesson [34], Larose [31],
Barto et Kozik [36], et Bulatov, Jeavons et Krokhin [27]) qui font l’objet de recherches
intenses à la fois en intelligence artificielle et en recherche opérationnelle (Russell et
Norvig [19]). Une relation n-aire r sur un ensemble U est rigide si elle n’admet aucun
automorphisme non-trivial ; elle est fortement rigide si elle n’est préservée que par les
projections. De plus r est dite projective si les seules opérations idempotentes qui la
préservent sont les projections.
Rosenberg (1973) a caracterisé toutes les relations fortement rigides sur un ensemble
à deux éléments, et a construit une relation binaire fortement rigide sur tout ensemble de
plus de deux éléments. Larose et Tardif (2001) ont étudié les graphes projectifs et fortement
rigides, et ont construit de grandes familles de graphes fortement rigides. Łuczak
et Nešetˇril (2004) ont démontré une conjecture de Larose and Tardif qui prévoyait que
la plupart des graphes avec suffisamment de sommets sont projectifs, et ont caractérisé
tous les graphes homogènes qui sont projectifs. Łuczak et Nešetˇril (2006) ont ensuite
confirmé une conjecture de Rosenberg qui prédisait que la plupart des relations sur un
ensemble suffisamment grand sont fortement rigides.
Le premier résultat principal de cette thése est une caractérisation des relations fortement
rigides sur un ensemble d’au moins 3 éléments, résolvant ainsi un problème ouvert
de Rosenberg (Rosenberg [7], Problème 6 de [13]). Ensuite nous montrons qu’à isomorphisme
près, il n’existe que 4 relations binaires rigides sur un ensemble à trois éléments,
parmi lesquelles deux seulement sont fortement rigides. De plus, nous déterminons, à
isomorphisme près, les 40 relations binaires rigides sur un univers à quatre éléments, et
montrons que 25 d’entre elles sont fortement rigides (Exemple 5.4 et Exemple 6.1 dans
Sun [41]). Nous généralisons une de ces relations pour construire une nouvelle relation
binaire fortement rigide sur tout ensemble d’au moins 4 éléments (Sun [43]), et décrivons
de plus une relation ternaire fortement rigide sur tout ensemble fini avec au moins 2 éléments
et conjecturons une relation k-aire fortement rigide sur tout domaine fini (Sun [42]).Strongly rigid relations play an important role in the study of the complexity of
Constraint Satisfaction Problems (CSPs) (Feder and Vardi [22], Schaefer [9], Jeavons
[23], Bulatov, Jeavons and Krokhin [24], Larose and Tesson [34], Larose [31], Barto
and Kozik [36], and Bulatov, Jeavons and Krokhin [27]) which are the subject of intense
research in both artificial intelligence and operations research (Russell and Norvig [19]).
An n-ary relation r on a set U is strongly rigid if it is preserved only by trivial operations.
It is projective if the only idempotent operations in Polr are projections.
Rosenberg (1973) characterized all strongly rigid relations on a set with two elements
and found a strongly rigid binary relation on every domain U of at least 3 elements.
Larose and Tardif (2001) studied the projective and strongly rigid graphs, and
constructed large families of strongly rigid graphs. Łuczak and Nešetˇril (2004) settled
in the affirmative a conjecture of Larose and Tardif that most graphs on a large set are
projective, and characterized all homogenous graphs that are projective. Łuczak and
Nešetˇril (2006) confirmed a conjecture of Rosenberg that most relations on a big set are
strongly rigid.
In this thesis we characterize all strongly rigid relations on a set with at least three
elements to answer an open question by Rosenberg (1973) (Rosenberg [7], Problem 6 in
Rosenberg [13]). We classify the binary relations on the 3-element domain and demonstrate
that there are merely 4 pairwise nonisomorphic rigid binary relations on the same
domain (among them 2 are pairwise nonisomorphic strongly rigid), and we classify the
binary relations on the 4-element domain and show that there are merely 40 pairwise
nonisomorphic rigid binary relations on the same domain (among them 25 are pairwise
nonisomorphic strongly rigid) (Example 5.4 and Example 6.1 in Sun [41]). We extend
a strongly rigid relation on a 4-element domain to any finite domain (Sun [43]). Finally,
we give a strongly rigid ternary relation on any finite domain and conjecture a strongly
rigid k-ary relation on any finite domain (Sun [42])
Some results on two conjectures of Schützenberger
No full textWe present some partial results concerning two conjectures of Schutzenberger on evacuations of Young tableaux
Retractions onto series-parallel posets
No full textThe poset retraction problem for a poset P is whether a given poset Q containing P as a subposet admits a retraction onto P, that is, whether there is a homomorphism from Q onto P which fixes every element of P. We study this problem for finite series-parallel posets . We present equivalent combinatorial, algebraic, and topological charaterisations of posets for which the problem is tractable, and, for such a poset P , we describe posets admitting a retraction onto P.Financial help of NSERC Canada is gratefully acknowledged. Víctor Dalmau is partially supported by the MEC under the program “Ramon y Cajal”, Grants TIN 2006-15387-C03-03, TIN 2004-04343, the EU PASCAL Network of Excellence IST-2002-506778, and the MODNET Marie Curie Research Training Network MRTN-CT-2004-512234. Andrei Krokhin is partially supported by the UK EPSRC Grants EP/C543831/1 and EP/C54384X/1. Benoit Larose is partially supported by NSERC, FQRNT, and CRM
Retractions onto series-parallel posets
No full textThe poset retraction problem for a poset P is whether a given poset Q containing P as a subposet admits a retraction onto P, that is, whether there is a homomorphism from Q onto P which fixes every element of P. We study this problem for finite series-parallel posets . We present equivalent combinatorial, algebraic, and topological charaterisations of posets for which the problem is tractable, and, for such a poset P , we describe posets admitting a retraction onto P.Financial help of NSERC Canada is gratefully acknowledged. Víctor Dalmau is partially supported by the MEC under the program “Ramon y Cajal”, Grants TIN 2006-15387-C03-03, TIN 2004-04343, the EU PASCAL Network of Excellence IST-2002-506778, and the MODNET Marie Curie Research Training Network MRTN-CT-2004-512234. Andrei Krokhin is partially supported by the UK EPSRC Grants EP/C543831/1 and EP/C54384X/1. Benoit Larose is partially supported by NSERC, FQRNT, and CRM
Surjective H-Colouring over Reflexive Digraphs
No full textThe Surjective H-Colouring problem is to test if a given graph allows a vertex-surjective homomorphism to a fixed graph H. The complexity of this problem has been well studied for undirected (partially) reflexive graphs. We introduce endo-triviality, the property of a structure that all of its endomorphisms that do not have range of size 1 are automorphisms, as a means to obtain complexity-theoretic classifications of Surjective H-Colouring in the case of reflexive digraphs.
Chen [2014] proved, in the setting of constraint satisfaction problems, that Surjective H-Colouring is NP-complete if H has the property that all of its polymorphisms are essentially unary. We give the first concrete application of his result by showing that every endo-trivial reflexive digraph H has this property. We then use the concept of endo-triviality to prove, as our main result, a dichotomy for Surjective H-Colouring when H is a reflexive tournament: if H is transitive, then Surjective H-Colouring is in NL, otherwise it is NP-complete.
By combining this result with some known and new results we obtain a complexity classification for Surjective H-Colouring when H is a partially reflexive digraph of size at most 3
Maximum constraint satisfaction on diamonds
No full textIn this paper we study the complexity of the (weighted) maximum constr aint satisfaction problem (Max CSP) over an arbitrary finite domain. In this pro blem, one is given a collection of weighted constraints on overlapping sets of v ariables, and the goal is to find an assignment of values to the variables so as to maximize the total weight of satisfied constraints. Max Cut is a typical exa mple of a Max CSP problem. Max CSP is NP-hard in general; however, some restrict ions on the form of constraints may ensure tractability. Recent results indicate that there is a connection between tractability of such restricted problems and supermodularity of the allowed constraint types with respect to some lattice or dering of the domain. We prove several results confirming this. Diamonds are the smallest lattices in terms of the number of comparabilities, and so are as unor dered as a lattice can possibly be. In the present paper, we study Max CSP on di amond-ordered domains. We show that if all allowed constraints are supermodular with respect to such an ordering then the problem can be solved in polynomial (i n fact, in cubic) time. We also prove a partial converse: if the set of allowed constraints includes a certain small family of binary supermodular constraints on such a lattice, then the problem is tractable if and only if all of the allowed constraints are supermodular; otherwise, it is NP-hard
Complexité des homomorphismes de graphes avec listes
No full textLes problèmes de satisfaction de contraintes, qui consistent à attribuer des valeurs à des variables en respectant un ensemble de contraintes, constituent une large classe de problèmes naturels. Pour étudier la complexité de ces problèmes, il est commode de les voir comme des problèmes d'homomorphismes vers des structures relationnelles. Un axe de recherche actuel est la caractérisation des classes de complexité auxquelles appartient le problème d'homomorphisme, ceci dans la perspective de confirmer des conjectures reliant les propriétés algébriques des structures relationelles à la complexité du problème d'homomorphisme.
Cette thèse propose dans un premier temps la caractérisation des digraphes pour lesquels le problème d'homomorphisme avec listes appartient à FO. On montre également que dans le cas du problèmes d'homomorphisme avec listes sur les digraphes télescopiques, les conjectures reliant algèbre et complexité sont confirmées.
Dans un deuxième temps, on caractérise les graphes pour lesquels le problème d'homomorphisme avec listes est résoluble par cohérence d'arc. On introduit la notion de polymorphisme monochromatique et on propose un algorithme simple qui résoud le problème d'homomorphisme avec listes si le graphe cible admet un polymorphisme monochromatique TSI d'arité k pour tout k ≥ 2.Constraint satisfaction problems, consisting in assigning values to variables while respecting a set of constraints, form a large class of natural problems. In order to study the complexity of these problems, it is convenient to see them as homomorphism problems on relational structures. One current research topic is to characterise complexity classes where the homomorphism problem belongs. The ultimate goal is to confirm conjectures that bind together algebraic properties of the relationnal structure and complexity of the homomorphism problem.
At first, the thesis characterizes digraphs which generate FO list-homomorphism problems. It is shown that in the particular case of telescopic digraphs, conjectures binding together algebra and complexity are confirmed.
Subsequently, we characterize graphs which generate arc-consistency solvable list-homomorphism problems. We introduce the notion of monochromatic polymorphism and we propose a simple algorithm which solves the list-homomorphism problem if the target graph admits a monochromatic TSI polymorphism of arity k for every k ≥ 2
Impact d'interventions non pharmacologiques sur les biomarqueurs émergents du risque cardiovasculaire : fonctionnalité des HDL et PCSK9
No full textLes maladies cardiovasculaires (MCV) représentent la première cause de mortalité au monde, malgré des progrès significatifs dans la prise en charge des facteurs de risque traditionnels. De nouveaux biomarqueurs émergent, par exemple la capacité d’efflux du cholestérol (CEC) des HDL qui serait associée aux MCV indépendamment des concentrations de cholestérol HDL. Un autre biomarqueur est la proprotéine convertase subtilisine/kexine de type 9 (PCSK9), qui augmente les niveaux de cholestérol LDL en dégradant le récepteur aux LDL. Les facteurs qui influencent la CEC des HDL et la quantité de PCSK9 sont peu connus. L’objectif général de cette thèse est de déterminer si des interventions non pharmacologiques telles que la modification des habitudes de vie et la chirurgie bariatrique peuvent influencer ces biomarqueurs. Afin de répondre à cet objectif, nous avons étudié trois populations à risque : (1) 86 patients avec maladie coronarienne, (2) 117 hommes avec obésité abdominale et dyslipidémie, et (3) 69 hommes et femmes atteints d’obésité sévère. Les deux premières cohortes ont suivi une intervention de modification des habitudes de vie pendant 1 an visant à instaurer un minimum de 150 minutes d’activité physique par semaine et à améliorer la qualité nutritionnelle de leur diète. La cohorte de patients avec obésité sévère a subi une chirurgie bariatrique de type dérivation biliopancréatique avec commutation duodénale. Les La CEC des HDL a été mesurée à l’aide d’essais cellulaires dans les deux premières cohortes, alors que le niveau de PCSK9 a été mesuré par ELISA dans les trois cohortes. Dans tous les cas, les mesures ont été effectuées au début et à la fin des interventions. Nous avons montré que chez les patients coronariens et les hommes avec obésité abdominale, l’intervention de modification des habitudes de vie a augmenté la CEC des HDL, mais a eu un effet mineur sur les niveaux de PCSK9. En revanche, chez les hommes avec obésité abdominale les niveaux de PCSK9 sont diminués par un repas riche en gras. Les améliorations de la CEC des HDL à la suite de l’intervention de modification des habitudes de vie étaient principalement expliquées par l’augmentation des concentrations de l’apolipoprotéine A1 et du cholestérol HDL. La chirurgie bariatrique a augmenté la concentration de PCSK9 en phase aiguë, mais l’a diminué à long terme. Ces résultats suggèrent que le niveau de PCSK9 seraient modifiés par des changements drastiques comme une importante perte de poids à la suite d’une chirurgie bariatrique. En conclusion, les interventions non pharmacologiques semblent moduler positivement la CEC des HDL et modestement la PCSK9 dans des populations à haut risque cardiovasculaire, ce qui pourrait en partie expliquer l’impact bénéfique de ces interventions sur le risque cardiovasculaire.Cardiovascular disease (CVD) is the leading cause of death in the world, despite significant progress in the management of traditional CVD risk factors. New biomarkers are emerging, such as the HDL cholesterol efflux capacities (HDL-CEC) that are associated with CVD, independently of HDL cholesterol levels. Another biomarker is proprotein convertase subtilisin/kexin type 9 (PCSK9), which increases low-density lipoprotein (LDL) cholesterol levels by degrading the LDL receptor. Little is known about the factors that influence HDLCEC and PCSK9 physiological variation. The overall objective of this thesis is to determine whether improvements in lifestyle and bariatric surgery can influence these biomarkers. To achieve this objective, we have studied three populations at risk: 86 patients with coronary artery disease, 117 men with abdominal obesity and dyslipidemic and 69 men and women with severe obesity. The first two cohorts followed a 1-year lifestyle modification program aimed at achieving a minimum of 150 minutes of aerobic physical activity weekly and improving diet quality. We measured HDL-CEC using cell assays in both populations before and after the interventions. The cohort of patients with severe obesity underwent bariatric surgery of biliopancreatic diversion with duodenal switch. PCSK9 levels were measured by ELISA in all cohorts, at the beginning and at the end of the interventions. Our results showed that in coronary patients and men with abdominal obesity, the interventions led to minor changes in PCSK9 levels. In contrast, in men with abdominal obesity PCSK9 levels were acutely decreased by a high-fat meal. Bariatric surgery increased PCSK9 concentration in the acute phase, but decreased in the long term. These results suggest that PCSK9 levels could be modified following drastic interventions such as significant weight loss following bariatric surgery. Lifestyle modification program induced significant increases of HDL-CEC in coronary patients and men with abdominal obesity. In addition, improvements in HDLCEC following lifestyle modification interventions were mainly explained by increases in apolipoprotein A1 and HDL cholesterol levels. In conclusion, non-pharmacological interventions appear to positively modulate HDL-CEC and modestly influence PCSK9 in populations at high cardiovascular risk, which could partly explain the beneficial impact of these interventions on cardiovascular risk
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