Benoit Larose

University of Quebec in Montreal | UQAM · Lacim

Call a finite relational structure $k$-Slupecki if its only surjective $k$-ary polymorphisms are essentially unary, and Slupecki if it is $k$-Slupecki for all $k \geq 2$. We present conditions, some necessary and some sufficient, for a reflexive digraph to be Slupecki. We prove that all digraphs that triangulate a 1-sphere are Slupecki, as are all...

A reflexive cycle is any reflexive digraph whose underlying undirected graph is a cycle. Call a relational structure Słupecki if its surjective polymorphisms are all essentially unary. We prove that all reflexive cycles of girth at least 4 have this property.

We give a complexity dichotomy for the Quantified Constraint Satisfaction Problem \( \mathrm{QCSP}(\mathrm{H}) \) when \( \mathrm{H} \) is a reflexive tournament. It is well known that reflexive tournaments can be split into a sequence of strongly connected components \( \mathrm{H}_1,\ldots ,\mathrm{H}_n \) so that there exists an edge from every v...

A reflexive cycle is any reflexive digraph whose underlying undirected graph is a cycle. Call a relational structure Slupecki if its surjective polymorphisms are all essentially unary. We prove that all reflexive cycles of girth at least 4 have this property.

We give a complexity dichotomy for the Quantified Constraint Satisfaction Problem QCSP(H) when H is a reflexive tournament. It is well-known that reflexive tournaments can be split into a sequence of strongly connected components H_1,...,H_n so that there exists an edge from every vertex of H_i to every vertex of H_j if and only if i<j. We prove th...

The Constraint Satisfaction Problem (CSP) and its counting counterpart appears under different guises in many areas of mathematics, computer science, and elsewhere. Its structural and algorithmic properties have been shown to play a crucial role in many of those applications. For instance, in the decision CSPs, structural properties of the relation...

In this paper we study alternative characterizations of dismantlability properties of relational structures in terms of various connectedness and mixing notions. We relate these results with earlier work of Brightwell and Winkler, providing a generalization from the graph case to the general relational structure context. In addition, we develop pro...

We find a set of generators of the variety of reflexive digraphs admitting k − NU polymorphisms. We do this, in spite of the fact that such digraphs do not have finite tree duality, by defining finite duals of infinite trees. As a result of this, we answer a question of Quackenbush, Rival, and Rosenberg, giving a finite family of generators of the...

The 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 auto...

The 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 auto...

The constraint satisfaction problem (CSP) involves deciding, given a set of variables and a set of constraints on the variables, whether or not there is an assignment to the variables satisfying all of the constraints. One formulation of the CSP is as the problem of deciding, given a pair (G,H) of relational structures, whether or not there is a ho...

The constraint satisfaction problem (CSP) involves deciding, given a set of variables and a set of constraints on the variables, whether or not there is an assignment to the variables satisfying all of the constraints. One formulation of the CSP is as the problem of deciding, given a pair (G,H) of relational structures, whether or not there is a ho...

We describe a generating set for the variety of simple graphs that admit a k-ary near-unanimity (NU) polymorphism. The result follows from an analysis of NU polymorphisms of strongly bipartite digraphs, i. e., whose vertices are either a source or a sink. We show that the retraction problem for a strongly bipartite digraph H has finite duality if a...

The Dichotomy Conjecture for constraint satisfaction problems (CSPs) states that every CSP is in P or is NP-complete (Feder-Vardi, 1993). It has been verified for conservative problems (also known as list homomorphism problems) by Bulatov (2003). We augment this result by showing that for digraph templates H, every conservative CSP, denoted LHOM(H)...

The Dichotomy Conjecture for constraint satisfaction problems (CSPs) states that every CSP is in P or is NP-complete (Feder-Vardi, 1993). It has been verified for conservative problems (also known as list homomorphism problems) by A. Bulatov (2003). We augment this result by showing that for digraph templates H, every conservative CSP, denoted LHOM...

We describe a generating set for the variety of reflexive graphs that admit a compatible k-ary near-unanimity (NU) operation. We further delineate a very simple subset that generates the variety of j-absolute retracts; in particular we show that the class of reflexive graphs with a 4-NU operation coincides with the class of 3-absolute retracts. Our...

We characterise the graphs (which may contain loops) whose list-homomorphism problem is solvable by arc consistency, or equivalently, that admit conservative totally symmetric idempotent operations of all arities. We prove that for every bipartite graph G, its list-homomorphism problem is tractable if and only if G admits a monochromatic conservati...

We characterise the graphs (which may contain loops) whose list-homomorphism problem is solvable by arc consistency, or equivalently, that admit conservative totally symmetric idempotent operations of all arities. We prove that for every bipartite graph , its list-homomorphism problem is tractable if and only if admits a monochromatic conservative...

We present some results concerning the projection property for finite ordered sets. We show that sums of non-trivial ramified ordered sets over a connected poset of at least two elements are projective. We construct a family of minimal automorphic posets of reach 2 and length 2 and show they are projective.

We completely classify the computational complexity of the list H-colouring problem for graphs (with possible loops) in combinatorial and algebraic terms: for every graph H, the problem is either NP-complete, NL-complete, L-complete or is first-order definable; descriptive complexity equivalents are given as well via Datalog and its fragments. Our...

We completely classify the computational complexity of the list H-colouring problem for graphs (with possible loops) in combinatorial and algebraic terms: for every graph H the problem is either NP-complete, NL-complete, L-complete or is first-order definable; descriptive complexity equivalents are given as well via Datalog and its fragments. Our a...

We say that a finite algebra A = hA; Fi has the ability to count if there are subalgebras C of A3 and Z of A such that the structure hA; C, Zi has the ability to count in the sense of Feder and Vardi. We show that for a core relational structure A the following conditions are equivalent: (i) the variety generated by the algebra A associated to A co...

We present algebraic conditions on constraint languages Γ that ensure the hardness of the constraint satisfaction problem for complexity classes L, NL, P, NP and ModpL. These criteria also give non-expressibility results for various restrictions of Datalog. Furthermore, we show that if is not first-order definable then it is L-hard. Our proofs rely...

We prove an analog of results by Erdős-Ko-Rado [P. Erdős, C. Ko, and R. Rao, “Intersection theorems for systems of finite sets,” Q. J. Math., Oxf. II. Ser. 12, 313–320 (1961; Zbl 0100.01902)] and Greenwell-Lovász [D. Greenwell and L. Lovász, “Applications of product colouring,” Acta Math. Acad. Sci. Hung. 25, 335–340 (1974; Zbl 0294.05108)] by char...

Let B be a finite, core relational structure and let A be the algebra associated to B, i.e. whose terms are the operations on the universe of B that preserve the relations of B. We show that if A generates a so-called arithmetical variety then CSP(B), the constraint satisfaction problem associated to B, is solvable in Logspace; in fact notCSP(B) is...

We show that the directed st-connectivity problem cannot be expressed in symmetric Datalog, a fragment of Datalog introduced in [5]. It was shown there that symmetric,Datalog programs can be eval- uated in logarithmic space and that this fragment of Datalog captures logspace when augmented with negation, and an auxiliary successor re- lation S toge...

The 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 P. We present equivalent combinatorial, algebraic, and topological charaterisation...

A sublattice in a lattice is called supermodular if, for every two elements, one of which belongs to the sublattice, at least one of their meet and join also belongs to the sublattice. In this note, we describe supermodular sublattices in products of relatively complemented lattices.

Recently, a strong link has been discovered between supermodularity on lattices and tractability of optimization problems known as maximum constraint satisfaction problems. This paper strengthens this link. We study the problem of maximizing a supermodular function which is defined on a product of $n$ copies of a fixed finite lattice and given by a...

In a nutshell, a duality for a constraint satisfaction problem equates the existence of one homomorphism to the non-existence of other homomorphisms. In this survey paper, we give an overview of logical, combinatorial, and algebraic aspects of the following forms of duality for constraint satisfaction problems: finite duality, bounded pathwidth dua...

The Annual European Meeting of the Association for Symbolic Logic, generally known as the Logic Colloquium, is the most prestigious annual meeting in the field. Many of the papers presented there are invited surveys of recent developments, and the rest of the papers are chosen to complement the invited talks. This volume includes surveys, tutorials...

We describe simple algebraic and combinatorial characterisations of finite relational core structures admitting finitely many obstructions. As a consequence, we show that it is decidable to determine whether a constraint satisfaction problem is first-order definable: we show the general problem to be NP-complete, and give a polynomial-time algorith...

We describe simple algebraic and combinatorial characterisations of finite relational core structures admitting finitely many obstructions. As a consequence, we show that it is decidable to determine whether a constraint satisfaction problem is first-order definable: we show the general problem to be NP-complete, and give a polynomial-time algorith...

Let \({\mathcal A}\) be finite relational structure of finite type, and let CSP \({(\mathcal A)}\) denote the following decision problem: if \({\mathcal I}\) is a given structure of the same type as \({\mathcal A}\) , is there a homomorphism from \({\mathcal I}\) to \({\mathcal A}\)? To each relational structure \({\mathcal A}\) is associated natur...

We introduce symmetric Datalog, a syntactic restriction of linear Datalog and show that its expressive power is exactly that of restricted symmetric Krom monotone SNP. The deep result of Reingold [17] on the complexity of undirected connectivity suffices to show that symmetric Datalog queries can be evaluated in logarithmic space. We show that for...

We study the complexity of counting the number of solutions to a system of equations over a fixed finite semigroup. We show that this problem is always either in FP or #P-complete and describe the borderline precisely. We use these results to convey some intuition about the conjectured dichotomy for the complexity of counting the number of solution...

We study the algorithmic complexity of determining whether a system of polynomial equations over a finite algebra admits a solution. We characterize, within various families of algebras, which of them give rise to an NP-complete problem and which yield a problem solvable in polynomial time. In particular, we prove a dichotomy result which encompass...

Communicated by R. McKenzie We study the algorithmic complexity of determining whether a system of polynomial equations over a finite algebra admits a solution. We characterize, within various families of algebras, which of them give rise to an NP-complete problem and which yield a problem solvable in polynomial time. In particular, we prove a dich...

In [25], a discrete homotopy theory for reflexive digraphs was de-veloped. In the present paper, we prove that if a finite, connected reflexive digraph X has non-trivial homotopy in some dimension but none of its proper retracts does, then the digraph is idempotent trivial. This is used to prove that if a structure admits an operation satisfying sp...

In this paper we study the complexity of the weighted maximum constraint satisfaction problem (Max CSP) over an arbitrary finite domain. In this problem, one is given a collection of weighted constraints on overlapping sets of variables, and the goal is to find an assignment of values to the variables so as to maximize the total weight of satisfied...

We present a simple polynomial-time algorithm that recognises reflexive, symmetric graphs admitting a near-unanimity operation. Several other characterisations of these graphs are also presented.

We prove that if a finite connected poset admits an order-preserving Taylor operation, then all of its homotopy groups are trivial. We use this to give new characterisations of locally finite varieties omitting type 1 in terms of the posets (or equivalently, finite topological spaces) in the variety. Similar variants of other omitting-type theorems...

We prove an analog of a lemma by Mal’tsev and deduce the following analog of a result of Rosenberg [11]: let Q be a finite poset with n elements, let k denote the k-element chain, and let h be an integer such that 2 h , assigns more than h colours to a homomorphic image of the poset Q, then there is such an image that lies in a subgrid G1 ◊ ... ◊ G...

We present a simple combinatorial construction of a sequence of functors σk from the category of pointed binary reflexive structures to the category of groups. We prove that if the relational structure is a poset P then the groups are (naturally) isomorphic to the homotopy groups of P when viewed as a topological space with the topology of ideals,...

A retraction from a structure P to its substructure Q is a homomorphism from P onto Q that is the identity on Q. We present an algebraic condition which completely characterises all posets and all reflexive graphs Q with the following property: the class of all posets or reflexive graphs, respectively, that admit a retraction onto Q is first-order...

We 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.

Abstract A retraction from a structure P to its substructure Q is a homomorphism,from P onto Q that is the identity on Q. We present an algebraic condition which completely characterises all posets and all reflexive graphs Q such that the class of all posets or reflexive graphs, respectively, that admit a retraction onto Q is first-order definable....

For a finite poset P let EXT(P ) denote the following decision problem. Given a finite poset Q and a partial map f from Q to P , decide whether f extends to a monotone total map from Q to P. It is easy to see that EXT(P ) is in the complexity class NP .I n (SIAM J. Comput., 28 (1998), pp. 57-104), Feder and Vardi define the classes of width 1 and o...

. We prove that the order-primal algebra of a non-trivial finite connected poset P generates a minimal variety if and only if P is dismantlable.

We combine methods of order theory, finite model theory, and universal algebra to study, within the constraint satisfaction framework, the complexity of some well-known combinatorial problems connected with a finite poset. We identify some conditions on a poset which guarantee solvability of the problems in (deterministic, symmetric, or non-determ...

We introduce the notion of strongly projective graph, and characterise these graphs in terms of their neighbourhood poset. We describe certain exponential graphs associated to complete graphs and odd cycles. We extend and generalise a result of Greenwell and Lovász [6]: if a connected graph $G$ does not admit a homomorphism to $K$ , where $K$ is an...

We investigate the relationship between projectivity and the structure of maximal independent sets in powers of circular graphs, Kneser graphs and truncated simplices. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 162–171, 2002

Let k denote a k-element chain, k3. Let M denote the clone generated by all unary isotone operations on k and let Pol denote the clone of all isotone operations on k. We investigate the interval of clones [MPol]. Among other results, we describe completely those clones which contain only join (or meet) homomorphisms, and describe the interval compl...

For our purposes, a graph is a relational structure 〈V,E〉 where E is an irreflexive symmetric binary relation on V. A graph is strongly rigid if the clone PolE is the clone of projections. It is projective if the only idempotent operations in PolE are projections. We show that a graph is projective if and only if the idempotent binary operations in...

We show that every core graph with a primitive automorphism group has the property that whenever it is a retract of a product of connected graphs, it is a retract of a factor. The example of Kneser graphs shows that the hypothesis that the factors are connected is essential. In the case of complete graphs, our result has already been shown in [4, 1...

Let A be a finite set, |A|≥2, let C be a clone on A and r a unary member of C satisfying r 2 =r. We investigate the relationship between the lattice of clones containing C and the lattice of clones on R=r(A). We obtain a criterion to determine if the function r is invertible in C by inspecting the relations preserved by all members of C. We give co...

Given a finite partially ordered set P, for subsets or, in other words coalitions X, Y of P let X Y mean that there exists an injection : X Y such that x (x) for all x X. The set L(P) of all subsets of P equipped with this relation is a partially ordered set. When L(P) is a lattice, it is called the coalition lattice of P. It is shown that P is det...

A graph G is said to be hom-idempotent if there is a homomorphism from G2 to G, and weakly hom-idempotent if for some n � 1 there is a homomorphism from Gn+1 to Gn. We characterise both classes of graphs in terms of a special class of Cayley graphs called normal Cayley graphs. This allows us to construct, for any integer n, a Cayley graph G such th...

We show that every finite connected poset which admits certain operations such as Gumm or Jónsson operations, or a near unanimity function is dismantlable. This result is used to prove that a finite poset admits Gumm operations if and only if it admits a near unanimity function. Finite connected posets satisfying these equivalent conditions are cha...

Let A, be a finite lattice-ordered set, and letC be the centralizer of the join (meet) operation of this lattice, i.e., the clone of all operations onA that commute with the join (meet) operation. We show thatC is a strictly meet irreducible element of the lattice of clones onA and is covered by the clone of isotone operations for this lattice-orde...

We show that quasiprojectivity and projectivity are equivalent properties for ordered sets of more than two elements.

We show that quasiprojectivity and projectivity are equivalent properties for finite ordered sets of more than two elements.

We present algebraic conditions on constraint languages Γ that ensure the hardness of the constraint satisfaction problem CSP (Γ) for complexity classes L, NL, P, NP and Mod p L. These criteria also give non-expressibility results for various restrictions of Datalog. Furthermore, we show that if CSP(Γ) is not first-order definable then it is L-hard...

We investigate the complexity of strong colouring problems of hy-pergraphs associated to groups of permutations initiated in [14]. By reformu-lating these as Constraint Satisfaction Problems (CSP's) we are able to exploit recent algebraic results to answer various questions posed by Haddad and Rödl [14]. In particular, we show that all known tracta...

For the moment, we outline how to characterise undirected, ir-reflexive graphs that admit a near-unanimity operation of arity at most k using the notion of dual of a tree; the same is done for reflexive graphs, and for directed bipartite graphs. We hope to complete this by showing that the generating sets for the varieties in question contain only...

Thèse (M. Sc.)--Université de Montréal, 1990. "Mémoire présenté à la Faculté des études supérieures en vue de l'obtention du grade de Maître ès sciences (M. Sc.) en mathématiques."

Thèse (Ph. D.)--Université de Montréal, 1993. Comprend des réf. bibliogr. Comprend du texte anglais.