On the Interval of Boolean Strong Partial Clones Containing Only Projections as Total Operations

International audienceA strong partial clone is a set of partial operations closed under composition and containing all partial projections. Let X be the set of all Boolean strong partial clones whose total operations are the projections. This set is of practical interest since it induces a partial order on the complexity of NP-complete constraint satisfaction problems. In this paper we study X from the algebraic point of view, and prove that there exists two intervals in X , corresponding to natural constraint satisfaction problems, such that one is at least countably infinite and the other has the cardinality of the continuum

The power of primitive positive definitions with polynomially many variables

Two well-studied closure operators for relations are based on existentially quantified conjunctive formulas, primitive positive (p.p.) definitions, and primitive positive formulas without existential quantification, quantifier-free primitive positive definitions (q.f.p.p.) definitions. Sets of relations closed under p.p. definitions are known as co-clones and sets of relations closed under q.f.p.p. definitions as weak partial co-clones. The latter do however have limited expressivity, and the corresponding lattice of strong partial clones is of uncountably infinite cardinality even for the Boolean domain. Hence, it is reasonable to consider the expressiveness of p.p. definitions where only a small number of existentially quantified variables are allowed. In this article, we consider p.p. definitions allowing only polynomially many existentially quantified variables, and say that a co-clone closed under such definitions is polynomially closed, and otherwise superpolynomially closed. We investigate properties of polynomially closed co-clones and prove that if the corresponding clone contains a k-ary near-unanimity operation for k amp;gt;= 3, then the co-clone is polynomially closed, and if the clone does not contain a k-edge operation for any k amp;gt;= 2, then the co-clone is superpolynomially closed. For the Boolean domain we strengthen these results and prove a complete dichotomy theorem separating polynomially closed co-clones from superpolynomially closed co-clones. Using these results, we then proceed to investigate properties of strong partial clones corresponding to superpolynomially closed co-clones. We prove that if Gamma is a finite set of relations over an arbitrary finite domain such that the clone corresponding to Gamma is essentially unary, then the strong partial clone corresponding to Gamma is of infinite order and cannot be generated by a finite set of partial functions

A Solution to a Problem of D. Lau: Complete Classification of Intervals in the Lattice of Partial Boolean Clones

International audienceThe following natural problem, first considered by D. Lau, has been tackled by several authors recently: Let C be a total clone on 2 := {0, 1}. Describe the interval I(C) of all partial clones on 2 whose total component is C. We establish some results in this direction and combine them with previous ones to show the following dichotomy result: For every total clone C on 2, the set I(C) is either finite or of continuum cardinality. 1. Preliminaries Let k ≥ 2 be an integer and let k be a k-element set. Without loss of generality we assume that k := {0,. .. , k − 1}. For a positive integer n, an n-ary partial function on k is a map f : dom (f) → k where dom (f) is a subset of k n , called the domain of f. Let Par (n) (k) denote the set of all n-ary partial functions on k and let Par(k) := n≥

Fine-Grained Complexity of Constraint Satisfaction Problems through Partial Polymorphisms: A Survey (Dedicated to the memory of Professor Ivo Rosenberg)

International audienceConstraint satisfaction problems (CSPs) are combi-natorial problems with strong ties to universal algebra and clone theory. The recently proved CSP dichotomy theorem states that finite-domain CSPs are always either tractable or NP-complete. However, among the intractable cases there is a seemingly large variance in complexity, which cannot be explained by the classical algebraic approach using polymorphisms. In this contribution we will survey an alternative approach based on partial polymorphisms, which is useful for studying the fine-grained complexity of NP-complete CSPs. Moreover, we will state some challenging open problems in the research field

A Survey on the Fine-grained Complexity of Constraint Satisfaction Problems Based on Partial Polymorphisms

International audienceConstraint satisfaction problems (CSPs) are combinatorial problems with strong ties to universal algebra and clone theory. The recently proved CSP dichotomy theorem states that each finite-domain CSP is either solvable in polynomial time, or that it is NP-complete. However, among the intractable CSPs there is a seemingly large variance in how fast they can be solved by exponential-time algorithms, which cannot be explained by the classical algebraic approach based on polymorphisms. In this contribution we will survey an alternative approach based on partial polymorphisms, which is useful for studying the fine-grained complexity of NP-complete CSPs. Moreover, we will state and discuss some challenging open problems in this research field

Klonovi nedeterminističkih operacija

This thesis is a survey of some well-known and several new results concerning lattices of total, partial, incompletely specified clones and hyper-clones. We assign to every partial, incompletely specified and hyperoperation a suitable total operation and investigate thereby induced embeddings of the three lattices into corresponding lattices of total clones. Next we modify the famous Galois connection (Pol,Inv) between relations and operations for partial operations, IS operations and hyperoperations and describe classes of clones of IS operations and hyperoperations which strongly and weakly preserve given relations. We also state some known results concerning the four lattices on a two-element set. Finally, we present completeness criteria for the lattices of total and partial clones, and in the case of hyperclones and IS clones we describe four classes of coatoms, determined by four classes of Rosenberg’s relations.Ова теза представља преглед неких познатих и неколико нових резултата везаних за мреже тоталних, парцијалних, непотпуно специфицираних клонова и хиперклонова. Свакој парцијалној, непотпуно специфицираној и хипероперацији придружијемо одговарајућу тоталну операцију, и испитујемо тиме индукована потапања три мреже у одговарајуће мреже тоталних клонова. Потом познату Галоаову везу (Pol,Inv) између релација и операција модификујемо за парцијалне операције, НС опера-ције и хипероперације и описујемо класе клонова непотпуно специфици-раних и хипероперација које јако и слабо чувају дате релације. Такође наводимо неке познате резултате о мрежама на двоелементном скупу. Коначно, наводимо критеријуме комплетности за мреже тоталних и парцијалних клонова, а у случају хиперклонова и НС клонова описујемо четири класе коатома, одређених са четири класе Розенбергових релација.Ova teza predstavlja pregled nekih poznatih i nekoliko novih rezultata vezanih za mreže totalnih, parcijalnih, nepotpuno specificiranih klonova i hiperklonova. Svakoj parcijalnoj, nepotpuno specificiranoj i hiperoperaciji pridružijemo odgovarajuću totalnu operaciju, i ispitujemo time indukovana potapanja tri mreže u odgovarajuće mreže totalnih klonova. Potom poznatu Galoaovu vezu (Pol,Inv) između relacija i operacija modifikujemo za parcijalne operacije, NS opera-cije i hiperoperacije i opisujemo klase klonova nepotpuno specifici-ranih i hiperoperacija koje jako i slabo čuvaju date relacije. Takođe navodimo neke poznate rezultate o mrežama na dvoelementnom skupu. Konačno, navodimo kriterijume kompletnosti za mreže totalnih i parcijalnih klonova, a u slučaju hiperklonova i NS klonova opisujemo četiri klase koatoma, određenih sa četiri klase Rozenbergovih relacija

The (Coarse) Fine-Grained Structure of NP-Hard SAT and CSP Problems

We study the fine-grained complexity of NP-complete satisfiability (SAT) problems and constraint satisfaction problems (CSPs) in the context of the strong exponential-time hypothesis (SETH), showing non-trivial lower and upper bounds on the running time. Here, by a non-trivial lower bound for a problem SAT(Gamma) (respectively CSP(Gamma)) with constraint language F, we mean a value c(0) &amp;gt; 1 such that the problem cannot be solved in time O(c(n)) for any c &amp;lt; c(0) unless SETH is false, while a non-trivial upper bound is simply an algorithm for the problem running in time O(c(n)) for some c &amp;lt; 2. Such lower bounds have proven extremely elusive, and except for cases where c(0) = 2 effectively no such previous bound was known. We achieve this by employing an algebraic framework, studying constraint languages r in terms of their algebraic properties. We uncover a powerful algebraic framework where a mild restriction on the allowed constraints offers a concise algebraic characterization. On the relational side we restrict ourselves to Boolean languages closed under variable negation and partial assignment, called sign-symmetric languages. On the algebraic side this results in a description via partial operations arising from system of identities, with a close connection to operations resulting in tractable CSPs, such as near unanimity operations and edge operations. Using this connection we construct improved algorithms for several interesting classes of sign-symmetric languages, and prove explicit lower bounds under SETH. Thus, we find the first example of an NP-complete SAT problem with a non-trivial algorithm which also admits a non-trivial lower bound under SETH. This suggests a dichotomy conjecture with a close connection to the CSP dichotomy theorem: an NP-complete SAT problem admits an improved algorithm if and only if it admits a non-trivial partial invariant of the above form.Funding Agencies|Swedish resourch council (VR) [2019-03690]</p

Tractability in Constraint Satisfaction Problems: A Survey

International audienceEven though the Constraint Satisfaction Problem (CSP) is NP-complete, many tractable classes of CSP instances have been identified. After discussing different forms and uses of tractability, we describe some landmark tractable classes and survey recent theoretical results. Although we concentrate on the classical CSP, we also cover its important extensions to infinite domains and optimisation, as well as #CSP and QCSP