Local consistency as a reduction between constraint satisfaction problems

We study the use of local consistency methods as reductions between constraint satisfaction problems (CSPs), and promise version thereof, with the aim to classify these reductions in a similar way as the algebraic approach classifies gadget reductions between CSPs. This research is motivated by the requirement of more expressive reductions in the scope of promise CSPs. While gadget reductions are enough to provide all necessary hardness in the scope of (finite domain) non-promise CSP, in promise CSPs a wider class of reductions needs to be used.We provide a general framework of reductions, which we call consistency reductions, that covers most (if not all) reductions recently used for proving NP-hardness of promise CSPs. We prove some basic properties of these reductions, and provide the first steps towards understanding the power of consistency reductions by characterizing a fragment associated to arc-consistency in terms of polymorphisms of the template. In addition to showing hardness, consistency reductions can also be used to provide feasible algorithms by reducing to a fixed tractable (promise) CSP, for example, to solving systems of affine equations. In this direction, among other results, we describe the well-known Sherali-Adams hierarchy for CSP in terms of a consistency reduction to linear programming

Functors on relational structures which admit both left and right adjoints

This paper describes several cases of adjunction in the homomorphism preorder of relational structures. We say that two functors $\Lambda$ and $\Gamma$ between thin categories of relational structures are adjoint if for all structures $\mathbf A$ and $\mathbf B$, we have that $\Lambda(\mathbf A)$ maps homomorphically to $\mathbf B$ if and only if $\mathbf A$ maps homomorphically to $\Gamma(\mathbf B)$. If this is the case $\Lambda$ is called the left adjoint to $\Gamma$ and $\Gamma$ the right adjoint to $\Lambda$. In 2015, Foniok and Tardif described some functors on the category of digraphs that allow both left and right adjoints. The main contribution of Foniok and Tardif is a construction of right adjoints to some of the functors identified as right adjoints by Pultr in 1970. We generalise results of Foniok and Tardif to arbitrary relational structures, and coincidently, we also provide more right adjoints on digraphs, and since these constructions are connected to finite duality, we also provide a new construction of duals to trees. Our results are inspired by an application in promise constraint satisfaction -- it has been shown that such functors can be used as efficient reductions between these problems

Right-Adjoints for Datalog Programs

A Datalog program can be viewed as a syntactic specification of a mapping from database instances over some schema to database instances over another schema. We establish a large class of Datalog programs for which this mapping admits a (generalized) right-adjoint. We employ these results to obtain new insights into the existence of, and methods for constructing, homomorphism dualities within restricted classes of instances. From this, we derive new results regarding the existence of uniquely characterizing data examples for database queries in the presence of integrity constraints.</p

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When do homomorphism counts help in query algorithms?

A query algorithm based on homomorphism counts is a procedure for determining whether a given instance satisfies a property by counting homomorphisms between the given instance and finitely many predetermined instances. In a left query algorithm, we count homomorphisms from the predetermined instances to the given instance, while in a right query algorithm we count homomorphisms from the given instance to the predetermined instances. Homomorphisms are usually counted over the semiring N of non-negative integers; it is also meaningful, however, to count homomorphisms over the Boolean semiring B, in which case the homomorphism count indicates whether or not a homomorphism exists. We first characterize the properties that admit a left query algorithm over B by showing that these are precisely the properties that are both first-order definable and closed under homomorphic equivalence. After this, we turn attention to a comparison between left query algorithms over B and left query algorithms over N. In general, there are properties that admit a left query algorithm over N but not over B. The main result of this paper asserts that if a property is closed under homomorphic equivalence, then that property admits a left query algorithm over B if and only if it admits a left query algorithm over N. In other words and rather surprisingly, homomorphism counts over N do not help as regards properties that are closed under homomorphic equivalence. Finally, we characterize the properties that admit both a left query algorithm over B and a right query algorithm over B.Comment: 24 page

Absorbing Subalgebras, Cyclic Terms, and the Constraint Satisfaction Problem

The Algebraic Dichotomy Conjecture states that the Constraint Satisfaction Problem over a fixed template is solvable in polynomial time if the algebra of polymorphisms associated to the template lies in a Taylor variety, and is NP-complete otherwise. This paper provides two new characterizations of finitely generated Taylor varieties. The first characterization is using absorbing subalgebras and the second one cyclic terms. These new conditions allow us to reprove the conjecture of Bang-Jensen and Hell (proved by the authors) and the characterization of locally finite Taylor varieties using weak near-unanimity terms (proved by McKenzie and Mar\'oti) in an elementary and self-contained way

Datalog and Constraint Satisfaction with Infinite Templates

On finite structures, there is a well-known connection between the expressive power of Datalog, finite variable logics, the existential pebble game, and bounded hypertree duality. We study this connection for infinite structures. This has applications for constraint satisfaction with infinite templates. If the template Gamma is omega-categorical, we present various equivalent characterizations of those Gamma such that the constraint satisfaction problem (CSP) for Gamma can be solved by a Datalog program. We also show that CSP(Gamma) can be solved in polynomial time for arbitrary omega-categorical structures Gamma if the input is restricted to instances of bounded treewidth. Finally, we characterize those omega-categorical templates whose CSP has Datalog width 1, and those whose CSP has strict Datalog width k.Comment: 28 pages. This is an extended long version of a conference paper that appeared at STACS'06. In the third version in the arxiv we have revised the presentation again and added a section that relates our results to formalizations of CSPs using relation algebra

Altered T-cell subset distribution in the viral reservoir in HIV-1-infected individuals with extremely low proviral DNA (LoViReTs)

HIV cure strategies aim to eliminate viral reservoirs that persist despite successful antiretroviral therapy (ART). We have previously described that 9% of HIV-infected individuals who receive ART harbor low levels of provirus (LoViReTs). We selected 22 LoViReTs matched with 22 controls ART suppressed for more than 3 years with fewer than 100 and more than 100 HIV-DNA copies/10 6 CD4 + T cells, respectively. We measured HIV reservoirs in blood and host genetic factors. Fourteen LoViReTs underwent leukapheresis to analyze replication-competent virus, and HIV-DNA in CD4 + T-cell subpopulations. Additionally, we measured HIV-DNA in rectum and/or lymph node biopsies from nine of them. We found that LoViReTs harbored not only lower levels of total HIV-DNA, but also significantly lower intact HIV-DNA, cell-associated HIV-RNA, and ultrasensitive viral load than controls. The proportion of intact versus total proviruses was similar in both groups. We found no differences in the percentage of host factors. In peripheral blood, 71% of LoViReTs had undetectable replication-competent virus. Minimum levels of total HIV-DNA were found in rectal and lymph node biopsies compared with HIV-infected individuals receiving ART. The main contributors to the reservoir were short-lived transitional memory and effector memory T cells (47% and 29%, respectively), indicating an altered distribution of the HIV reservoir in the peripheral T-cell subpopulations of LoViReTs. In conclusion, LoViReTs are characterized by low levels of viral reservoir in peripheral blood and secondary lymphoid tissues, which might be explained by an altered distribution of the proviral HIV-DNA towards more short-lived memory T cells. LoViReTs can be considered exceptional candidates for future interventions aimed at curing HIV