Maltsev digraphs have a majority polymorphism

We prove that when a digraph $G$ has a Maltsev polymorphism, then $G$ also has a majority polymorphism. We consider the consequences of this result for the structure of Maltsev graphs and the complexity of the Constraint Satisfaction Problem.Comment: 8 pages, 4 figures; minor changes (stylistics, elsarticle LaTeX style, citations); submitted to European Journal of Combinatoric

The complexity of global cardinality constraints

In a constraint satisfaction problem (CSP) the goal is to find an assignment of a given set of variables subject to specified constraints. A global cardinality constraint is an additional requirement that prescribes how many variables must be assigned a certain value. We study the complexity of the problem CCSP(G), the constraint satisfaction problem with global cardinality constraints that allows only relations from the set G. The main result of this paper characterizes sets G that give rise to problems solvable in polynomial time, and states that the remaining such problems are NP-complete

The power of linear programming for general-valued CSPs

Let $D$, called the domain, be a fixed finite set and let $\Gamma$, called the valued constraint language, be a fixed set of functions of the form $f:D^m\to\mathbb{Q}\cup\{\infty\}$, where different functions might have different arity $m$. We study the valued constraint satisfaction problem parametrised by $\Gamma$, denoted by VCSP$(\Gamma)$. These are minimisation problems given by $n$ variables and the objective function given by a sum of functions from $\Gamma$, each depending on a subset of the $n$ variables. Finite-valued constraint languages contain functions that take on only rational values and not infinite values. Our main result is a precise algebraic characterisation of valued constraint languages whose instances can be solved exactly by the basic linear programming relaxation (BLP). For a valued constraint language $\Gamma$, BLP is a decision procedure for $\Gamma$ if and only if $\Gamma$ admits a symmetric fractional polymorphism of every arity. For a finite-valued constraint language $\Gamma$, BLP is a decision procedure if and only if $\Gamma$ admits a symmetric fractional polymorphism of some arity, or equivalently, if $\Gamma$ admits a symmetric fractional polymorphism of arity 2. Using these results, we obtain tractability of several novel classes of problems, including problems over valued constraint languages that are: (1) submodular on arbitrary lattices; (2) $k$-submodular on arbitrary finite domains; (3) weakly (and hence strongly) tree-submodular on arbitrary trees.Comment: A full version of a FOCS'12 paper by the last two authors (arXiv:1204.1079) and an ICALP'13 paper by the first author (arXiv:1207.7213) to appear in SIAM Journal on Computing (SICOMP

The wonderland of reflections

A fundamental fact for the algebraic theory of constraint satisfaction problems (CSPs) over a fixed template is that pp-interpretations between at most countable \omega-categorical relational structures have two algebraic counterparts for their polymorphism clones: a semantic one via the standard algebraic operators H, S, P, and a syntactic one via clone homomorphisms (capturing identities). We provide a similar characterization which incorporates all relational constructions relevant for CSPs, that is, homomorphic equivalence and adding singletons to cores in addition to pp-interpretations. For the semantic part we introduce a new construction, called reflection, and for the syntactic part we find an appropriate weakening of clone homomorphisms, called h1 clone homomorphisms (capturing identities of height 1). As a consequence, the complexity of the CSP of an at most countable $\omega$-categorical structure depends only on the identities of height 1 satisfied in its polymorphism clone as well as the the natural uniformity thereon. This allows us in turn to formulate a new elegant dichotomy conjecture for the CSPs of reducts of finitely bounded homogeneous structures. Finally, we reveal a close connection between h1 clone homomorphisms and the notion of compatibility with projections used in the study of the lattice of interpretability types of varieties.Comment: 24 page

Existentially Restricted Quantified Constraint Satisfaction

The quantified constraint satisfaction problem (QCSP) is a powerful framework for modelling computational problems. The general intractability of the QCSP has motivated the pursuit of restricted cases that avoid its maximal complexity. In this paper, we introduce and study a new model for investigating QCSP complexity in which the types of constraints given by the existentially quantified variables, is restricted. Our primary technical contribution is the development and application of a general technology for proving positive results on parameterizations of the model, of inclusion in the complexity class coNP

SNP Assay Development for Linkage Map Construction, Anchoring Whole-Genome Sequence, and Other Genetic and Genomic Applications in Common Bean.

A total of 992,682 single-nucleotide polymorphisms (SNPs) was identified as ideal for Illumina Infinium II BeadChip design after sequencing a diverse set of 17 common bean (Phaseolus vulgaris L) varieties with the aid of next-generation sequencing technology. From these, two BeadChips each with >5000 SNPs were designed. The BARCBean6K_1 BeadChip was selected for the purpose of optimizing polymorphism among market classes and, when possible, SNPs were targeted to sequence scaffolds in the Phaseolus vulgaris 14× genome assembly with sequence lengths >10 kb. The BARCBean6K_2 BeadChip was designed with the objective of anchoring additional scaffolds and to facilitate orientation of large scaffolds. Analysis of 267 F2 plants from a cross of varieties Stampede × Red Hawk with the two BeadChips resulted in linkage maps with a total of 7040 markers including 7015 SNPs. With the linkage map, a total of 432.3 Mb of sequence from 2766 scaffolds was anchored to create the Phaseolus vulgaris v1.0 assembly, which accounted for approximately 89% of the 487 Mb of available sequence scaffolds of the Phaseolus vulgaris v0.9 assembly. A core set of 6000 SNPs (BARCBean6K_3 BeadChip) with high genotyping quality and polymorphism was selected based on the genotyping of 365 dry bean and 134 snap bean accessions with the BARCBean6K_1 and BARCBean6K_2 BeadChips. The BARCBean6K_3 BeadChip is a useful tool for genetics and genomics research and it is widely used by breeders and geneticists in the United States and abroad