A Galois Connection for Weighted (Relational) Clones of Infinite Size

A Galois connection between clones and relational clones on a fixed finite domain is one of the cornerstones of the so-called algebraic approach to the computational complexity of non-uniform Constraint Satisfaction Problems (CSPs). Cohen et al. established a Galois connection between finitely-generated weighted clones and finitely-generated weighted relational clones [SICOMP'13], and asked whether this connection holds in general. We answer this question in the affirmative for weighted (relational) clones with real weights and show that the complexity of the corresponding valued CSPs is preserved

The complexity of Boolean surjective general-valued CSPs

Valued constraint satisfaction problems (VCSPs) are discrete optimisation problems with a $(\mathbb{Q}\cup\{\infty\})$-valued objective function given as a sum of fixed-arity functions. In Boolean surjective VCSPs, variables take on labels from $D=\{0,1\}$ and an optimal assignment is required to use both labels from $D$. Examples include the classical global Min-Cut problem in graphs and the Minimum Distance problem studied in coding theory. We establish a dichotomy theorem and thus give a complete complexity classification of Boolean surjective VCSPs with respect to exact solvability. Our work generalises the dichotomy for $\{0,\infty\}$-valued constraint languages (corresponding to surjective decision CSPs) obtained by Creignou and H\'ebrard. For the maximisation problem of $\mathbb{Q}_{\geq 0}$-valued surjective VCSPs, we also establish a dichotomy theorem with respect to approximability. Unlike in the case of Boolean surjective (decision) CSPs, there appears a novel tractable class of languages that is trivial in the non-surjective setting. This newly discovered tractable class has an interesting mathematical structure related to downsets and upsets. Our main contribution is identifying this class and proving that it lies on the borderline of tractability. A crucial part of our proof is a polynomial-time algorithm for enumerating all near-optimal solutions to a generalised Min-Cut problem, which might be of independent interest.Comment: v5: small corrections and improved presentatio

On the valued constraint satisfaction problem

The Valued Constraint Satisfaction Problem (VCSP) is a framework which captures many natural decision and optimisation problems. An instance of the VCSP consists of a set of variables, which are to be assigned labels from a finite domain, and a collection of local constraints, each specified by a weighted relation mapping labellings of the variables in the constraint's scope to values. The objective is to minimise the total value from all constraints. The VCSP is commonly parameterised by a language, i.e. a set of weighted relations that are available for use in the constraints. Languages are classified according to the computational complexity of the VCSP as tractable, for which the problem can be solved in polynomial time, and intractable, for which the problem is NP-hard. The recently proved VCSP dichotomy theorem established a classification of all languages into these two categories. Additionally, various structural restrictions can be imposed to limit the set of admissible instances further, thus potentially changing the complexity of the VCSP. Our first contribution relates to the algebraic approach to the VCSP, which proved instrumental in recent advances in the field. We generalise the Galois connection between weighted relational clones and weighted clones so that it applies to infinite sets as well. Second, we study a structural restriction requiring that the incidence graph be planar. In this setting, we establish a complexity classification of conservative languages (i.e. languages containing all {0,1}-valued unary weighted relations) and a necessary tractability condition for Boolean languages (i.e. languages over a two-element domain). Third, we study the surjective variant of the VCSP, in which labellings are required to assign every domain element to at least one variable. We establish a complexity classification of Boolean languages, which encompasses a new tractable class of problems.</p

A Galois connection for valued constraint languages of infinite size

A Galois connection between clones and relational clones on a fixed finite domain is one of the cornerstones of the so-called algebraic approach to the computational complexity of non-uniform Constraint Satisfaction Problems (CSPs). Cohen et al. established a Galois connection between finitely-generated weighted clones and finitely-generated weighted relational clones [SICOMP’13], and asked whether this connection holds in general. We answer this question in the affirmative for weighted (relational) clones with real weights and show that the complexity of the corresponding Valued CSPs is preserved

A Galois connection for valued constraint languages of infinite size

A Galois connection between clones and relational clones on a fixed finite domain is one of the cornerstones of the so-called algebraic approach to the computational complexity of non-uniform Constraint Satisfaction Problems (CSPs). Cohen et al. established a Galois connection between finitely-generated weighted clones and finitely-generated weighted relational clones [SICOMP’13], and asked whether this connection holds in general. We answer this question in the affirmative for weighted (relational) clones with real weights and show that the complexity of the corresponding Valued CSPs is preserved

The High Energy Density Scientific Instrument at the European XFEL

The European XFEL delivers up to 27000 intense (>1012 photons) pulses per second, of ultrashort (≤50 fs) and transversely coherent X-ray radiation, at a maximum repetition rate of 4.5 MHz. Its unique X-ray beam parameters enable groundbreaking experiments in matter at extreme conditions at the High Energy Density (HED) scientific instrument. The performance of the HED instrument during its first two years of operation, its scientific remit, as well as ongoing installations towards full operation are presented. Scientific goals of HED include the investigation of extreme states of matter created by intense laser pulses, diamond anvil cells, or pulsed magnets, and ultrafast X-ray methods that allow their diagnosis using self-amplified spontaneous emission between 5 and 25 keV, coupled with X-ray monochromators and optional seeded beam operation. The HED instrument provides two target chambers, X-ray spectrometers for emission and scattering, X-ray detectors, and a timing tool to correct for residual timing jitter between laser and X-ray pulses