186 research outputs found
Constraint Satisfaction and Semilinear Expansions of Addition over the Rationals and the Reals
A semilinear relation is a finite union of finite intersections of open and
closed half-spaces over, for instance, the reals, the rationals, or the
integers. Semilinear relations have been studied in connection with algebraic
geometry, automata theory, and spatiotemporal reasoning. We consider semilinear
relations over the rationals and the reals. Under this assumption, the
computational complexity of the constraint satisfaction problem (CSP) is known
for all finite sets containing R+={(x,y,z) | x+y=z}, <=, and {1}. These
problems correspond to expansions of the linear programming feasibility
problem. We generalise this result and fully determine the complexity for all
finite sets of semilinear relations containing R+. This is accomplished in part
by introducing an algorithm, based on computing affine hulls, which solves a
new class of semilinear CSPs in polynomial time. We further analyse the
complexity of linear optimisation over the solution set and the existence of
integer solutions.Comment: 22 pages, 1 figur
Constraint Satisfaction Problems over Numeric Domains
We present a survey of complexity results for constraint satisfaction problems (CSPs) over the integers, the rationals, the reals, and the complex numbers. Examples of such problems are feasibility of linear programs, integer linear programming, the max-atoms problem, Hilbert\u27s tenth problem, and many more. Our particular focus is to identify those CSPs that can be solved in polynomial time, and to distinguish them from CSPs that are NP-hard. A very helpful tool for obtaining complexity classifications in this context is the concept of a polymorphism from universal algebra
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Homogeneous Structures: Model Theory meets Universal Algebra (online meeting)
The workshop "Homogeneous Structures: Model Theory meets Universal
Algebra'' was centred around transferring recently obtained advances
in universal algebra from the finite to the infinite. As it turns out,
the notion of homogeneity together with other model-theoretic concepts
like -categoricity and the Ramsey property play an
indispensable role in this endeavour
Characterizing the integer points in 2-decomposable polyhedra by closedness under operations
Characterizing the solution sets in a problem by closedness under operations
is recognized as one of the key aspects of algorithm development, especially in
constraint satisfaction. An example from the Boolean satisfiability problem is
that the solution set of a Horn conjunctive normal form (CNF) is closed under
the minimum operation, and this property implies that minimizing a nonnegative
linear function over a Horn CNF can be done in polynomial time. In this paper,
we focus on the set of integer points (vectors) in a polyhedron, and study the
relation between these sets and closedness under operations from the viewpoint
of 2-decomposability. By adding further conditions to the 2-decomposable
polyhedra, we show that important classes of sets of integer vectors in
polyhedra are characterized by 2-decomposability and closedness under certain
operations, and in some classes, by closedness under operations alone. The most
prominent result we show is that the set of integer vectors in a
unit-two-variable-per-inequality polyhedron can be characterized by closedness
under the median and directed discrete midpoint operations, each of these
operations was independently considered in constraint satisfaction and discrete
convex analysis.Comment: 22 page
On the consistency of a spatial-type interval-valued median for random intervals
The sample -median is a robust estimator of the central tendency or
location of an interval-valued random variable. While the interval-valued
sample mean can be highly influenced by outliers, this spatial-type
interval-valued median remains much more reliable. In this paper, we show that
under general conditions the sample -median is a strongly consistent
estimator of the -median of an interval-valued random variable.Comment: 14 page
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