15,589 research outputs found
Quasi-cluster algebras from non-orientable surfaces
With any non necessarily orientable unpunctured marked surface (S,M) we
associate a commutative algebra, called quasi-cluster algebra, equipped with a
distinguished set of generators, called quasi-cluster variables, in bijection
with the set of arcs and one-sided simple closed curves in (S,M). Quasi-cluster
variables are naturally gathered into possibly overlapping sets of fixed
cardinality, called quasi-clusters, corresponding to maximal non-intersecting
families of arcs and one-sided simple closed curves in (S,M). If the surface S
is orientable, then the quasi-cluster algebra is the cluster algebra associated
with the marked surface (S,M) in the sense of Fomin, Shapiro and Thurston. We
classify quasi-cluster algebras with finitely many quasi-cluster variables and
prove that for these quasi-cluster algebras, quasi-cluster monomials form a
linear basis. Finally, we attach to (S,M) a family of discrete integrable
systems satisfied by quasi-cluster variables associated to arcs in the
quasi-cluster algebra and we prove that solutions of these systems can be
expressed in terms of cluster variables of type A.Comment: 38 pages, 14 figure
Closed classes of functions, generalized constraints and clusters
Classes of functions of several variables on arbitrary non-empty domains that
are closed under permutation of variables and addition of dummy variables are
characterized in terms of generalized constraints, and hereby Hellerstein's
Galois theory of functions and generalized constraints is extended to infinite
domains. Furthermore, classes of operations on arbitrary non-empty domains that
are closed under permutation of variables, addition of dummy variables and
composition are characterized in terms of clusters, and a Galois connection is
established between operations and clusters.Comment: 21 page
Searching edges in the overlap of two plane graphs
Consider a pair of plane straight-line graphs, whose edges are colored red
and blue, respectively, and let n be the total complexity of both graphs. We
present a O(n log n)-time O(n)-space technique to preprocess such pair of
graphs, that enables efficient searches among the red-blue intersections along
edges of one of the graphs. Our technique has a number of applications to
geometric problems. This includes: (1) a solution to the batched red-blue
search problem [Dehne et al. 2006] in O(n log n) queries to the oracle; (2) an
algorithm to compute the maximum vertical distance between a pair of 3D
polyhedral terrains one of which is convex in O(n log n) time, where n is the
total complexity of both terrains; (3) an algorithm to construct the Hausdorff
Voronoi diagram of a family of point clusters in the plane in O((n+m) log^3 n)
time and O(n+m) space, where n is the total number of points in all clusters
and m is the number of crossings between all clusters; (4) an algorithm to
construct the farthest-color Voronoi diagram of the corners of n axis-aligned
rectangles in O(n log^2 n) time; (5) an algorithm to solve the stabbing circle
problem for n parallel line segments in the plane in optimal O(n log n) time.
All these results are new or improve on the best known algorithms.Comment: 22 pages, 6 figure
Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs
We study the family of intersection graphs of low density objects in low
dimensional Euclidean space. This family is quite general, and includes planar
graphs. We prove that such graphs have small separators. Next, we present
efficient -approximation algorithms for these graphs, for
Independent Set, Set Cover, and Dominating Set problems, among others. We also
prove corresponding hardness of approximation for some of these optimization
problems, providing a characterization of their intractability in terms of
density
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