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Clin d'Oeil on L_1Embeddable Planar Graphs
, 1996
"... In this note we present some properties of L1embeddable planar garphs. We show that every such graph G has a scale 2 embedding into a hypercube. Further, under some additional conditions we show that for a simple circuit C of G the subgraph H of G bounded by C is also L1embeddable. In many importa ..."
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Cited by 19 (2 self)
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In this note we present some properties of L1embeddable planar garphs. We show that every such graph G has a scale 2 embedding into a hypercube. Further, under some additional conditions we show that for a simple circuit C of G the subgraph H of G bounded by C is also L1embeddable. In many important cases, the length of C is the dimension of the smallest cube in which H has a scale embedding. Using these facts we establish the L1embeddability of a list of planar graphs.
Continuous Weber and kMedian Problems
, 2000
"... We give the first exact algorithmic study of facility location problems that deal with finding a median for a continuum of demand points. In particular, we consider versions of the "continuous kmedian (Weber) problem" where the goal is to select one or more center points that minimize the ..."
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Cited by 16 (3 self)
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We give the first exact algorithmic study of facility location problems that deal with finding a median for a continuum of demand points. In particular, we consider versions of the "continuous kmedian (Weber) problem" where the goal is to select one or more center points that minimize the average distance to a set of points in a demand region. In such problems, the average is computed as an integral over the relevant region, versus the usual discrete sum of distances. The resulting facility location problems are inherently geometric, requiring analysis techniques of computational geometry. We provide polynomialtime algorithms for various versions of the L1 1median (Weber) problem. We also consider the multiplecenter version of the L1 kmedian problem, which we prove is NPhard for large k.
Discrete Convexity and Polynomial Solvability in Minimum 0Extension Problems
, 2012
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Discrete Convexity for Multiflows and 0extensions
"... This paper addresses an approach to extend the submodularity and Lconvexity concepts to more general structures than integer lattices. The main motivations are to give a solution of the tractability classication in the minimum 0extension problem and to give a discreteconvexanalysis view to comb ..."
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Cited by 3 (2 self)
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This paper addresses an approach to extend the submodularity and Lconvexity concepts to more general structures than integer lattices. The main motivations are to give a solution of the tractability classication in the minimum 0extension problem and to give a discreteconvexanalysis view to combinatorial multiflow dualities.
WEAKLY MODULAR GRAPHS AND NONPOSITIVE CURVATURE
, 2014
"... This article investigates structural, geometrical, and topological characterizations and properties of weakly modular graphs and of cell complexes derived from them. The unifying themes of our investigation are various “nonpositive curvature” and “localtoglobal” properties and characterizations o ..."
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This article investigates structural, geometrical, and topological characterizations and properties of weakly modular graphs and of cell complexes derived from them. The unifying themes of our investigation are various “nonpositive curvature” and “localtoglobal” properties and characterizations of weakly modular graphs and their subclasses. Weakly modular graphs have been introduced as a farreaching common generalization of median graphs (and more generally, of modular and orientable modular graphs), Helly graphs, bridged graphs, and dual polar spaces occurring under different disguises (1–skeletons, collinearity graphs, covering graphs, domains, etc.) in several seeminglyunrelated fields of mathematics: • Metric graph theory • Geometric group theory • Incidence geometries and buildings • Theoretical computer science and combinatorial optimization We give a localtoglobal characterization of weakly modular graphs and their subclasses in terms of simple connectedness of associated trianglesquare complexes and specific local combinatorial conditions. In particular, we revisit characterizations of dual polar graphs by Cameron
COLLAPSIBLE POLYHEDRA AND MEDIAN SPACES
, 1998
"... It is shown that a collapsible, compact, connected, simplicial polyhedron admits a cubical subdivision and a median convexity, such that all cubes are convex subspaces with a convexity of subcubes. Conversely, a compact, connected, cubical polyhedron with a convexity as described admits a collapsib ..."
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It is shown that a collapsible, compact, connected, simplicial polyhedron admits a cubical subdivision and a median convexity, such that all cubes are convex subspaces with a convexity of subcubes. Conversely, a compact, connected, cubical polyhedron with a convexity as described admits a collapsible simplicial subdivision. Such a convexity, when it exists, is uniquely determined by the corresponding cubical presentation. Some related open problems have been formulated. .