202 research outputs found

    Sparsity-Certifying Graph Decompositions

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    We describe a new algorithm, the (k, ℓ)-pebble game with colors, and use it to obtain a characterization of the family of (k, ℓ)-sparse graphs and algorithmic solutions to a family of problems concerning tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and have received increased attention in recent years. In particular, our colored pebbles generalize and strengthen the previous results of Lee and Streinu [12] and give a new proof of the Tutte-Nash-Williams characterization of arboricity. We also present a new decomposition that certifies sparsity based on the (k, ℓ)-pebble game with colors. Our work also exposes connections between pebble game algorithms and previous sparse graph algorithms by Gabow [5], Gabow and Westermann [6] and Hendrickson [9]

    Natural realizations of sparsity matroids

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    A hypergraph G with n vertices and m hyperedges with d endpoints each is (k,l)-sparse if for all sub-hypergraphs G' on n' vertices and m' edges, m'\le kn'-l. For integers k and l satisfying 0\le l\le dk-1, this is known to be a linearly representable matroidal family. Motivated by problems in rigidity theory, we give a new linear representation theorem for the (k,l)-sparse hypergraphs that is natural; i.e., the representing matrix captures the vertex-edge incidence structure of the underlying hypergraph G.Comment: Corrected some typos from the previous version; to appear in Ars Mathematica Contemporane

    Generic rigidity with forced symmetry and sparse colored graphs

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    We review some recent results in the generic rigidity theory of planar frameworks with forced symmetry, giving a uniform treatment to the topic. We also give new combinatorial characterizations of minimally rigid periodic frameworks with fixed-area fundamental domain and fixed-angle fundamental domain.Comment: 21 pages, 2 figure

    Algorithms for detecting dependencies and rigid subsystems for CAD

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    Geometric constraint systems underly popular Computer Aided Design soft- ware. Automated approaches for detecting dependencies in a design are critical for developing robust solvers and providing informative user feedback, and we provide algorithms for two types of dependencies. First, we give a pebble game algorithm for detecting generic dependencies. Then, we focus on identifying the "special positions" of a design in which generically independent constraints become dependent. We present combinatorial algorithms for identifying subgraphs associated to factors of a particular polynomial, whose vanishing indicates a special position and resulting dependency. Further factoring in the Grassmann- Cayley algebra may allow a geometric interpretation giving conditions (e.g., "these two lines being parallel cause a dependency") determining the special position.Comment: 37 pages, 14 figures (v2 is an expanded version of an AGD'14 abstract based on v1

    Slider-pinning Rigidity: a Maxwell-Laman-type Theorem

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    We define and study slider-pinning rigidity, giving a complete combinatorial characterization. This is done via direction-slider networks, which are a generalization of Whiteley's direction networks.Comment: Accepted, to appear in Discrete and Computational Geometr

    The Slider-Pinning Problem

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    A Laman mechanism is a flexible planar bar-and-joint framework with m ≤ 2n-3 edges and exactly k = 2n-m degrees of freedom. The slider-pinning problem is to eliminate all the degrees of freedom of a Laman mechanism, in an optimal fashion, by individually fixing x or y coordinates of vertices. We describe two easy to implement O(n2) time algorithms

    Sparsest Cut on Bounded Treewidth Graphs: Algorithms and Hardness Results

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    We give a 2-approximation algorithm for Non-Uniform Sparsest Cut that runs in time nO(k)n^{O(k)}, where kk is the treewidth of the graph. This improves on the previous 22k2^{2^k}-approximation in time \poly(n) 2^{O(k)} due to Chlamt\'a\v{c} et al. To complement this algorithm, we show the following hardness results: If the Non-Uniform Sparsest Cut problem has a ρ\rho-approximation for series-parallel graphs (where ρ1\rho \geq 1), then the Max Cut problem has an algorithm with approximation factor arbitrarily close to 1/ρ1/\rho. Hence, even for such restricted graphs (which have treewidth 2), the Sparsest Cut problem is NP-hard to approximate better than 17/16ϵ17/16 - \epsilon for ϵ>0\epsilon > 0; assuming the Unique Games Conjecture the hardness becomes 1/αGWϵ1/\alpha_{GW} - \epsilon. For graphs with large (but constant) treewidth, we show a hardness result of 2ϵ2 - \epsilon assuming the Unique Games Conjecture. Our algorithm rounds a linear program based on (a subset of) the Sherali-Adams lift of the standard Sparsest Cut LP. We show that even for treewidth-2 graphs, the LP has an integrality gap close to 2 even after polynomially many rounds of Sherali-Adams. Hence our approach cannot be improved even on such restricted graphs without using a stronger relaxation
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