135 research outputs found

    Products of Foldable Triangulations

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    Regular triangulations of products of lattice polytopes are constructed with the additional property that the dual graphs of the triangulations are bipartite. The (weighted) size difference of this bipartition is a lower bound for the number of real roots of certain sparse polynomial systems by recent results of Soprunova and Sottile [Adv. Math. 204(1):116-151, 2006]. Special attention is paid to the cube case.Comment: new title; several paragraphs reformulated; 23 page

    Counting Small Induced Subgraphs Satisfying Monotone Properties

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    Given a graph property Ī¦\Phi, the problem #IndSub(Ī¦)\#\mathsf{IndSub}(\Phi) asks, on input a graph GG and a positive integer kk, to compute the number of induced subgraphs of size kk in GG that satisfy Ī¦\Phi. The search for explicit criteria on Ī¦\Phi ensuring that #IndSub(Ī¦)\#\mathsf{IndSub}(\Phi) is hard was initiated by Jerrum and Meeks [J. Comput. Syst. Sci. 15] and is part of the major line of research on counting small patterns in graphs. However, apart from an implicit result due to Curticapean, Dell and Marx [STOC 17] proving that a full classification into "easy" and "hard" properties is possible and some partial results on edge-monotone properties due to Meeks [Discret. Appl. Math. 16] and D\"orfler et al. [MFCS 19], not much is known. In this work, we fully answer and explicitly classify the case of monotone, that is subgraph-closed, properties: We show that for any non-trivial monotone property Ī¦\Phi, the problem #IndSub(Ī¦)\#\mathsf{IndSub}(\Phi) cannot be solved in time f(k)ā‹…āˆ£V(G)āˆ£o(k/logā”1/2(k))f(k)\cdot |V(G)|^{o(k/ {\log^{1/2}(k)})} for any function ff, unless the Exponential Time Hypothesis fails. By this, we establish that any significant improvement over the brute-force approach is unlikely; in the language of parameterized complexity, we also obtain a #W[1]\#\mathsf{W}[1]-completeness result

    Counting Small Induced Subgraphs Satisfying Monotone Properties

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    Given a graph property Ī¦\Phi, the problem #IndSub(Ī¦)\#\mathsf{IndSub}(\Phi) asks, on input a graph GG and a positive integer kk, to compute the number of induced subgraphs of size kk in GG that satisfy Ī¦\Phi. The search for explicit criteria on Ī¦\Phi ensuring that #IndSub(Ī¦)\#\mathsf{IndSub}(\Phi) is hard was initiated by Jerrum and Meeks [J. Comput. Syst. Sci. 15] and is part of the major line of research on counting small patterns in graphs. However, apart from an implicit result due to Curticapean, Dell and Marx [STOC 17] proving that a full classification into "easy" and "hard" properties is possible and some partial results on edge-monotone properties due to Meeks [Discret. Appl. Math. 16] and D\"orfler et al. [MFCS 19], not much is known. In this work, we fully answer and explicitly classify the case of monotone, that is subgraph-closed, properties: We show that for any non-trivial monotone property Ī¦\Phi, the problem #IndSub(Ī¦)\#\mathsf{IndSub}(\Phi) cannot be solved in time f(k)ā‹…āˆ£V(G)āˆ£o(k/logā”1/2(k))f(k)\cdot |V(G)|^{o(k/ {\log^{1/2}(k)})} for any function ff, unless the Exponential Time Hypothesis fails. By this, we establish that any significant improvement over the brute-force approach is unlikely; in the language of parameterized complexity, we also obtain a #W[1]\#\mathsf{W}[1]-completeness result.Comment: 33 pages, 2 figure

    Counting small induced subgraphs satisfying monotone properties

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    Given a graph property Ī¦\Phi, the problem #IndSub(Ī¦)\#\mathsf{IndSub}(\Phi) asks, on input a graph GG and a positive integer kk, to compute the number of induced subgraphs of size kk in GG that satisfy Ī¦\Phi. The search for explicit criteria on Ī¦\Phi ensuring that #IndSub(Ī¦)\#\mathsf{IndSub}(\Phi) is hard was initiated by Jerrum and Meeks [J. Comput. Syst. Sci. 15] and is part of the major line of research on counting small patterns in graphs. However, apart from an implicit result due to Curticapean, Dell and Marx [STOC 17] proving that a full classification into "easy" and "hard" properties is possible and some partial results on edge-monotone properties due to Meeks [Discret. Appl. Math. 16] and D\"orfler et al. [MFCS 19], not much is known. In this work, we fully answer and explicitly classify the case of monotone, that is subgraph-closed, properties: We show that for any non-trivial monotone property Ī¦\Phi, the problem #IndSub(Ī¦)\#\mathsf{IndSub}(\Phi) cannot be solved in time f(k)ā‹…āˆ£V(G)āˆ£o(k/logā”1/2(k))f(k)\cdot |V(G)|^{o(k/ {\log^{1/2}(k)})} for any function ff, unless the Exponential Time Hypothesis fails. By this, we establish that any significant improvement over the brute-force approach is unlikely; in the language of parameterized complexity, we also obtain a #W[1]\#\mathsf{W}[1]-completeness result

    Counting small induced subgraphs satisfying monotone properties

    Get PDF
    Given a graph property Ī¦\Phi, the problem #IndSub(Ī¦)\#\mathsf{IndSub}(\Phi) asks, on input a graph GG and a positive integer kk, to compute the number of induced subgraphs of size kk in GG that satisfy Ī¦\Phi. The search for explicit criteria on Ī¦\Phi ensuring that #IndSub(Ī¦)\#\mathsf{IndSub}(\Phi) is hard was initiated by Jerrum and Meeks [J. Comput. Syst. Sci. 15] and is part of the major line of research on counting small patterns in graphs. However, apart from an implicit result due to Curticapean, Dell and Marx [STOC 17] proving that a full classification into "easy" and "hard" properties is possible and some partial results on edge-monotone properties due to Meeks [Discret. Appl. Math. 16] and D\"orfler et al. [MFCS 19], not much is known. In this work, we fully answer and explicitly classify the case of monotone, that is subgraph-closed, properties: We show that for any non-trivial monotone property Ī¦\Phi, the problem #IndSub(Ī¦)\#\mathsf{IndSub}(\Phi) cannot be solved in time f(k)ā‹…āˆ£V(G)āˆ£o(k/logā”1/2(k))f(k)\cdot |V(G)|^{o(k/ {\log^{1/2}(k)})} for any function ff, unless the Exponential Time Hypothesis fails. By this, we establish that any significant improvement over the brute-force approach is unlikely; in the language of parameterized complexity, we also obtain a #W[1]\#\mathsf{W}[1]-completeness result

    Lower Bounds for Real Solutions to Sparse Polynomial Systems

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    We show how to construct sparse polynomial systems that have non-trivial lower bounds on their numbers of real solutions. These are unmixed systems associated to certain polytopes. For the order polytope of a poset P this lower bound is the sign-imbalance of P and it holds if all maximal chains of P have length of the same parity. This theory also gives lower bounds in the real Schubert calculus through sagbi degeneration of the Grassmannian to a toric variety, and thus recovers a result of Eremenko and Gabrielov.Comment: 31 pages. Minor revision

    Spherical and Hyperbolic Toric Topology-Based Codes On Graph Embedding for Ising MRF Models: Classical and Quantum Topology Machine Learning

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    The paper introduces the application of information geometry to describe the ground states of Ising models by utilizing parity-check matrices of cyclic and quasi-cyclic codes on toric and spherical topologies. The approach establishes a connection between machine learning and error-correcting coding. This proposed approach has implications for the development of new embedding methods based on trapping sets. Statistical physics and number geometry applied for optimize error-correcting codes, leading to these embedding and sparse factorization methods. The paper establishes a direct connection between DNN architecture and error-correcting coding by demonstrating how state-of-the-art architectures (ChordMixer, Mega, Mega-chunk, CDIL, ...) from the long-range arena can be equivalent to of block and convolutional LDPC codes (Cage-graph, Repeat Accumulate). QC codes correspond to certain types of chemical elements, with the carbon element being represented by the mixed automorphism Shu-Lin-Fossorier QC-LDPC code. The connections between Belief Propagation and the Permanent, Bethe-Permanent, Nishimori Temperature, and Bethe-Hessian Matrix are elaborated upon in detail. The Quantum Approximate Optimization Algorithm (QAOA) used in the Sherrington-Kirkpatrick Ising model can be seen as analogous to the back-propagation loss function landscape in training DNNs. This similarity creates a comparable problem with TS pseudo-codeword, resembling the belief propagation method. Additionally, the layer depth in QAOA correlates to the number of decoding belief propagation iterations in the Wiberg decoding tree. Overall, this work has the potential to advance multiple fields, from Information Theory, DNN architecture design (sparse and structured prior graph topology), efficient hardware design for Quantum and Classical DPU/TPU (graph, quantize and shift register architect.) to Materials Science and beyond.Comment: 71 pages, 42 Figures, 1 Table, 1 Appendix. arXiv admin note: text overlap with arXiv:2109.08184 by other author
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