27,323 research outputs found

    New Planar P-time Computable Six-Vertex Models and a Complete Complexity Classification

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    We discover new P-time computable six-vertex models on planar graphs beyond Kasteleyn's algorithm for counting planar perfect matchings. We further prove that there are no more: Together, they exhaust all P-time computable six-vertex models on planar graphs, assuming #P is not P. This leads to the following exact complexity classification: For every parameter setting in C{\mathbb C} for the six-vertex model, the partition function is either (1) computable in P-time for every graph, or (2) #P-hard for general graphs but computable in P-time for planar graphs, or (3) #P-hard even for planar graphs. The classification has an explicit criterion. The new P-time cases in (2) provably cannot be subsumed by Kasteleyn's algorithm. They are obtained by a non-local connection to #CSP, defined in terms of a "loop space". This is the first substantive advance toward a planar Holant classification with not necessarily symmetric constraints. We introduce M\"obius transformation on C{\mathbb C} as a powerful new tool in hardness proofs for counting problems.Comment: 61 pages, 16 figures. An extended abstract appears in SODA 202

    Approximability of the Eight-Vertex Model

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    We initiate a study of the classification of approximation complexity of the eight-vertex model defined over 4-regular graphs. The eight-vertex model, together with its special case the six-vertex model, is one of the most extensively studied models in statistical physics, and can be stated as a problem of counting weighted orientations in graph theory. Our result concerns the approximability of the partition function on all 4-regular graphs, classified according to the parameters of the model. Our complexity results conform to the phase transition phenomenon from physics. We introduce a quantum decomposition of the eight-vertex model and prove a set of closure properties in various regions of the parameter space. Furthermore, we show that there are extra closure properties on 4-regular planar graphs. These regions of the parameter space are concordant with the phase transition threshold. Using these closure properties, we derive polynomial time approximation algorithms via Markov chain Monte Carlo. We also show that the eight-vertex model is NP-hard to approximate on the other side of the phase transition threshold

    DHLP 1&2: Giraph based distributed label propagation algorithms on heterogeneous drug-related networks

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    Background and Objective: Heterogeneous complex networks are large graphs consisting of different types of nodes and edges. The knowledge extraction from these networks is complicated. Moreover, the scale of these networks is steadily increasing. Thus, scalable methods are required. Methods: In this paper, two distributed label propagation algorithms for heterogeneous networks, namely DHLP-1 and DHLP-2 have been introduced. Biological networks are one type of the heterogeneous complex networks. As a case study, we have measured the efficiency of our proposed DHLP-1 and DHLP-2 algorithms on a biological network consisting of drugs, diseases, and targets. The subject we have studied in this network is drug repositioning but our algorithms can be used as general methods for heterogeneous networks other than the biological network. Results: We compared the proposed algorithms with similar non-distributed versions of them namely MINProp and Heter-LP. The experiments revealed the good performance of the algorithms in terms of running time and accuracy.Comment: Source code available for Apache Giraph on Hadoo
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