1,068 research outputs found

    A Combinatorial, Strongly Polynomial-Time Algorithm for Minimizing Submodular Functions

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    This paper presents the first combinatorial polynomial-time algorithm for minimizing submodular set functions, answering an open question posed in 1981 by Grotschel, Lovasz, and Schrijver. The algorithm employs a scaling scheme that uses a flow in the complete directed graph on the underlying set with each arc capacity equal to the scaled parameter. The resulting algorithm runs in time bounded by a polynomial in the size of the underlying set and the largest length of the function value. The paper also presents a strongly polynomial-time version that runs in time bounded by a polynomial in the size of the underlying set independent of the function value.Comment: 17 page

    Polytopal realizations of finite type g\mathbf{g}-vector fans

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    This paper shows the polytopality of any finite type g\mathbf{g}-vector fan, acyclic or not. In fact, for any finite Dynkin type Γ\Gamma, we construct a universal associahedron Assoun(Γ)\mathsf{Asso}_{\mathrm{un}}(\Gamma) with the property that any g\mathbf{g}-vector fan of type Γ\Gamma is the normal fan of a suitable projection of Assoun(Γ)\mathsf{Asso}_{\mathrm{un}}(\Gamma).Comment: 27 pages, 9 figures; Version 2: Minor changes in the introductio

    A New Framework for Distributed Submodular Maximization

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    A wide variety of problems in machine learning, including exemplar clustering, document summarization, and sensor placement, can be cast as constrained submodular maximization problems. A lot of recent effort has been devoted to developing distributed algorithms for these problems. However, these results suffer from high number of rounds, suboptimal approximation ratios, or both. We develop a framework for bringing existing algorithms in the sequential setting to the distributed setting, achieving near optimal approximation ratios for many settings in only a constant number of MapReduce rounds. Our techniques also give a fast sequential algorithm for non-monotone maximization subject to a matroid constraint

    Efficient Algorithms for Searching the Minimum Information Partition in Integrated Information Theory

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    The ability to integrate information in the brain is considered to be an essential property for cognition and consciousness. Integrated Information Theory (IIT) hypothesizes that the amount of integrated information (Φ\Phi) in the brain is related to the level of consciousness. IIT proposes that to quantify information integration in a system as a whole, integrated information should be measured across the partition of the system at which information loss caused by partitioning is minimized, called the Minimum Information Partition (MIP). The computational cost for exhaustively searching for the MIP grows exponentially with system size, making it difficult to apply IIT to real neural data. It has been previously shown that if a measure of Φ\Phi satisfies a mathematical property, submodularity, the MIP can be found in a polynomial order by an optimization algorithm. However, although the first version of Φ\Phi is submodular, the later versions are not. In this study, we empirically explore to what extent the algorithm can be applied to the non-submodular measures of Φ\Phi by evaluating the accuracy of the algorithm in simulated data and real neural data. We find that the algorithm identifies the MIP in a nearly perfect manner even for the non-submodular measures. Our results show that the algorithm allows us to measure Φ\Phi in large systems within a practical amount of time
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