100 research outputs found

    Coalition structure generation over graphs

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    We give the analysis of the computational complexity of coalition structure generation over graphs. Given an undirected graph G = (N,E) and a valuation function v : P(N) → R over the subsets of nodes, the problem is to find a partition of N into connected subsets, that maximises the sum of the components values. This problem is generally NP-complete; in particular, it is hard for a defined class of valuation functions which are independent of disconnected members — that is, two nodes have no effect on each others marginal contribution to their vertex separator. Nonetheless, for all such functions we provide bounds on the complexity of coalition structure generation over general and minor free graphs. Our proof is constructive and yields algorithms for solving corresponding instances of the problem. Furthermore, we derive linear time bounds for graphs of bounded treewidth. However, as we show, the problem remains NP-complete for planar graphs, and hence, for any Kk minor free graphs where k ≥ 5. Moreover, a 3-SAT problem with m clauses can be represented by a coalition structure generation problem over a planar graph with O(m2) nodes. Importantly, our hardness result holds for a particular subclass of valuation functions, termed edge sum, where the value of each subset of nodes is simply determined by the sum of given weights of the edges in the induced subgraph

    Determination of partition coefficients and aqueous solubilities by reverse phase chromatography--II : Evaluation of partitioning and solubility models

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    Part I of this series examined the chemical and thermodynamic principles underlying the solubility and partitioning characteristics of non-electrolyte organic compounds in aqueous systems. It further considered those concepts which interrelate solubility and partitioning phenomena with retentive behavior in reverse phase liquid chromatography. Conceptual and predictive models which intercorrelate solubility, octanol/water partitioning, and reverse phase retention times were developed. Part II of the series evaluates the relative predictive abilities of these models for a wide range of different classes of organic pollutants. The results reveal that the reverse phase liquid chromatography models developed in Part I provide good estimates of octanol/water coefficients and aqueous solubilities from experimentally determined chromatographic retention times. Models structured to predict aqueous solubilities from octanol/water partition coefficient data were also found to provide reasonable estimates, but require as input physical and chemical parameters which are not readily accessible.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/26006/1/0000073.pd

    Sorption of hydrophobic compounds by sediments, soils and suspended solids--II. Sorbent evaluation studies

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    Concepts underlying the sorption of hydrophobic compounds and models for representation of observed equilibrium relationships were presented in Part I of this series. The earlier paper also summarized and evaluated major factors which affect the sorption of pollutants in natural environmental systems. This second part of the series presents a detailed summary and evaluation of the sorption of a particular class of hydrophobic pollutants, polychlorinated biphenyls (PCB), on a variety of different types of sediments, soils, suspended solids, and microorganisms. Equilibrium models described in Part I are used here to describe and analyze the PCB sorption data.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/25453/1/0000903.pd
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