In works dealing with capacities (fuzzy measures) and the Choquet integral on finite spaces, it is usually considered that all subsets of the universe are measureable. Hence, all functions are integrable in the sense ...
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(纸本)9789899507968
In works dealing with capacities (fuzzy measures) and the Choquet integral on finite spaces, it is usually considered that all subsets of the universe are measureable. Hence, all functions are integrable in the sense of Choquet. We consider the situation where some subsets are not measurable (not feasible), so that there are non-integrable functions. Since this is a severe limitation in applications, we study how to extend the Choquet integral to any function. Our results mainly deal with the case where the set of feasible subsets is a distributive lattice.
A model for a Choquet integral for arbitrary finite set systems is presented. The model includes in particular the classical model on the system of all subsets of a finite set. The general model associates canonical n...
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A model for a Choquet integral for arbitrary finite set systems is presented. The model includes in particular the classical model on the system of all subsets of a finite set. The general model associates canonical non-negative and positively homogeneous superadditive functionals with generalized belief functions relative to an ordered system, which are then extended to arbitrary valuations on the set system. It is shown that the general Choquet integral can be computed by a simple monge-type algorithm for so-called intersection systems, which include as a special case weakly union-closed families. Generalizing Lovasz' classical characterization, we give a characterization of the superadditivity of the Choquet integral relative to a capacity on a union-closed system in terms of an appropriate model of supermodularity of such capacities. (C) 2010 Elsevier B.V. All rights reserved.
An algebraic model generalizing submodular polytopes is presented, where modular functions on partially ordered sets take over the role of vectors in R-n. This model unifies various generalizations of combinatorial mo...
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An algebraic model generalizing submodular polytopes is presented, where modular functions on partially ordered sets take over the role of vectors in R-n. This model unifies various generalizations of combinatorial models in which the greedy algorithm and the monge algorithm are successful and generalizations of the notions of core and Weber set in cooperative game theory. As a further application, we show that an earlier model of ours as well as the algorithmic model of Queyranne, Spieksma and Tardella for the monge algorithm can be treated within the framework of usual matroid theory (on unordered ground-sets), which permits also the efficient algorithmic solution of the intersection problem within this model.
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