a complete complexity classification of the role assignment problem in social network theory a society is often represented by a simple graph g where vertices stand for individuals and edges represent relationships between those individuals the description of the social network is tried to be simplified by assigning roles to the individuals such that the neighborhood relation is preserved formally for a fixed graph r we ask for a vertex mapping r vg vr such that r ng u nr r u for all u vg if such a mapping exists the graph g is called r role assignable and the corresponding decision problem is called the r role assignment problem kristiansen and telle conjectured that the r role assignment problem is an np complete problem for any simple connected graph r on at least three vertices in this paper we prove their conjecture in addition we determine the computational complexity of the role assignment problem for nonsimple and disconnected role graphs as these are considered in social network theory as well

on the f divergence for non additive measures the f divergence evaluates the dissimilarity between two probability distributions defined in terms of the radon nikodym derivative of these two probabilities the f divergence generalizes the hellinger distance and the kullback leibler divergence among other divergence functions in this paper we define an analogous function for non additive measures we discuss them for distorted lebesgue measures and give examples examples focus on the hellinger distance

an axiomatic approach to the definition of the entropy of a discrete choquet capacity to extend the classical shannon entropy to nonadditive measures marichal recently introduced the concept of generalized entropy for discrete choquet capacities we provide a first axiomatization of this new concept on the basis of three axioms the symmetry property a boundary condition for which the entropy reduces to the shannon entropy and a generalized version of the well known recursivity property we also show that this generalized entropy fulfills several properties considered as requisites for defining an entropy like measure lastly we provide an interpretation of it in the framework of aggregation by the discrete choquet integral

compromise principle based methods of identifying capacities in the framework of multicriteria decision analysis abstract the basic aim of the compromise principle employed in this paper is to seek the capacity by which each alternative has a relatively equal chance to reach as close as possible to its highest reachable overall evaluation according to the compromise principle three types of capacity identification methods the simple arithmetic average based compromise method the least squares based compromise method and the linear programming based compromise method are proposed the input information required for the compromise principle based identification methods consists of the preference information with respect to the decision criteria and the partial evaluations of the alternatives provided by the decision maker an illustrative example is given to show the processes of the proposed methods and a comparison analysis with the maximum entropy principle based identification method is also presented

determination of weights of interacting criteria from a reference set abstract in this paper we present a model allowing to determine the weights related to interacting criteria this is done on the basis of the knowledge of a partial ranking over a reference set of alternatives prototypes a partial ranking over the set of criteria and a partial ranking over the set of interactions between pairs of criteria

2 role assignments on triangulated graphs if g is a graph a k role assignment is a function mapping each vertex into a role a positive integer 1 2 k so that if x and y have the same role then the sets of roles assigned to their neighbors are the same a graph is called a triangulated graph if it contains no chord less cycle of four or more vertices one interesting type of triangulated graph is the indifference graph that is a graph for which we can find a function on its vertex set so that if x and y are adjacent then their assigned function values are close we study 2 role assignments for triangulated graphs we provide a greedy algorithm for finding a 2 role assignment on a connected nonbipartite triangulated graph with at most one pendant vertex we characterize indifference graphs that have a 2 role assignment

variational inequalities for combined control and stopping we study the variational inequality associated with the combined stochastic control problem and establish the existence of a unique viscosity solution without uniform ellipticity an application to the quasi variational inequality is given

on equivalence classes of fuzzy connectives the case of fuzzy integrals an equivalence relation between connectives or operators in the framework of multiple criteria decision making is introduced two operators are said to be equivalent if they lead to the same ranking of alternatives some general results to find equivalence classes of operators are given through the concept of level surfaces in a second part these results are applied to the case of discrete fuzzy integrals which are considered here as n place operators first some general results on the situation of fuzzy integrals among fuzzy operators are given and then the equivalence classes of fuzzy integrals

k intolerant capacities and choquet integrals we define an aggregation function to be at most k intolerant if it is bounded from above by its kth lowest input value applying this definition to the discrete choquet integral and its underlying capacity we introduce the concept of k intolerant capacities which when varying k from 1 to n cover all the possible capacities on n objects just as the concepts of k additive capacities and p symmetric capacities have been previously introduced essentially to overcome the problem of computational complexity of capacities k intolerant capacities are proposed here for the same purpose but also for dealing with intolerant or tolerant behaviors of aggregation we also introduce axiomatically indices to appraise the extent to which a given capacity is k intolerant and we apply them on a particular recruiting problem

computing role assignments of chordal graphs in social network theory a simple graph g is called k role assignable if there is a surjective mapping that assigns a number from 1 k called a role to each vertex of g such that any two vertices with the same role have the same sets of roles assigned to their neighbors the decision problem whether such a mapping exists is called the k role assignment problem this problem is known to be np complete for any fixed k 2 in this paper we classify the computational complexity of the k role assignment problem for the class of chordal graphs we show that for this class the problem can be solved in linear time for k 2 but remains np complete for any k 3 this generalizes earlier results by sheng and answers her open problem

indicators for the characterization of discrete choquet integrals ordered weighted averaging owa operators and their extensions are powerful tools used in numerous decision making problems this class of operator belongs to a more general family of aggregation operators where each operator of this family is understood as a discrete choquet integral aggregation operators are usually characterized by indicators in this article four indicators usually associated with the owa operator are extended to the discrete choquet integral namely the degree of balance the divergence the variance indicator and renyi entropies all of these indicators are considered from a local and a global perspective linearity of indicators for linear combinations of capacities is investigated and to illustrate the usefulness of results indicators of the probabilistic ordered weighted averaging powa operator are derived finally a numerical example is provided to discuss the results in a specific context

entropy of capacities on lattices and set systems we propose a definition for the entropy of capacities defined on lattices classical capacities are monotone set functions and can be seen as a generalization of probability measures capacities on lattices address the general case where the family of subsets is not necessarily the boolean lattice of all subsets our definition encompasses the classical definition of shannon for probability measures as well as the entropy of marichal defined for classical capacities some properties and examples are given

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

graph classes with structured neighborhoods and algorithmic applications given a graph in any of the following graph classes trapezoid graphs circular permutation graphs convex graphs dilworth k graphs k polygon graphs circular arc graphs and complements of k degenerate graphs we show how to compute decompositions with the number of d neighborhoods bounded by a polynomial of the input size combined with results of bui xuan telle and vatshelle 2013 1 this leads to polynomial time algorithms for a large class of locally checkable vertex subset and vertex partitioning problems on all of these graph classes the boolean width of a graph is related to the number of 1 neighborhoods and our results imply that any of these graph classes have boolean width o logn

equivalent representations of set functions this paper introduces four alternative representations of a set function the ma bius transformation the co ma bius transformation and the interactions between elements of any subset of a given set as extensions of shapley and banzhaf values the links between the five equivalent representations of a set function are emphasized in this presentation

entropy of bi capacities in the context of multicriteria decision making whose aggregation process is based on the choquet integral bi capacities can be regarded as a natural extension of capacities when the underlying evaluation scale is bipolar the notion of entropy recently generalized to capacities to measure their uniformity is now extended to bi capacities we show that the resulting entropy measure has a very natural interpretation in terms of the choquet integral and satisfies many natural properties that one would expect from an entropy measure

martingale approach to stochastic differential games of control and stopping we develop a martingale approach for studying continuous time stochastic differential games of control and stopping in a non markovian framework and with the control affecting only the drift term of the state process under appropriate conditions we show that the game has a value and construct a saddle pair of optimal control and stopping strategies crucial in this construction is a characterization of saddle pairs in terms of pathwise and martingale properties of suitable quantities

utility maximization with discretionary stopping utility maximization problems of mixed optimal stopping control type are considered which can be solved by reduction to a family of related pure optimal stopping problems sufficient conditions for the existence of optimal strategies are provided in the context of continuous time ito process models for complete markets the mathematical tools used are those of optimal stopping theory continuous time martingales convex analysis and duality theory several examples are solved explicitly including one which demonstrates that optimal strategies need not always exist

random generation of k interactive capacities abstract the theory of capacities provides powerful formal methodology to account for criteria dependencies in multiple criteria decision problems the discrete choquet and sugeno integrals aggregate criteria valuations accounting for criteria synergies and redundancies we address an important problem of randomly generating capacities of special classes for simulation studies and for capacity learning through evolutionary algorithms we discuss two efficient methods suitable for k interactive capacities the results are supported by the extensive numerical evidence and provide a useful tool for large scale simulations

minimum variance capacity identification abstract in the framework of multi criteria decision making whose aggregation process is based on the choquet integral we present a maximum entropy like method enabling to determine if it exists the least specific capacity compatible with the initial preferences of the decision maker the proposed approach consists in solving a strictly convex quadratic program whose objective function is equivalently either the opposite of a generalized entropy measure or the variance of the capacity the application of the proposed approach is illustrated on two examples

the application of fuzzy integrals in multicriteria decision making abstract this paper presents a synthesis on the application of fuzzy integrals as an innovative tool for criteria aggregation in decision problems the main point is that fuzzy integrals are able to model interaction between criteria in a flexible way the methodology has been elaborated mainly in japan and has been applied there successfully in various fields such as design reliability evaluation of goods etc it seems however that this technique is still very little known in europe it is one of the aims of this review to disseminate this emerging technology in many industrial fields

an axiomatization of entropy of capacities on set systems we present an axiomatization of the entropy of capacities defined on set systems which are not necessarily the whole power set but satisfy a condition of regularity this entropy encompasses the definition of marichal and roubens for the entropy of capacities its axiomatization is in the spirit of the one of faddeev for the classical shannon entropy in addition we present also an axiomatization of the entropy for capacities proposed by dukhovny

fuzzy integral based information fusion for classification of highly confusable non speech sounds acoustic event classification may help to describe acoustic scenes and contribute to improve the robustness of speech technologies in this work fusion of different information sources with the fuzzy integral fi and the associated fuzzy measure fm are applied to the problem of classifying a small set of highly confusable human non speech sounds as fi is a meaningful formalism for combining classifier outputs that can capture interactions among the various sources of information it shows in our experiments a significantly better performance than that of any single classifier entering the fi fusion module actually that fi decision level fusion approach shows comparable results to the high performing svm feature level fusion and thus it seems to be a good choice when feature level fusion is not an option we have also observed that the importance and the degree of interaction among the various feature types given by the fm can be used for feature selection and gives a valuable insight into the problem

on the entropy of fuzzy measures fuzzy measures provides a structure for modeling the knowledge available about variables whose values are unknown and uncertain a large class of different types of uncertainty can be represented in this framework in this work we provide a measure of entropy that can be used to calculate the amount of uncertainty associated with a fuzzy measure

computing role assignments of split graphs a k role assignment of a simple graph g is a surjective function r v g 1 k where r u u n u r v v n v for every pair u v v g with r u r v this concept appears in the context of social networks it is known that the problem of finding a 2 role assignment of chordal graphs can be solved in linear time and that deciding whether a graph of this class admits a k role assignment for any fixed k 3 is np complete in this work we investigate this problem on split graphs a subclass of chordal graphs we characterize the split graphs admitting a 3 role assignment such result leads to a linear time algorithm for this case furthermore we prove that deciding whether a split graph has a k role assignment is np complete for any fixed k 4

entropy of discrete choquet capacities we introduce a measure of entropy for any discrete choquet capacity and we interpret it in the setting of aggregation by the choquet integral

2 additive capacity identification methods from multicriteria correlation preference information the essential role of the particular families of capacities and the capacity identification methods is to help the decision maker to deal with the exponential complexity inherent in the construction process of the capacity the 2 additive capacities appear to be the most popular among the particular families of capacities since they permit to model interactions between criteria while preserving simplicity besides the preference with respect to the decision criteria most of the capacity identification methods also need to provide the desired overall evaluations of the decision alternatives in the learning set which is a time consuming task for the decision maker in this paper we propose some models to identify 2 additive capacities only from a kind of refined preference information with respect to the decision criteria called the multicriteria correlation preference information mccpi the mccpi is a group of 2 d preference information which can be described and obtained by the refined diamond diagram the common principle of the proposed models is to minimize the different kinds of deviations between the mccpi and the most desired 2 additive capacity ies a multicriteria decision making example is presented to show the feasibility of the proposed methods and a 2 d scale of the mccpi is also introduced in the further discussion of the illustrative example

an axiomatic approach of the discrete choquet integral as a tool to aggregate interacting criteria the most often used operator to aggregate criteria in decision making problems is the classical weighted arithmetic mean however in many problems the criteria considered interact and a substitute to the weighted arithmetic mean has to be adopted we show that under rather natural conditions the discrete choquet integral is an adequate aggregation operator that extends the weighted arithmetic mean by taking into consideration of the interaction among criteria the axiomatic that supports the choquet integral is presented and an intuitive approach is proposed as well

an axiomatic approach of the discrete sugeno integral as a tool to aggregate interacting criteria in a qualitative framework we present a model allowing us to aggregate decision criteria when the available information is of a qualitative nature the use of the sugeno integral as an aggregation function is justified by an axiomatic approach it is also shown that the mutual preferential independence of criteria reduces the sugeno integral to a dictatorial aggregation

a nonconvex singular stochastic control problem and its related optimal stopping boundaries equivalences are known between problems of singular stochastic control ssc with convex performance criteria and related questions of optimal stopping see for example karatzas and shreve siam j control optim 22 1984 pp 856 877 the aim of this paper is to investigate how far connections of this type generalize to a nonconvex problem of purchasing electricity where the classical equivalence breaks down we provide alternative connections to optimal stopping problems we consider a nonconvex infinite time horizon ssc problem whose state consists of an uncontrolled diffusion representing a real valued commodity price and a controlled increasing bounded process representing an inventory we analyze the geometry of the action and inaction regions by characterizing their optimal boundaries unlike the case of convex ssc problems we find that the optimal boundaries may be both reflecting and repelling and it is natural to interpret the problem as one of ssc with discretionary stopping

a theory of fuzzy measures representations the choquet integral and null sets abstract some theorems of t murofushi and m sugeno fuzzy sets and systems 29 1989 201 227 concerning representation of fuzzy measures and the choquet integral are generalized it is shown that if there is a certain relation between two measurable functions then the choquet integral is additive for these two functions in addition this article discusses null sets with respect to fuzzy measures and also fuzzy measures defined on a class which is not a algebra