drawing graphs in two layers abstract let g u l e be a bipartite graph with vertex set u l and edge set e u x l a typical convention for drawing g is to put the vertices of u on a line and the vertices of l on a separate parallel line and then to represent edges by placing open straight line segments between the vertices that determine them in this convention a drawing is biplanar if edges do not cross and a subgraph of g is biplanar if it has a biplanar drawing the main results of this paper are the following 1 it is np complete to determine whether g has a biplanar subgraph with at least k edges 2 it is also np complete to determine whether g has such a subgraph when the positions for the vertices in either u or l are specified 3 when the positions of the vertices in both u and l are specified the problem can be solved in polynomial time by transformation to the longest ascending subsequence problem

solving the patient zero inverse problem by using generalized simulated annealing abstract identifying patient zero the initially infected source of a given outbreak is an important step in epidemiological investigations of both existing and emerging infectious diseases here the use of the generalized simulated annealing algorithm gsa to solve the inverse problem of finding the source of an outbreak is studied the classical disease natural histories susceptible infected si susceptible infected susceptible sis susceptible infected recovered sir and susceptible infected recovered susceptible sirs in a regular lattice are addressed both the position of patient zero and its time of infection are considered unknown the algorithm performance with respect to the generalization parameter q v and the fraction of infected nodes for whom infection was ascertained is assessed numerical experiments show the algorithm is able to retrieve the epidemic source with good accuracy even when is small but present no evidence to support that gsa performs better than its classical version our results suggest that simulated annealing could be a helpful tool for identifying patient zero in an outbreak where not all cases can be ascertained

an axiomatic approach to fuzzy cardinalities of finite fuzzy sets an axiomatic definition of fuzzy cardinalities for finite fuzzy sets defined by means of a convex fuzzy set on the natural numbers is presented in such a way that it includes the fuzzy cardinalities defined by authors like zadeh ralescu dubois wygralak and characterizes the cardinalities that fulfill the additivity property by means of the extended addition of fuzzy numbers such cardinalities result defined by two functions one nondecreasing and the other nonincreasing in a similar way to the scalar cardinality fuzzy sets and systems 110 2000 175

a branch and bound algorithm for minimizing the number of crossing arcs in bipartite graphs abstract acyclic directed graphs are widely used in many fields of economic and social sciences this has generated considerable interest in algorithms for drawing good maps of acyclic diagraphs the most important criterion to obtain a readable map of an acyclic graph is that of minimizing the number of crossing arcs in this paper we present a branch and bound algorithm for solving the problem of minimizing the number of crossing arcs in a bipartite graph computational results are reported on a set of randomly generated test problems

questions of cardinality of finite fuzzy sets abstract in this paper we focus our attention on finite fuzzy sets a complete simple and easily applicable cardinality theory for them is presented questions of equipotency and non classically understood cardinal numbers of finite fuzzy sets are discussed in detail also problems of arithmetical operations addition subtraction multiplication division and exponentiation on as well as ordering relation for those cardinals are carefully investigated

type fuzzy quantifiers determined by fuzzy measures defined on residuated lattices part ii permutation and isomorphism invariances abstract we study the properties of permutation invariance and isomorphism invariance of fuzzy quantifiers of type 1 1 defined using fuzzy measures and integrals these properties hold if fuzzy quantifiers are invariant with respect to permutations bijective mappings on the universe of discourse permutation invariance and with respect to bijections between different universes of discourse isomorphism invariance in other words fuzzy quantifiers with these properties are sensitive only to the cardinality of subsets of the universe of discourse and not to the individual nature of the elements of these subsets we characterize these properties by means of the corresponding properties of functionals used in the definition of the fuzzy quantifiers

on bipartite drawings and the linear arrangement problem the bipartite crossing number problem is studied and a connection between this problem and the linear arrangement problem is established a lower bound and an upper bound for the optimal number of crossings are derived where the main terms are the optimal arrangement values two polynomial time approximation algorithms for the bipartite crossing number are obtained the performance guarantees are o log n and o log2 n times the optimal respectively for a large class of bipartite graphs on n vertices no polynomial time approximation algorithm which could generate a provably good solution had been known for a tree a formula is derived that expresses the optimal number of crossings in terms of the optimal value of the linear arrangement and the degrees resulting in an o n1 6 time algorithm for computing the bipartite crossing number the problem of computing a maximum weight biplanar subgraph of an acyclic graph is also studied and a linear time algorithm for solving it is derived no polynomial time algorithm for this problem was known and the unweighted version of the problem had been known to be np hard even for planar bipartite graphs of degree at most 3

a probabilistic definition of a nonconvex fuzzy cardinality the existing methods to assess the cardinality of a fuzzy set with finite support are intended to preserve the properties of classical cardinality in particular the main objective of researchers in this area has been to ensure the convexity of fuzzy cardinalities in order to preserve some properties based on the addition of cardinalities such as the additivity property we have found that in order to solve many real world problems such as the induction of fuzzy rules in data mining convex cardinalities are not always appropriate in this paper we propose a possibilistic and a probabilistic cardinality of a fuzzy set with finite support these cardinalities are not convex in general but they are most suitable for solving problems and contrary to the generalizing opinion they are found to be more intuitive for humans their suitability relies mainly on the fact that they assume dependency among objects with respect to the property to be in a fuzzy set the cardinality measures are generalized to relative ones among pairs of fuzzy sets we also introduce a definition of the entropy of a fuzzy set by using one of our probabilistic measures finally a fuzzy ranking of the cardinality of fuzzy sets is proposed and a definition of graded equipotency is introduced

epidemic models and percolation the authors argue that local epidemic models with immunisation are in the same universality class as percolation cluster growth models and show that the static exponents are equal to all orders in the epsilon expansion they calculate the dynamic exponent nu t 1 1 28 epsilon o epsilon 2 and relate this to other exponents involving the chemical distance

inference by aggregation of evidence with applications to fuzzy probabilities abstract optimal aggregation plays an important role towards developing a theory of aggregation of fuzzy concepts based on non additive set functions and nonlinear choquet or fuzzy integrals this paper presents a framework supporting such aggregations and argues that such a framework is useful to decision making based on distributed sources of evidence

field theoretic formulation of an epidemic process with immunisation the author describes a model of the spread of an epidemic on a lattice in which sites may be infected by their neighbours after which they recover and are subsequently immune it corresponds to a modification of the directed percolation problem and also to a growth model of clusters the model is analysed using a continuum field theory both within a self consistent approximation and using an epsilon expansion below six transverse dimensions

stochastic lattice gas model describing the dynamics of the sirs epidemic process abstract we study a stochastic process describing the onset of spreading dynamics of an epidemic in a population composed of individuals of three classes susceptible s infected i and recovered r the stochastic process is defined by local rules and involves the following cyclic process s i r s sirs the open process s i r sir is studied as a particular case of the sirs process the epidemic process is analyzed at different levels of description by a stochastic lattice gas model and by a birth and death process by means of monte carlo simulations and dynamical mean field approximations we show that the sirs stochastic lattice gas model exhibit a line of critical points separating the two phases an absorbing phase where the lattice is completely full of s individuals and an active phase where s i and r individuals coexist which may or may not present population cycles the critical line that corresponds to the onset of epidemic spreading is shown to belong in the directed percolation universality class by considering the birth and death process we analyze the role of noise in stabilizing the oscillations

fuzzy integral representation abstract in this paper we give representation results for nonlinear functionals in arbitrary spaces as fuzzy integrals the main assumption is that of comonotonic maxitivity i e i f g if ig for comonotonic f g under the more restrictive condition of stochastic dominance we prove a more specific representation as a fuzzy integral with respect to a distorted probability measure we discuss the particular cases of possibility measures and upper fuzzy expectations some partial randon nikodym results are proved for both the choquet and the fuzzy integrals finally an example is shown where the fuzzy integral representation is useful

on the defect of complementarity of fuzzy measures it is well known that the fuzzy measures are non additive this means that the sum between the measure of a set and the measure of its complement is not equal with the measure of space as a consequence in this paper we introduce the concept of defect of complementarity also for large classes of fuzzy measures this defect of complementarity is calculated or estimated finally some applications and interpretations are presented

fuzzy sets the approximation of semisets abstract this paper deals with alternative set theory which substantially departs from classical set theory the main notion of it a semiset is proposed to model the same intuitive notion as fuzzy set but it is more general the reasonable tool for grasping semisets could be provided by fuzzy sets whose applicability is beyond discussion in the paper reasons for such an approximation are given and two ways how to provide it are proposed the fact that the membership function is modeled within the theory should be stressed at the end some problems and an indication for further investigation are discussed

fuzzy cardinality and the modeling of imprecise quantification abstract the concept of cardinality of a fuzzy set has received attention from several researchers and has been defined in several apparently independent manners a systematic investigation of this notion is performed which unifies and improves previous attempts the cardinality of a fuzzy set viewed as a fuzzy integer is related to scalar cardinality indices the closely related question of the probability of a fuzzy event is dealt with lastly the usefulness of fuzzy cardinality for meaning representation of statements or queries involving fuzzy linguistic quantifiers is emphasized

the set of fuzzy rational numbers and flexible querying a fuzzy bag is a bag in which each occurrence of an element is associated with a grade of membership this notion can be viewed as a generalization of the concepts of set fuzzy set and bag the set of fuzzy integers n f provides a general characterization in which all these different concepts are treated in a uniform way and can then be composed in the field of databases the use of fuzzy bags is motivated by their ability to manage both quantities and preferences however n f becomes too restricted a framework when dealing with queries based on difference or division operations so a more general structure based on the set of fuzzy relative integers z f in which exact differences can be performed has been first developed in this paper we carry on with this approach and we extend z f to the set of fuzzy rational numbers q f this context leads to define a closed system of multiplicative operations and allows to perform exact divisions applied to flexible querying of databases q f and the notion of division on fuzzy numbers allow to generalize the relational division they define a sound basis for queries involving ratios between quantities

fuzzy quantifiers defined over fuzzy domains abstract in this paper we introduce a generalization of the concept of fuzzy quantifier called fuzzy quantifiers over fuzzy domains the development of this generalization is motivated by the impossibility to satisfactorily define the important operation of relativization for standard fuzzy quantifiers we provide basic examples of these quantifiers and define and investigate also several semantic properties of them finally we prove a theorem saying among other things that all these two argument fuzzy quantifiers that possess properties of extension and conservativity are relativizations of one argument fuzzy quantifiers of this kind

critical behavior of the susceptible infected recovered model on a square lattice by means of numerical simulations and epidemic analysis the transition point of the stochastic asynchronous susceptible infected recovered model on a square lattice is found to be c0 0 176 500 510 where c is the probability a chosen infected site spontaneously recovers rather than tries to infect one neighbor this point corresponds to an infection recovery rate of c 1 c0 c0 4 665 713 and a net transmissibility of 1 c0 1 3 c0 0 538 4102 which falls between the rigorous bounds of the site and bond thresholds the critical behavior of the model is consistent with the two dimensional percolation universality class but local growth probabilities differ from those of dynamic percolation cluster growth as is demonstrated explicitly

a new lower bound for the bipartite crossing number with applications let g be a connected bipartite graph on n vertices we give a short proof using a variation of menger s theorem for a new lower bound which relates the bipartite crossing number of g denoted by bcr g to the edge connectivity properties of g the general lower bound implies a weaker version of a very recent result establishing a bisection based lower bound on bcr g which has algorithmic consequences moreover we show further applications of our general method to estimate bcr g for well structured families of graphs for which tight isoperimetric inequalities are available for hypercubes and 2 dimensional meshes the upper bounds asymptotically are within multiplicative factors of 4 and 2 from the lower bounds respectively the general lower bound also implies a lower bound involving eigenvalues of g

generalized cardinal numbers and operations on them abstract in this paper we present a cardinality theory for so called vaguely defined objects which are mild generalizations of fuzzy sets obtained by introducing lower and upper approximations of the membership functions the theory makes use of the sentential calculus in the infinite valued ukasiewicz logic and can be applied to fuzzy sets with arbitrary supports some kinds of twofold fuzzy sets partial sets etc the notion of equipotency of vaguely defined objects is studied in detail the resulting generalized cardinal numbers are convenient tools for describing the powers of vaguely defined objects in the second part of the paper basic operations on the generalized cardinals are defined and carefully investigated similarities and anomalies in comparison with the classical arithmetic of the usual cardinals are indicated

connectives and quantifiers in fuzzy sets abstract we discuss some of the issues involved in the selection of appropriate operators for implementing the union and intersection of fuzzy subsets we discuss the extension principles used in extending operators to fuzzy subsets we provide a structure for aggregating linguistic variables in the theory of approximate reasoning we look at mean operators on fuzzy subsets as well as the inclusion of importances in aggregation we discuss non monotone set operators we introduce the concept of linguistic quantifiers and look at some applications of these structures we are particularly concerned with their application to linguistic summarizers syllogistic reasoning and multi criteria decision making

a linear edge kernel for two layer crossing minimization the two layer crossing minimization problem tlcm given a bipartite graph g with n vertices and a positive integer k asks whether g has a two layer drawing with at most k crossings we consider a simple generalization of tlcm called leaf edge weighted tlcm lew tlcm where we allow positive weights on edges incident to leaves and show that this problem admits a kernel with o k edges provided that the given graph is connected as a straightforward consequence lew tlcm and hence tlcm has a fixed parameter algorithm that runs in 2 o k log k n o 1 time which improves on the previously best known algorithm with running time 2 o k 3 n

a graded approach to cardinal theory of finite fuzzy sets part i in this article we propose a fuzzy class relation assigning to each pair of finite fuzzy sets a degree to which they are equipollent which indicates that they have the same number of elements the concepts of fuzzy sets and fuzzy classes in the class of all sets in zfc are introduced and several standard relations and constructions such as the fuzzy power set and exponentiation are defined a functional approach to the cardinal theory of finite fuzzy sets based on graded equipollence is shown and a relation to generalized cardinals and wygralak s cardinal theory of finite fuzzy sets defined over triangular norms is demonstrated

fuzzy cardinals based on the generalized equality of fuzzy subsets the purpose of this paper is to present a new concept of the cardinality of finite fuzzy subsets which is based on the notion of the generalized equality the proposed idea will be compared with several approaches which were earlier offered in fuzzy literature using the new definition of the fuzzy cardinality we next introduce a notion of fuzzy semicardinals i e we construct fuzzy natural numbers defining degrees to which a finite fuzzy subset has respectively at most least and less than more than k elements various properties of different fuzzy cardinals and semicardinals as well as some interdependences between them are also given

on triangular norm based generalized cardinals and singular fuzzy sets there are three basic types of triangular norm based generalized cardinals of fuzzy sets namely generalized fgcounts flcounts and fecounts all of them are convex fuzzy sets of usual cardinal numbers our attention will be focused on generalized fecounts if nonstrict archimedean triangular norms are involved generalized fecounts of many fuzzy sets become the zero function those singular fuzzy sets being totally dissimilar to any set of any cardinality and their properties are the subject of this paper a relationship between singularity and fuzziness measures suited to fuzzy sets with complements induced by a strict negation will also be discussed

measuring properties of fuzzy sets a general technique and its use in fuzzy query evaluation abstract a fuzzy set can be viewed as a weighted max combination or as a convex combination of a family of nested ordinary sets the latter point of view allows the interpretation of a fuzzy set as a consonant random set then any numerical set function applied to a fuzzy set leads to a result which is a random number the expectation attached to this random number provides a scalar evaluation of the fuzzy set with respect to the considered set function this approach is very general and can be used for various kinds of measurements cardinality confidence measures attached to the occurrence of events diameter or perimeter of a fuzzy region distance between two sets etc also the fuzziness of the set to evaluate naturally induces a fuzzy set of possible evaluations and expected values of the lower and upper bounds of this fuzzy set of numbers can be computed the weighted max combination view which leads to another type of scalar evaluation is also briefly discussed in this paper the above approach is more particularly applied to the treatment of queries asking for some global evaluation of a set of items specified by means of a fuzzy property this kind of query can be managed by languages like sql in case of a crisp set of items when this set is fuzzily specified a scalar answer can still be provided as well as a fuzzy valued or interval valued one if we want to keep track of the effect of the fuzziness of the query the case where the data base contains precise information but also the more general case of fuzzy information data bases are considered

cardinality quantifiers and the aggregation of fuzzy criteria abstract in this paper we are interested in counting the number of elements of a fuzzy set we study two concepts of fuzzy cardinality one which gives the answer as a fuzzy set or fuzzy number and the other which gives an ordinary integer relationships between these concepts and others which have been studied in the literature e g sigma count are also investigated some properties of the new concepts are studied such as fuzzy convexity monotonicity and additivity the concepts of fuzzy cardinality are important for a better understanding of fuzzy quantified rules of the form qx s are a or qa s are b where q is a fuzzy quantifier and a b are fuzzy sets we study such rules as well as aggregation rules of the form q of a1 ap contain x

the threshold of coexistence and critical behaviour of a predator prey cellular automaton we study a probabilistic cellular automaton to describe two population biology problems the threshold of species coexistence in a predator prey system and the spreading of an epidemic in a population by carrying out mean field approximations and numerical simulations we obtain the phase boundaries thresholds related to the transition between an active state where prey and predators present a stable coexistence and a prey absorbing state the numerical estimates for the critical exponents show that the transition belongs to the directed percolation universality class in the limit where the cellular automaton maps into a model for the spreading of an epidemic with immunization we observe a crossover from directed percolation class to the dynamic percolation class patterns of growing clusters related to species coexistence and spreading of epidemic are shown and discussed

fuzzy sets with triangular norms and their cardinality theory the most advanced and adequate approach to the question of cardinality of a fuzzy set seems to be that offering a fuzzy perception of cardinality the resulting convex fuzzy sets of usual cardinals of nonnegative integers in the finite case are then called generalized cardinal numbers three types of them are of special interest and importance namely fgcounts flcounts and fecounts in this paper first we show that their original forms are suitable only for fuzzy sets with the classical min and max operations second we propose an appropriate generalization to fuzzy sets with triangular norms and conorms further we investigate the resulting generalized fgcounts flcounts and fecounts from the viewpoint of the corresponding equipotency relations inequalities and arithmetical operations

cardinality concepts for type two fuzzy sets in this paper we define generalized concepts of cardinality of a fuzzy valued function and obtained some properties of these new concepts also we investigate examples for the calculation of two generalized cardinality of fuzzy valued functions and compared with concepts of cardinality of a fuzzy set and a fuzzy valued function

on optimal linear arrangements of trees abstract in this paper we investigate the problem of finding a labeling of the vertices of an undirected tree with distinct integers such that the sum of weights of all edges is minimized the weight of an edge is the absolute value of the difference between the labelings of its end points we give an o n algorithm where is approximately log 3 log2 1 6 for solving this problem

aggregation of owa operators inspired by the real needs of group decision problems aggregation of ordered weighted averaging owa operators is studied and discussed our results can be applied for data acting on any real interval such as the standard scales 0 1 and 0 infty bipolar scales 1 1 and mathbb r infty infty etc a direct aggregation is shown to be rather restrictive allowing the convex combinations to be considered only except the case of dimension n 2 more general is the approach based on the aggregation of related cumulative weighting vectors the piecewise linearity of owa operators allows us to consider bilinear forms of aggregation of related weighting vectors several interesting examples yielding the link between the aggregation of owa operators and the related andness and orness measures are also included some possible applications and generalizations of our results are also discussed

a discussion on fuzzy cardinality and quantification some applications in image processing in this paper we discuss on some different representations of the cardinality of a fuzzy set and their use in fuzzy quantification we have considered the widely employed sigma count fuzzy numbers and gradual numbers gradual numbers assign numbers to values of a relevance scale typically 0 1 contrary to sigma count and fuzzy numbers they provide a precise representation of the cardinality of a fuzzy set we illustrate our claims by calculating the cardinality of the fuzzy set of pixels that match a certain fuzzy color in an image for that purpose we consider fuzzy color spaces previously defined by the authors consisting of a collection of fuzzy sets providing a suitable conceptual quantization with soft boundaries of crisp color spaces finally we show the suitability of our approaches to fuzzy quantification for different applications in image processing first the calculation of histograms second the definition of the notion of dominant fuzzy color and the calculation of the degree to which we can say that a certain color is dominant in an image

epidemic outbreaks on random voronoi delaunay triangulations abstract we study epidemic outbreaks on random delaunay triangulations by applying the asynchronous sir susceptible infected removed dynamics coupled to two dimensional voronoi delaunay triangulations in order to investigate the critical behavior of the model we obtain the cluster size distribution by using newman ziff algorithm allowing to simulate random inhomogeneous lattices and measure any desired observable related to percolation we numerically calculate the order parameter defined as the wrapping cluster density the mean cluster size and binder cumulant ratio defined for percolation in order to estimate the epidemic threshold our findings suggest that the system falls into two dimensional dynamic percolation universality class and the quenched random disorder is irrelevant in agreement with results for classical percolation

a framework for fuzzy quantification models analysis a framework for description of fuzzy quantification models is presented within this framework the fuzzy quantified statements evaluation problem is described as the compatibility between the fuzzy quantifier and a fuzzy cardinality or a fuzzy aggregation measure a list of desirable properties for quantification models is presented and those models that fit the framework are confronted with it

on the relation between fuzzy and generalized quantifiers fuzzy quantified sentences have been widely applied for modeling statements in natural language since they allow to represent a broad type of intrinsically imprecise quantified statements that are usually linguistically expressed two different frameworks arise in order to model this imprecision in quantification zadeh s fuzzy quantifiers and the fuzzification of generalized quantification for both models several approaches have been developed for assessing the degree of satisfiability of fuzzy quantified sentences we present and compare both models concluding that although the concept of quantifier in each model has not the same semantics we can establish a connection between them and between the evaluation methods in both theories

a graded approach to cardinal theory of finite fuzzy sets part ii fuzzy cardinality measures and their relationship to graded equipollence abstract in this article we propose an axiomatic system for fuzzy cardinality measures referred to as fuzzy c measures for short assigning to each finite fuzzy set a generalized cardinal that expresses the number of elements that the fuzzy set contains the system generalizes an axiomatic system introduced by j casasnovas and j torrens 2003 we show that each fuzzy c measure is determined by two appropriate homomorphisms between the reducts of residuated dually residuated rdr lattices for linearly ordered rdr lattices we prove that each fuzzy c measure is a product of a non decreasing and a non increasing fuzzy c measure which indicates that there is a close relation between fuzzy c measures and fgcount flcount and fecount provided by l a zadeh 1983 and generalized by m wygralak 2001 finally the relationship of fuzzy c measures to graded equipollence introduced in the first part of this contribution is analyzed

susceptible infected recovered and susceptible exposed infected models two stochastic epidemic lattice models the susceptible infected recovered and the susceptible exposed infected models are studied on a cayley tree of coordination number k the spreading of the disease in the former is found to occur when the infection probability b is larger than bc k 2 k 1 in the latter which is equivalent to a dynamic site percolation model the spreading occurs when the infection probability p is greater than pc 1 k 1 we set up and solve the time evolution equations for both models and determine the final and time dependent properties including the epidemic curve we show that the two models are closely related by revealing that their relevant properties are exactly mapped into each other when p b k k 1 b these include the cluster size distribution and the density of individuals of each type quantities that have been determined in closed forms

susceptible infected recovered model with recurrent infection abstract we analyze a stochastic lattice model describing the spreading of a disease among a community composed by susceptible infected and removed individuals a susceptible individual becomes infected catalytically an infected individual may spontaneously either become recovered that is acquire a permanent immunization or become again susceptible the critical properties including the phase diagram is obtained by means of mean field theories as well as numerical simulations the model is found to belong to the universality class of dynamic percolation except when the recovering rate vanishes in which case the model belongs to the directed percolation universality class

twofold fuzzy sets and rough sets some issues in knowledge representation abstract this paper deals with the representation of sets where the membership of some elements may be ill known rather than just a matter of degree as in a fuzzy set the notion of a twofold fuzzy set is introduced when the relevant information for determining the membership status is incomplete a twofold fuzzy set is made of a nested pair of fuzzy sets the one which gathers the elements which more or less necessarily belong and the one which gathers the elements which more or less possibly belong twofold fuzzy sets are compared from a frontal and from a semantical point of view with other proposals and particularly with the notion of a rough set recently introduced by pawlak set operations of twofold fuzzy sets are discussed and the cardinality of a twofold fuzzy set is defined twofold fuzzy relations are also introduced finally various applications of twofold fuzzy sets in knowledge representation are briefly discussed

two layer planarization parameterized by feedback edge set given an undirected graph g and an integer k 0 the np hard 2 layer planarization problem asks whether g can be transformed into a forest of caterpillar trees by removing at most k edges 2 layer planarization was known to be fixed parameter tractable with respect to the parameter k the state of the art is an o 3 562 k k g time search tree algorithm and an o k size problem kernel since transforming g into a forest of caterpillar trees requires breaking every cycle the size f of a minimum feedback edge set is a natural parameter with f k we improve on previous fixed parameter tractability results with respect to k by presenting new polynomial time data reduction rules leading to a problem kernel with o f vertices and edges and a new search tree based algorithm we expect f to be significantly smaller than k for a wide range of input instances