fixed parameter algorithms for kemeny rankings the computation of kemeny rankings is central to many applications in the context of rank aggregation given a set of permutations votes over a set of candidates one searches for a consensus permutation that is closest to the given set of permutations unfortunately the problem is np hard we provide a broad study of the parameterized complexity for computing optimal kemeny rankings besides the three obvious parameters number of votes number of candidates and solution size called kemeny score we consider further structural parameterizations more specifically we show that the kemeny score and a corresponding kemeny ranking of an election can be computed efficiently whenever the average pairwise distance between two input votes is not too large in other words kemeny score is fixed parameter tractable with respect to the parameter average pairwise kendall tau distance d a we describe a fixed parameter algorithm with running time 16 d a email protected poly moreover we extend our studies to the parameters maximum range and average range of positions a candidate takes in the input votes whereas kemeny score remains fixed parameter tractable with respect to the parameter maximum range it becomes np complete in the case of an average range of two this excludes fixed parameter tractability with respect to the parameter average range unless p np finally we extend some of our results to votes with ties and incomplete votes where in both cases one no longer has permutations as input

oscillation for solutions of nonlinear neutral differential equations with impulses abstract this paper is concerned with nonlinear neutral differential equations with impulses of the form some oscillation criteria for solutions of this equation are established an interesting example is also given which illustrates that impulses play an important role in giving rise to the oscillation of equations

using extension sets to aggregate partial rankings in a flexible setting this paper deals with the rank aggregation problem in a general setting in particular we approach the problem for any kind of ranking complete or incomplete and with or without ties the underlying idea behind our approach is to take into account the so called extension set of a ranking that is the set of permutations that are compatible with the given ranking in this way we aim to manage the uncertainty inherent to this problem when not all the items are ranked and or when some items are equally preferred we develop two approaches a general one based on this idea and a constrained version in which the extension set is limited by not allowing non ranked items to be placed in an existent bucket of tied items we test our proposal by coupling it with two different algorithms we formalize our approaches mathematically and also carry out an extensive experimental evaluation by using 22 datasets the results show that the use of our extension sets based approaches to compute the precedence matrix clearly outperforms the standard way of computing the preference matrix by only using the information explicitly provided by the rankings in the sample

statistical ranking and combinatorial hodge theory we propose a technique that we call hodgerank for ranking data that may be incomplete and imbalanced characteristics common in modern datasets coming from e commerce and internet applications we are primarily interested in cardinal data based on scores or ratings though our methods also give specific insights on ordinal data from raw ranking data we construct pairwise rankings represented as edge flows on an appropriate graph our statistical ranking method exploits the graph helmholtzian which is the graph theoretic analogue of the helmholtz operator or vector laplacian in much the same way the graph laplacian is an analogue of the laplace operator or scalar laplacian we shall study the graph helmholtzian using combinatorial hodge theory which provides a way to unravel ranking information from edge flows in particular we show that every edge flow representing pairwise ranking can be resolved into two orthogonal components a gradient flow that represents the l 2 optimal global ranking and a divergence free flow cyclic that measures the validity of the global ranking obtained if this is large then it indicates that the data does not have a good global ranking this divergence free flow can be further decomposed orthogonally into a curl flow locally cyclic and a harmonic flow locally acyclic but globally cyclic these provides information on whether inconsistency in the ranking data arises locally or globally when applied to statistical ranking problems hodge decomposition sheds light on whether a given dataset may be globally ranked in a meaningful way or if the data is inherently inconsistent and thus could not have any reasonable global ranking in the latter case it provides information on the nature of the inconsistencies an obvious advantage over the np hardness of kemeny optimization is that hodgerank may be easily computed via a linear least squares regression we also discuss connections with well known ordinal ranking techniques such as kemeny optimization and borda count from social choice theory

the attenuated space poset aq n m abstract in this paper we study the incidence algebra t of the attenuated space poset a q n m we consider the following topics we consider some generators of t the raising matrix r the lowering matrix l and a certain diagonal matrix k we describe some relations among r l k we put these relations in an attractive form using a certain matrix s in t we characterize the center z t using z t we relate t to the quantum group u sl 2 with 2 q we consider two elements a a in t of a certain form we find necessary and sufficient conditions for a a to satisfy the tridiagonal relations let w denote an irreducible t module we find necessary and sufficient conditions for the above a a to act on w as a leonard pair

linearized oscillation theory for a nonlinear nonautonomous delay differential equation oscillation properties of two following equations are compared a scalar nonlinear delay differential equation y t k 1m rk t f k y hk t 0 with rk t o hk t t and a linear delay differential equation x t k 1m rk t x hk t 0 coefficients rk t and delays are not assumed to be continuous as an application explicit oscillation and nonoscillation conditions are established for nonlinear equations arising in population dynamics

two linear transformations each tridiagonal with respect to an eigenbasis of the other let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v satisfying both conditions below 1 i there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal 2 ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal we call such a pair a leonard pair on v refining this notion a bit we introduce the concept of a leonard system we give a complete classification of leonard systems integral to our proof is the following result we show that for any leonard pair a a on v there exists a sequence of scalars taken from k such that both 0 a a2a aa a a a2 aa a a a 0 a a 2a a aa aa 2 a a aa a where r s means rs sr the sequence is uniquely determined by the leonard pair if the dimension of v is at least 4 we conclude by showing how leonard systems correspond to q racah and related polynomials from the askey scheme

deterministic pivoting algorithms for constrained ranking and clustering problems we consider ranking and clustering problems related to the aggregation of inconsistent information in particular rank aggregation weighted feedback arc set in tournaments consensus and correlation clustering and hierarchical clustering ailon et al ailon n m charikar a newman 2005 aggregating inconsistent information ranking and clustering proc 37th annual acm sympos theory comput stoc 05 684 693 ailon and charikar ailon n m charikar 2005 fitting tree metrics hierarchical clustering and phylogeny proc 46th annual ieee sympos foundations comput sci focs 05 73 82 and ailon ailon n 2007 aggregation of partial rankings p ratings and top m lists proc 18th annual acm siam sympos discrete algorithms soda 07 415 424 proposed randomized constant factor approximation algorithms for these problems which recursively generate a solution by choosing a random vertex as pivot and dividing the remaining vertices into two groups based on the pivot vertex in this paper we answer an open question in these works by giving deterministic approximation algorithms for these problems the analysis of our algorithms is simpler than the analysis of the randomized algorithms in addition we consider the problem of finding minimum cost rankings and clusterings that must obey certain constraints e g an input partial order in the case of ranking problems which were introduced by hegde and jain hegde r k jain 2006 personal communication we show that the first type of algorithms we propose can also handle these constrained problems in addition we show that in the case of a rank aggregation or consensus clustering problem if the input rankings or clusterings obey the constraints then we can always ensure that the output of any algorithm obeys the constraints without increasing the objective value of the solution

hybrid sampling based clustering ensemble with global and local constitutions among a number of ensemble learning techniques boosting and bagging are the most popular sampling based ensemble approaches for classification problems boosting is considered stronger than bagging on noise free data set with complex class structures whereas bagging is more robust than boosting in cases where noise data are present in this paper we extend both ensemble approaches to clustering tasks and propose a novel hybrid sampling based clustering ensemble by combining the strengths of boosting and bagging in our approach the input partitions are iteratively generated via a hybrid process inspired by both boosting and bagging then a novel consensus function is proposed to encode the local and global cluster structure of input partitions into a single representation and applies a single clustering algorithm to such representation to obtain the consolidated consensus partition our approach has been evaluated on 2 d synthetic data collection of benchmarks and real world facial recognition data sets which show that the proposed technique outperforms the existing benchmarks for a variety of clustering tasks

linearized oscillation of nonlinear impulsive delay differential equations abstract the main result of this paper is to show oscillations of nonlinear impulsive delay differential equations are equivalent to those of corresponding linear impulsive delay differential equations the results of this paper generalize some well known results in the literature

a linearized oscillation result for odd order neutral delay differential equations abstract consider the odd order nonlinear neutral equation d n dt n x t p t g x t q t h x t 0 we establish a linearized oscillation result of this equation when p t 1 which answers partially an open problem proposed by gyori and ladas

predicting link directions via a recursive subgraph based ranking abstract link directions are essential to the functionality of networks and their prediction is helpful toward a better knowledge of directed networks from incomplete real world data we study the problem of predicting the directions of some links by using the existence and directions of the rest of links we propose a solution by first ranking nodes in a specific order and then predicting each link as stemming from a lower ranked node and pointing toward a higher ranked one the proposed ranking method works recursively by utilizing local indicators on multiple scales each corresponding to a subgraph extracted from the original network experiments on real networks show that the directions of a substantial fraction of links can be correctly recovered by our method which outperforms either purely local or global methods

some optimal inapproximability results we prove optimal up to an arbitrary e 0 inapproximability results for max e k sat for k 3 maximizing the number of satisfied linear equations in an over determined system of linear equations modulo a prime p and set splitting as a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously in particular for max e2 sat max cut max di cut and vertex cover

aggregating inconsistent information ranking and clustering we address optimization problems in which we are given contradictory pieces of input information and the goal is to find a globally consistent solution that minimizes the extent of disagreement with the respective inputs specifically the problems we address are rank aggregation the feedback arc set problem on tournaments and correlation and consensus clustering we show that for all these problems and various weighted versions of them we can obtain improved approximation factors using essentially the same remarkably simple algorithm additionally we almost settle a long standing conjecture of bang jensen and thomassen and show that unless np bpp there is no polynomial time algorithm for the problem of minimum feedback arc set in tournaments

evaluating combinational illumination estimation methods on real world images illumination estimation is an important component of color constancy and automatic white balancing a number of methods of combining illumination estimates obtained from multiple subordinate illumination estimation methods now appear in the literature these combinational methods aim to provide better illumination estimates by fusing the information embedded in the subordinate solutions the existing combinational methods are surveyed and analyzed here with the goals of determining 1 the effectiveness of fusing illumination estimates from multiple subordinate methods 2 the best method of combination 3 the underlying factors that affect the performance of a combinational method and 4 the effectiveness of combination for illumination estimation in multiple illuminant scenes the various combinational methods are categorized in terms of whether or not they require supervised training and whether or not they rely on high level scene content cues e g indoor versus outdoor extensive tests and enhanced analyzes using three data sets of real world images are conducted for consistency in testing the images were labeled according to their high level features 3d stages indoor outdoor and this label data is made available on line the tests reveal that the trained combinational methods direct combination by support vector regression in particular clearly outperform both the non combinational methods and those combinational methods based on scene content cues

positive solutions and asymptotic behavior of delay differential equations with nonlinear impulses abstract consider the delay differential equation dde with nonlinear impulses x t i 1 n p i t x t i 0 t t j t t 0 x t j x t j i j x t j j 1 2 wherepi c t0 r i 0 fori 1 2 n andij c r r forj 1 2 the purpose of this paper is to obtain a necessary and sufficient condition for the existence of positive solutions of dde without impulses and to establish a kind of order persistence of solutions of eq and criteria of the asymptotic behavior of eq which can be used to improve and develop some of the known results in the literature

linearized oscillation theory for a nonlinear delay impulsive equation for a scalar nonlinear impulsive delay differential equation y t k 1m rk t fk y hk t 0 t j y j ij y j with rk t 0 hk t t limj j such an auxiliary linear impulsive delay differential equation x t k 1m rk t ak t x hk t 0 x j bjx j is constructed that oscillation nonoscillation of the nonlinear equation can be deduced from the corresponding properties of the linear equation coefficients rk t and delays are not assumed to be continuous explicit oscillation and nonoscillation conditions are established for some nonlinear impulsive models of population dynamics such as the impulsive logistic equation and the impulsive generalized lasota wazewska equation which describes the survival of red blood cells it is noted that unlike nonimpulsive delay logistic equations a solution of a delay impulsive logistic equation may become negative

a randomized ptas for the minimum consensus clustering with a fixed number of clusters the consensus clustering problem has been introduced as an effective way to analyze the results of different microarray experiments filkov and skiena 2004a b 1 2 the problem asks for a partition that summarizes a set of input partitions each corresponding to a different microarray experiment under a simple and intuitive cost the problem on instances with two input partitions has a simple polynomial time algorithm but it becomes apx hard on instances with three input partitions the quest for defining the boundary between tractable and intractable instances leads to the investigation of the restriction of consensus clustering when the output partition contains a fixed number of sets in this paper we give a randomized polynomial time approximation scheme for such problems while proving its np hardness even for 2 output partitions therefore definitively settling the approximation complexity of the problem

correlation clustering with noisy partial information in this paper we propose and study a semi random model for the correlation clustering problem on arbitrary graphs g we give two approximation algorithms for correlation clustering instances from this model the first algorithm finds a solution of value 1 opt cost o nlog 3 n with high probability where opt cost is the value of the optimal solution for every 0 the second algorithm finds the ground truth clustering with an arbitrarily small classific ation error under some additional assumptions on the instance

clustering with local restrictions we study a family of graph clustering problems where each cluster has to satisfy a certain local requirement formally let m be a function on the subsets of vertices of a graph g in the m p q partition problem the task is to find a partition of the vertices into clusters where each cluster c satisfies the requirements that 1 at most q edges leave c and 2 m c

the tetrahedron algebra the onsager algebra and the sl2 loop algebra let k denote a field with characteristic 0 and let t denote an indeterminate we give a presentation for the three point loop algebra sl2 k t t 1 t 1 1 via generators and relations this presentation displays s4 symmetry using this presentation we obtain a decomposition of the above loop algebra into a direct sum of three subalgebras each of which is isomorphic to the onsager algebra

the kendall and mallows kernels for permutations we show that the widely used kendall tau correlation coefficient and the related mallows kernel are positive definite kernels for permutations they offer computationally attractive alternatives to more complex kernels on the symmetric group to learn from rankings or learn to rank we show how to extend these kernels to partial rankings multivariate rankings and uncertain rankings examples are presented on how to formulate typical problems of learning from rankings such that they can be solved with state of the art kernel algorithms we demonstrate promising results on clustering heterogeneous rank data and high dimensional classification problems in biomedical applications

on the approximability of dodgson and young elections the voting rules proposed by dodgson and young are both designed to find an alternative closest to being a condorcet winner according to two different notions of proximity the score of a given alternative is known to be hard to compute under either rule in this paper we put forward two algorithms for approximating the dodgson score a combinatorial greedy algorithm and an lp based algorithm both of which yield an approximation ratio of h m 1 where m is the number of alternatives and h m 1 is the m 1 st harmonic number we also prove that our algorithms are optimal within a factor of 2 unless problems in np have quasi polynomial time algorithms despite the intuitive appeal of the greedy algorithm we argue that the lp based algorithm has an advantage from a social choice point of view further we demonstrate that computing any reasonable approximation of the ranking produced by dodgson s rule is np hard this result provides a complexity theoretic explanation of sharp discrepancies that have been observed in the social choice theory literature when comparing dodgson elections with simpler voting rules finally we show that the problem of calculating the young score is np hard to approximate by any factor this leads to an inapproximability result for the young ranking

an active learning algorithm for ranking from pairwise preferences with an almost optimal query complexity given a set v of n elements we wish to linearly order them given pairwise preference labels which may be non transitive due to irrationality or arbitrary noise the goal is to linearly order the elements while disagreeing with as few pairwise preference labels as possible our performance is measured by two parameters the number of disagreements loss and the query complexity number of pairwise preference labels our algorithm adaptively queries at most o e 6n log5 n preference labels for a regret of e times the optimal loss as a function of n this is asymptotically better than standard non adaptive learning bounds achievable for the same problem our main result takes us a step closer toward settling an open problem posed by learning to rank from pairwise information theoreticians and practitioners what is a provably correct way to sample preference labels to further show the power and practicality of our solution we analyze a typical test case in which a large margin linear relaxation is used for efficiently solving the simpler learning problems in our decomposition

oscillation theorems for first order nonlinear neutral functional differential equations abstract this paper discusses a class of first order nonlinear neutral differential equations with variable coefficients and variable deviations sharp conditions are established for all solutions of the equations to be oscillatory linearized oscillation criteria of the equations are also given

oscillation criteria for second order nonlinear differential equations with impulses abstract the present paper is devoted to the investigation of the oscillatory behavior of a kind of extensively studied second order nonlinear delay differential equations with impulses some interesting results are obtained which illustrate that impulses play a very important role in giving rise to the oscillations of equations

dual polar graphs the quantum algebra uq sl2 and leonard systems of dual q krawtchouk type abstract in this paper we consider how the following three objects are related i the dual polar graphs ii the quantum algebra u q sl 2 iii the leonard systems of dual q krawtchouk type for convenience we first describe how ii and iii are related for a given leonard system of dual q krawtchouk type we obtain two u q sl 2 module structures on its underlying vector space we now describe how i and iii are related let denote a dual polar graph fix a vertex x of and let t t x denote the corresponding subconstituent algebra by definition t is generated by the adjacency matrix a of and a certain diagonal matrix a a x called the dual adjacency matrix that corresponds to x by construction the algebra t is semisimple we show that for each irreducible t module w the restrictions of a and a to w induce a leonard system of dual q krawtchouk type we now describe how i and ii are related we obtain two u q sl 2 module structures on the standard module of we describe how these two u q sl 2 module structures are related each of these u q sl 2 module structures induces a c algebra homomorphism u q sl 2 t we show that in each case t is generated by the image together with the center of t using the combinatorics of we obtain a generating set l f r k of t along with some attractive relations satisfied by these generators

a generalization of the terwilliger algebra abstract p m terwilliger 1992 j algebraic combin 1 363 388 considered the c algebra generated by a given bose mesner algebra m and the associated dual bose mesner algebra m this algebra is now known as the terwilliger algebra and is usually denoted by t terwilliger showed that each vanishing intersection number and krein parameter of m gives rise to a relation on certain generators of t these relations determine much of the structure of t thought not all of it in general to illuminate the role these relations play we consider a certain generalization t of t to go from t to t we replace m and m with a pair of dual character algebras c and c we define t by generators and relations intuitively t is the associative c algebra with identity generated by c and c subject to the analogues of terwilliger s relations t is infinite dimensional and noncommutative in general we construct an irreducible t module which we call the primary module the dimension of this module is the same as that of c and c we find two bases of the primary module one diagonalizes c and the other diagonalizes c we compute the action of the generators of t on these bases we show t is a direct sum of two sided ideals t 0 and t 1 with t 0 isomorphic to a full matrix algebra we show that the irreducible module associated with t 0 is isomorphic to the primary module we compute the central primitive idempotent of t associated with t 0 in terms of the generators of t

leonard triples the racah algebra and some distance regular graphs of racah type by a leonard triple we mean a triple of diagonalizable operators on a finite dimensional vector space such that for each operator there is an ordering of an eigenbasis for the selected operator with respect to which the other two operators are irreducible tridiagonal let c denote the field of complex numbers and let d denote an integer at least 3 let 12h 2d 1 2 denote the halved graph of the 2d 1 cube with respect to the original p polynomial structure r0 r1 rd and another q polynomial structure e0 e2 e4 e3 e1 in terms of the original ones let 12h 4d 2 denote the folded halved graph of the 4d cube and let 12h 4d 2 2 denote the folded halved graph of the 4d 2 cube note that they are all distance regular graphs of racah type in this paper we consider the relations between the above three graphs and the leonard triples or the racah algebra over c our results are described as follows 1 fix a vertex of 12h 2d 1 2 and let t1 denote the corresponding terwilliger algebra with respect to this vertex we first construct three elements u1 u1 and u1e of t1 then we show that the triple u1 u1 u1e acts on each irreducible t1 module as a leonard triple moreover let k1 be a racah algebra with its generators and real parameters satisfying certain conditions we display a c algebra homomorphism from k1 to t1 2 fix a vertex of 12h 4d 2 and let t2 denote the terwilliger algebra of 12h 4d 2 with respect to this vertex we construct three elements u2 u2 u2e of t2 and show that the triple u2 u2 u2e not only acts on each irreducible t2 module as a leonard triple but also satisfies some very appealing equations moreover let w denote an irreducible t2 module with type and let k be a racah algebra with respect to then there exists a k module structure on w 3 fix a vertex of 12h 4d 2 2 and let t3 denote the terwilliger algebra of 12h 4d 2 2 with respect to this vertex we construct three elements u3 u3 u3e of t3 and show that the triple u3 u3 u3e not only acts on each irreducible t3 module as a leonard triple but also satisfies some very appealing equations moreover let w denote an irreducible t3 module with auxiliary parameter e and let ke be a racah algebra with respect to e then there exists a ke module structure on w

structure of thin irreducible modules of a q polynomial distance regular graph let be a q polynomial distance regular graph with vertex set x diameter d 3 and adjacency matrix a fix x x and let a a x be the corresponding dual adjacency matrix recall that the terwilliger algebra t t x is the subalgebra of matx c generated by a and a let w denote a thin irreducible t module it is known that the action of a and a on w induces a linear algebraic object known as a leonard pair over the past decade many results have been obtained concerning leonard pairs in this paper we apply these results to obtain a detailed description of w in our description we do not assume that the reader is familiar with leonard pairs everything will be proved from the point of view of our results are summarized as follows let ei i 0d be a q polynomial ordering of the primitive idempotents of and let ei i 0d be the dual primitive idempotents of with respect to x let r t and d be the endpoint dual endpoint and diameter of w respectively let uandv be nonzero vectors in etw and er w respectively we show that er i aiv i 0d and et ia iu i 0d are bases for w that are orthogonal with respect to the standard hermitian dot product we display the matrix representations of a and a with respect to these bases we associate with w two sequences of polynomials pi i 0dand pi i 0d we show that for 0 i d pi a v er i aiv and pi a u et ia iu next we show that er i u i 0d and et iv i 0d are orthogonal bases for w we call these the standard basis and dual standard basis for w respectively we display the matrix representations of a and a with respect to these bases the entries in these matrices will play an important role in our theory we call these the intersection numbers and dual intersection numbers of w using these numbers we compute all inner products involving the standard and dual standard bases we also use these numbers to define two normalizations ui vi resp ui vi for pi resp pi using the orthogonality of the standard and dual standard bases we show that for each of the sequences pi i 0d pi i 0d ui i 0d ui i 0d vi i 0d vi i 0d the polynomials involved are orthogonal and we display the orthogonality relations we also show that each of the sequences satisfy a three term recurrence and a relation known as the askey wilson duality we then turn our attention to two more bases for w we find the matrix representations of a and a with respect to these bases from the entries of these matrices we obtain two sequences of scalars known as the first split sequence and second split sequence of w we associate with w a sequence of scalars called the parameter array this sequence consists of the eigenvalues of the restriction of a to w the eigenvalues of the restriction of a to w the first split sequence of w and the second split sequence of w we express all the scalars and polynomials associated with w in terms of its parameter array we show that the parameter array of w is determined by r t d and one more free parameter from this we conclude that the isomorphism class of w is determined by these four parameters finally we apply our results to the case in which has q racah type or classical parameters

on the hardness of labeled correlation clustering problem motivated by practical applications the labeled correlation clustering problem a variant of correlation clustering problem is formally defined and studied in this paper since the problem is np complete we consider the parameterized complexities three different parameterizations are considered and the corresponding parameterized complexities are studied for the two parameterized problems which are fixed parameter tractable the lower bounds of them are analyzed under seth strong exponential time hypothesis

uniform asymptotic stability of impulsive delay differential equations abstract in this paper criteria on uniform asymptotic stability are established for impulsive delay differential equations using lyapunov functions and razumikhin techniques it is shown that impulses do contribute to yield stability properties even when the underlying system does not enjoy any stability behavior some examples are also discussed to illustrate the theorems

the terwilliger algebras of grassmann graphs abstract let j q n m denote the grassmann graph with vertex set x and diameter min m n m fix a vertex x x let t t x denote the terwilliger algebra of j q n m corresponding to x in this paper we study the structure of t under the assumption that m 3 and n 2 m let u q sl 2 be the quantum enveloping algebra of sl 2 and let q be the q tetrahedron algebra we first obtain an action of u q sl 2 on the standard module of j q n m then we display a c algebra homomorphism u q sl 2 t and show that t is generated by the image of and some central elements of t as an application we also display an action of q on the standard module of j q n m these results are obtained by using the theory of leonard pairs

the terwilliger algebras of johnson graphs abstract let j n m denote the johnson graph with vertex set x fix a vertex x x let t t x denote the terwilliger algebra of j n m corresponding to x in this paper we study the structure of t under the assumption that m 3 and n 4 m let u sl 2 be the universal enveloping algebra of sl 2 we display a c algebra homomorphism u sl 2 t we show that t is generated by the image of and some central elements of t we obtain these results using the theory of leonard pairs

an approximation algorithm for feedback vertex sets in tournaments we obtain a necessary and sufficient condition in terms of forbidden structures for tournaments to possess the min max relation on packing and covering directed cycles together with strongly polynomial time algorithms for the feedback vertex set problem and the cycle packing problem in this class of tournaments applying the local ratio technique of bar yehuda and even to the forbidden structures we find a 2 5 approximation polynomial time algorithm for the feedback vertex set problem in any tournament

on oscillation of a second order impulsive linear delay differential equation abstract for the delay differential equation x t k 1 m a k t x g k t 0 g k t t with impulsive conditions x j a j x j 0 x j b j x j 0 a connection between the following properties is established nonoscillation of the differential equation and the corresponding differential inequality positiveness of the fundamental function and the existence of a solution of a generalized riccati inequality explicit conditions for nonoscillation and oscillation and comparison theorems are presented

rank centrality ranking from pairwise comparisons the question of aggregating pairwise comparisons to obtain a global ranking over a collection of objects has been of interest for a very long time be it ranking of online gamers e g msr s trueskill system and chess players aggregating social opinions or deciding which product to sell based on transactions in most settings in addition to obtaining a ranking finding scores for each object e g player s rating is of interest for understanding the intensity of the preferences in this paper we propose rank centrality an iterative rank aggregation algorithm for discovering scores for objects or items from pairwise comparisons the algorithm has a natural random walk interpretation over the graph of objects with an edge present between a pair of objects if they are compared the score which we call rank centrality of an object turns out to be its stationary probability under this random walk to study the efficacy of the algorithm we consider the popular bradley terry luce btl model equ

kernel and fast algorithm for dense triplet inconsistency we study the parameterized complexity of inferring supertrees from sets of rooted triplets an important problem in phylogenetics for a set l of labels and a dense set t of triplets distinctly leaf labeled by 3 subsets of l we seek a tree distinctly leaf labeled by l and containing all but at most k triplets from t as homeomorphic subtree our results are the first polynomial kernel for this problem with o k 2 labels and a subexponential fixed parameter algorithm running in time o l 4 2 o k 1 3 l o g k

correlation clustering with constrained cluster sizes and extended weights bounds we consider the problem of correlation clustering on graphs with constraints on both the cluster sizes and the positive and negative weights of edges our contributions are twofold first we introduce the problem of correlation clustering with bounded cluster sizes second we extend the regime of weight values for which the clustering may be performed with constant approximation guarantees in polynomial time and apply the results to the bounded cluster size problem

crc handbook of combinatorial designs balanced incomplete block designs and t designs2 v k l designs of small orderbibds with small block sizet designs t 3steiner systemssymmetric designsresolvable and near resolvable designslatin squares mols and orthogonal arrayslatin squaresmutually orthogonal latin squares mols incomplete molsorthogonal arrays of index more than oneorthogonal arrays of strength more than twopairwise balanced designspbds and gdds the basicspbds recursive constructionspbd closurepairwise balanced designs as linear spacespbds and gdds of higher indexpbds frames and resolvabilityother combinatorial designsassociation schemesbalanced part ternary designsbalanced tournament designsbhaskar rao designscomplete mappings and sequencings of finite groupsconfigurationscostas arrayscoveringscycle systemsdifference familiesdifference matricesdifference sets abeliandifference sets nonabeliandifference triangle setsdirected designsd optimal matricesembedding partial quasigroupsequidistant permutation arraysfactorial designsfrequency squaresgeneralized quadranglesgraph decompositions and designsgraphical designshadamard matrices and designshall triple systemshowell designsmaximal sets of molsmendelsohn designsthe oberwolfach problemordered designs and perpendicular arraysorthogonal designsorthogonal main effect planspackingspartial geometriespartially balanced incomplete block designsquasigroupsquasi symmetric designs r l designsroom squaresself orthogonal latin squares sols sols with a symmetric orthogonal mate solssom sequences with zero autocorrelationskolem sequencesspherical t designsstarterstrades and defining sets t m s netstuscan squarest wise balanced designsuniformly resolvable designsvector space designsweighing matrices and conference matriceswhist tournamentsyouden designs generalizedyouden squaresapplicationscodescomputer science selected applicationsapplications of designs to cryptographyderandomizationoptimality and efficiency comparing block designsgroup testingscheduling a tournamentwinning the lotteryrelated mathematics and computational methodsfinite groups and designsnumber theory and finite fieldsgraphs and multigraphsfactorizations of graphsstrongly regular graphstwo graphsclassical geometriesprojective planes nondesarguesiancomputational methods in design theoryindex

active learning using smooth relative regret approximations with applications the disagreement coeffcient of hanneke has become a central data independent invariant in proving active learning rates it has been shown in various ways that a concept class with low complexity together with a bound on the disagreement coeffcient at an optimal solution allows active learning rates that are superior to passive learning ones we present a different tool for pool based active learning which follows from the existence of a certain uniform version of low disagreement coeffcient but is not equivalent to it in fact we present two fundamental active learning problems of significant interest for which our approach allows nontrivial active learning bounds however any general purpose method relying on the disagreement coeffcient bounds only fails to guarantee any useful bounds for these problems the applications of interest are learning to rank from pairwise preferences and clustering with side information a k a semi supervised clustering the tool we use is based on the learner s ability to compute an estimator of the difference between the loss of any hypothesis and some fixed pivotal hypothesis to within an absolute error of at most e times the disagreement measure l1 distance between the two hypotheses we prove that such an estimator implies the existence of a learning algorithm which at each iteration reduces its in class excess risk to within a constant factor each iteration replaces the current pivotal hypothesis with the minimizer of the estimated loss difference function with respect to the previous pivotal hypothesis the label complexity essentially becomes that of computing this estimator

a more effective linear kernelization for cluster editing in the np hard cluster editing problem we have as input an undirected graph g and an integer k 0 the question is whether we can transform g by inserting and deleting at most k edges into a cluster graph that is a union of disjoint cliques we first confirm a conjecture by michael fellows iwpec 2006 that there is a polynomial time kernelization for cluster editing that leads to a problem kernel with at most 6k vertices more precisely we present a cubic time algorithm that given a graph g and an integer k 0 finds a graph g and an integer k k such that g can be transformed into a cluster graph by at most k edge modifications iff g can be transformed into a cluster graph by at most k edge modifications and the problem kernel g has at most 6k vertices so far only a problem kernel of 24k vertices was known second we show that this bound for the number of vertices of g can be further improved to 4k vertices finally we consider the variant of cluster editing where the number of cliques that the cluster graph can contain is stipulated to be a constant d 0 we present a simple kernelization for this variant leaving a problem kernel of at most d 2 k d vertices

clustering algorithms for chains we consider the problem of clustering a set of chains to k clusters a chain is a totally ordered subset of a finite set of items chains are an intuitive way to express preferences over a set of alternatives as well as a useful representation of ratings in situations where the item specific scores are either difficult to obtain too noisy due to measurement error or simply not as relevant as the order that they induce over the items first we adapt the classical k means for chains by proposing a suitable distance function and a centroid structure we also present two different approaches for mapping chains to a vector space the first one is related to the planted partition model while the second one has an intuitive geometrical interpretation finally we discuss a randomization test for assessing the significance of a clustering to this end we present an mcmc algorithm for sampling random sets of chains that share certain properties with the original data the methods are studied in a series of experiments using real and artificial data results indicate that the methods produce interesting clusterings and for certain types of inputs improve upon previous work on clustering algorithms for orders

oscillation of nonlinear delay impulsive differential equations and inequalities the oscillatory and asymptotic behavior of first order nonlinear delay impulsive differential equations and inequalities is studied some new sharp sufficient conditions for oscillation and nonoscillation of solutions of the equations and inequalities are obtained many known results are improved in particular the conditions are necessary and sufficient if the equations and inequalities are autonomous

a new correlation coefficient for comparing and aggregating non strict and incomplete rankings abstract we introduce a correlation coefficient that is designed to deal with a variety of ranking formats including those containing non strict i e with ties and incomplete i e unknown preferences the correlation coefficient is designed to enforce a neutral treatment of incompleteness whereby no assumptions are made about individual preferences involving unranked objects the new measure which can be regarded as a generalization of the seminal kendall tau correlation coefficient is proven to satisfy a set of metric like axioms and to be equivalent to a recently developed ranking distance function associated with kemeny aggregation in an effort to further unify and enhance both robust ranking methodologies this work proves the equivalence of an additional distance and correlation coefficient pairing in the space of non strict incomplete rankings these connections induce new exact optimization methodologies a specialized branch and bound algorithm and an exact integer programming formulation moreover the bridging of these complementary theories reinforces the singular suitability of the featured correlation coefficient to solve the general consensus ranking problem the latter premise is bolstered by an accompanying set of experiments on random instances which are generated via a herein developed sampling technique connected with the classic mallows distribution of ranking data associated experiments with the branch and bound algorithm demonstrate that as data becomes noisier the featured correlation coefficient yields relatively fewer alternative optimal solutions and that the aggregate rankings tend to be closer to an underlying ground truth shared by a majority

on oscillation of a generalized logistic equation with several delays abstract for a scalar delay generalized logistic equation y t k 1 m r k t y t 1 y h k t k 1 y h k t k k 1 explicit oscillation conditions are established for all the cases of parameters k i e sublinear superlinear and also mixed ones coefficients rk t and delays hk t are not assumed to be continuous

cost optimal constrained correlation clustering via weighted partial maximum satisfiability abstract integration of the fields of constraint solving and data mining and machine learning has recently been identified within the ai community as an important research direction with high potential this work contributes to this direction by providing a first study on the applicability of state of the art boolean optimization procedures to cost optimal correlation clustering under constraints in a general similarity based setting we develop exact formulations of the correlation clustering task as maximum satisfiability maxsat the optimization version of the boolean satisfiability sat problem for obtaining cost optimal clusterings we apply a state of the art maxsat solver for solving the resulting maxsat instances optimally resulting in cost optimal clusterings we experimentally evaluate the maxsat based approaches to cost optimal correlation clustering both on the scalability of our method and the quality of the clusterings obtained furthermore we show how the approach extends to constrained correlation clustering where additional user knowledge is imposed as constraints on the optimal clusterings of interest we show experimentally that added user knowledge allows clustering larger datasets and at the same time tends to decrease the running time of our approach we also investigate the effects of maxsat level preprocessing symmetry breaking and the choice of the maxsat solver on the efficiency of the approach

linear programming based approximation algorithms for feedback set problems in bipartite tournaments we consider the feedback vertex set and feedback arc set problems on bipartite tournaments we improve on recent results by giving a 2 approximation algorithm for the feedback vertex set problem we show that this result is the best that we can attain when using optimal solutions to a certain linear program as a lower bound on the optimal value for the feedback arc set problem on bipartite tournaments we show that a recent 4 approximation algorithm proposed by gupta 2008 8 is incorrect we give an alternative 4 approximation algorithm based on an algorithm for the feedback arc set on non bipartite tournaments given by van zuylen and williamson 2009 14

comparing and aggregating partially resolved trees partially resolved that is non binary trees arise frequently in the analysis of species evolution non binary nodes also called multifurcations must be treated carefully since they can be interpreted as reflecting either lack of information or actual evolutionary history while several distance measures exist for comparing trees none of them deal explicitly with this dichotomy here we introduce two kinds of distance measures between rooted and unrooted partially resolved phylogenetic trees over the same set of species the measures address multifurcations directly for rooted trees the measures are based on the topologies the input trees induce on triplets that is on three element subsets of the set of species for unrooted trees the measures are based on quartets four element subsets the first class of measures are parametric distances where there is a parameter that weighs the difference between an unresolved triplet quartet topology and a resolved one the second class of measures are based on the hausdorff distance where each tree is viewed as a set of all possible ways in which the tree can be refined to eliminate unresolved nodes we give efficient algorithms for computing parametric distances and give conditions under which hausdorff distances can be calculated approximately in polynomial time additionally we i derive the expected value of the parametric distance between two random trees ii characterize the conditions under which parametric distances are near metrics or metrics iii study the computational and algorithmic properties of consensus tree methods based on the measures and iv analyze the interrelationships among hausdorff and parametric distances