sampling decomposable graphs using a markov chain on junction trees this paper makes two contributions to the computational geometry of decomposable graphs aimed primarily at facilitating statistical inference about such graphs where they arise as assumed conditional independence structures in stochastic models the rst of these provides sucient conditions under which it is possible to completely connect two disconnected cliques of vertices or perform the reverse procedure yet maintain decomposability of the graph the second is a new markov chain monte carlo sampler for arbitrary positive distributions on decomposable graphs taking a junction tree representing the graph as its state variable the resulting methodology is illustrated with two numerical experiments eld model determination

learning equivalence classes of bayesian network structures two bayesian network structures are said to be equivalent if the set of distributions that can be represented with one of those structures is identical to the set of distributions that can be represented with the other many scoring criteria that are used to learn bayesian network structures from data are score equivalent that is these criteria do not distinguish among networks that are equivalent in this paper we consider using a score equivalent criterion in conjunction with a heuristic search algorithm to perform model selection or model averaging we argue that it is often appropriate to search among equivalence classes of network structures as opposed to the more common approach of searching among individual bayesian network structures we describe a convenient graphical representation for an equivalence class of structures and introduce a set of operators that can be applied to that representation by a search algorithm to move among equivalence classes we show that our equivalence class operators can be scored locally and thus share the computational efficiency of traditional operators defined for individual structures we show experimentally that a greedy model selection algorithm using our representation yields slightly higher scoring structures than the traditional approach without any additional time overhead and we argue that more sophisticated search algorithms are likely to benefit much more

characteristic polynomials and controlability of partially prescribed matrices abstract in a previous paper it was proved that n 1 arbitrary entries and the characteristic polynomial of a n n matrix over a field f can be arbitrarily prescribed except if all the nonprincipal entries of a row or column are prescribed equal to zero and the characteristic polynomial does not have a root in f this paper describes the possible characteristic polynomials of a pk pk matrix partitioned into k k blocks of size p p when k 1 blocks are fixed and the others vary it also studies the possibility of a pair of matrices a 1 a 2 where a 1 is square and a 1 a 2 is partitioned into k k 1 blocks of size p p being completely controllable when some of the blocks are prescribed and the others vary

fast bayes and the dynamic junction forest abstract it has been shown that junction tree algorithms can provide a quick and efficient method for propagating probabilities in complex multivariate problems when they can be described by a fixed conditional independence structure in this paper we formalise and illustrate with two practical examples how these probabilistic propagation algorithms can be applied to high dimensional processes whose conditional independence structure as well as their underlying distributions are augmented through the passage of time

a recursive method for structural learning of directed acyclic graphs in this paper we propose a recursive method for structural learning of directed acyclic graphs dags in which a problem of structural learning for a large dag is first decomposed into two problems of structural learning for two small vertex subsets each of which is then decomposed recursively into two problems of smaller subsets until none subset can be decomposed further in our approach search for separators of a pair of variables in a large dag is localized to small subsets and thus the approach can improve the efficiency of searches and the power of statistical tests for structural learning we show how the recent advances in the learning of undirected graphical models can be employed to facilitate the decomposition simulations are given to demonstrate the performance of the proposed method

small schemes and varieties of minimal degree we prove that if x p r is any 2 regular scheme in the sense of castelnuovo mumford then x is small this means that if l is a linear space and y l x is finite then y is linearly independent in the sense that the dimension of the linear span of y is deg y 1 the converse is true and well known for finite schemes but false in general the main result of this paper is that the converse small implies 2 regular is also true for reduced schemes algebraic sets this is proven by means of a delicate geometric analysis leading to a complete classification we show that the components of a small algebraic set are varieties of minimal degree meeting in a particularly simple way from the classification one can show that if x p r is 2 regular then so is xred and so also is the projection of x from any point of x our results extend the del pezzo bertini classification of varieties of minimal degree the charac terization of these as the varieties of regularity 2 by eisenbud goto and the construction of 2 regular square free monomial ideals by froberg 0 introduction throughout this paper we will work with projective schemes x p r over an algebraically closed field k the castelnuovo mumford regularity of x p r is a basic homological measure of the complexity of x and its embedding in p r that gives a bound for the degrees of the generators of the defining ideal ix of x and for many other invariants the only schemes of regular ity 1 are the linear spaces but no classification is known for projective schemes of regularity 2 in this paper we prove a structure theorem for reduced 2 regular schemes showing that their irreducible components are varieties of minimal degree and characterizing how these components can meet we also show that the reduced structure on any 2 regular scheme is 2 regular and thus we obtain a complete description of the reduced structures on 2 regular schemes since a high veronese re embedding of any zero dimensional scheme is 2 regular one cannot hope to characterize the isomorphism types of all 2 regular nonreduced schemes before stating our results we review some basic notions for any subscheme x p r we write span x for the smallest linear subspace of p r containing x recall that every variety reduced irreducible scheme x p r satisfies the condition deg x 1 codim x span x

network based heuristics for constraint satisfaction problems abstract many ai tasks can be formulated as constraint satisfaction problems csp i e the assignment of values to variables subject to a set of constraints while some csps are hard those that are easy can often be mapped into sparse networks of constraints which in the extreme case are trees this paper identifies classes of problems that lend themselves to easy solutions and develops algorithms that solve these problems optimally the paper then presents a method of generating heuristic advice to guide the order of value assignments based on both the sparseness found in the constraint network and the simplicity of tree structured csps the advice is generated by simplifying the pending subproblems into trees counting the number of consistent solutions in each simplified subproblem and comparing these counts to decide among the choices pending in the original problem

approximate locality for quantum systems on graphs in this letter we make progress on a long standing open problem of aaronson and ambainis theory comput 1 47 2005 we show that if u is a sparse unitary operator with a gap delta in its spectrum then there exists an approximate logarithm h of u which is also sparse the sparsity pattern of h gets more dense as 1 delta increases this result can be interpreted as a way to convert between local continuous time and local discrete time quantum processes as an example we show that the discrete time coined quantum walk can be realized stroboscopically from an approximately local continuous time quantum walk

counting and exploring sizes of markov equivalence classes of directed acyclic graphs when learning a directed acyclic graph dag model via observational data one generally cannot identify the underlying dag but can potentially obtain a markov equivalence class the size the number of dags of a markov equivalence class is crucial to infer causal effects or to learn the exact causal dag via further interventions given a set of markov equivalence classes the distribution of their sizes is a key consideration in developing learning methods however counting the size of an equivalence class with many vertices is usually computationally infeasible and the existing literature reports the size distributions only for equivalence classes with ten or fewer vertices in this paper we develop a method to compute the size of a markov equivalence class we first show that there are five types of markov equivalence classes whose sizes can be formulated as five functions of the number of vertices respectively then we introduce a new concept of a rooted sub class the graph representations of rooted subclasses of a markov equivalence class are used to partition this class recursively until the sizes of all rooted subclasses can be computed via the five functions the proposed size counting is efficient for markov equivalence classes of sparse dags with hundreds of vertices finally we explore the size and edge distributions of markov equivalence classes and find experimentally that in general 1 most markov equivalence classes are half completed and their average sizes are small and 2 the sizes of sparse classes grow approximately exponentially with the numbers of vertices

locality for quantum systems on graphs depends on the number field adapting a definition of aaronson and ambainis 2005 theory comput 1 47 79 we call a quantum dynamics on a digraph saturated z local if the nonzero transition amplitudes specifying the unitary evolution are in exact correspondence with the directed edges including loops of the digraph this idea appears recurrently in a variety of contexts including angular momentum quantum chaos and combinatorial matrix theory complete characterization of the digraph properties that allow such a process to exist is a long standing open question that can also be formulated in terms of minimum rank problems we prove that saturated z local dynamics involving complex amplitudes occur on a proper superset of the digraphs that allow restriction to the real numbers or even further the rationals consequently among these fields complex numbers guarantee the largest possible choice of topologies supporting a discrete quantum evolution a similar construction separates complex numbers from the skew field of quaternions the result proposes a concrete ground for distinguishing between complex and quaternionic quantum mechanics

a structural markov property for decomposable graph laws that allows control of clique intersections summary we present a new kind of structural markov property for probabilistic laws on decomposable graphs which allows the explicit control of interactions between cliques and so is capable of encoding some interesting structure we prove the equivalence of this property to an exponential family assumption and discuss identifiability modelling inferential and computational implications

extracting constrained 2 interval subsets in 2 interval sets 2 interval sets were used in s vialette pattern matching over 2 intervals sets in proc 13th annual symposium combinatorial pattern matching cpm 2002 in lecture notes in computer science vol 2373 springer verlag 2002 pp 53 63 s vialette on the computational complexity of 2 interval pattern matching theoret comput sci 312 2 3 2004 223 249 for establishing a general representation for macroscopic describers of rna secondary structures in this context we have a 2 interval for each legal local fold in a given rna sequence and a constrained pattern made of disjoint 2 intervals represents a putative rna secondary structure we focus here on the problem of extracting a constrained pattern in a set of 2 intervals more precisely given a set of 2 intervals and a model r describing if two disjoint 2 intervals in a solution can be in precedence order be allowed to nest and or be allowed to cross we consider the problem of finding a maximum cardinality subset of disjoint 2 intervals such that any two 2 intervals in agree with r the different combinations of restrictions on model r alter the computational complexity of the problem and need to be examined separately in this paper we improve the time complexity of s vialette on the computational complexity of 2 interval pattern matching theoret comput sci 312 2 3 2004 223 249 for model r by giving an optimal o nlogn time algorithm where n is the cardinality of the 2 interval set we also give a graph like relaxation for model r that is solvable in time finally we prove that the considered problem is np complete for model r even for same length intervals and give a fixed parameter tractability result based on the crossing structure of

decomposing constraint satisfaction problems using database techniques abstract there is a very close relationship between constraint satisfaction problems and the satisfaction of join dependencies in a relational database which is due to a common underlying structure namely a hypergraph by making that relationship explicit we are able to adapt techniques previously developed for the study of relational databases to obtain new results for constraint satisfaction problems in particular we prove that a constraint satisfaction problem may be decomposed into a number of subproblems precisely when the corresponding hypergraph satisfies a simple condition we show that combining this decomposition approach with existing algorithms can lead to a significant improvement in efficiency

simple linear time algorithms to test chordality of graphs test acyclicity of hypergraphs and selectively reduce acyclic hypergraphs chordal graphs arise naturally in the study of gaussian elimination on sparse symmetric matrices acyclic hypergraphs arise in the study of relational data bases rose tarjan and lueker siam j c

interlacing inequalities and control theory abstract given a p by p matrix a we solve the problem of the existence of a p by q matrix b such that a b has prescribed controllability indices and i p a b has prescribed invariant polynomials the solution of this problem together with an earlier theorem of the author s is used to provide a new proof of the sa thompson interlacing inequalities for invariant polynomials

a remark on the cluster theorem an improved version of the cluster theorem of relativistic quantum field theory with the correct exponential decay is derived by function theoretical methods

sequential and parallel triangulating algorithms for elimination game and new insights on minimum degree elimination game is a well known algorithm that simulates gaussian elimination of matrices on graphs and it computes a triangulation of the input graph the number of fill edges in the computed triangulation is highly dependent on the order in which elimination game processes the vertices and in general the produced triangulations are neither minimum nor minimal in order to obtain a triangulation which is close to minimum the minimum degree heuristic is widely used in practice but until now little was known on the theoretical mechanisms involved in this paper we show some interesting properties of elimination game in particular that it is able to compute a partial minimal triangulation of the input graph regardless of the order in which the vertices are processed this results in a new algorithm to compute minimal triangulations that are sandwiched between the input graph and the triangulation resulting from elimination game one of the strengths of the new approach is that it is easily parallelizable and thus we are able to present the first parallel algorithm to compute such sandwiched minimal triangulations in addition the insight that we gain through elimination game is used to partly explain the good behavior of the minimum degree algorithm we also give a new algorithm for producing minimal triangulations that is able to use the minimum degree idea to a wider extent

on the number of invariant polynomials of partially prescribed matrices abstract we study the number of invariant polynomials of a i j i j 1 2 when the submatrices a 1 1 a 1 2 a 2 2 are prescribed

matrices with prescribed rows and invariant factors abstract in this paper we solve the problem of the existence of an n n matrix over an arbitrary field when its invariant polynomials and either some rows or columns are prescribed the solution is given in terms of invariant factor inequalities and of majorization inequalities involving controllability indices and the degrees of the invariant polynomials

the number of invariant polynomials of a matrix with prescribed complementary principal submatrices abstract we study the number of invariant polynomials of a i j i j 1 2 when the submatrices a 1 1 and a 2 2 are prescribed

on the number of invariant factors of partially prescribed matrices and control theory abstract let a and b be matrices of sizes m m and m u respectively over an infinite field let i a b be the number of invariant factors of xi a b different from 1 this paper gives a necessary and sufficient condition for the existence of a matrix a b with a prescribed submatrix on the right lower corner such that i a b in particular this paper studies the possibility of a linear system t a t b t being completely controllable when a submatrix on the right lower corner of a b is unknown and all the other entries are prescribed

on the invariant polynomials of jordan products let f be an algebraically closed field this paper describes the possible numbers of nonconstant invariant polynomials of the jordan product xa ax when a is fixed and x varies

on feedback equivalence and completion problems abstract in 1987 i zaballa characterized the possible similarity classes of a square matrix with some prescribed rows in 1988 the same author characterized the possible feedback equivalence classes of a b where a is a fixed square matrix and b varies firstly in this paper we observe that these results are equivalent that is each one of them can be obtained as a corollary of the other then we apply similar arguments to other inverse problems in particular we study the possible invariant polynomials and the possible characteristic polynomials of a bx yc when x and y vary and we study the linear systems that can become completely observable with a suitable choice of linear state feedback control

a note on minimal d separation trees for structural learning structural learning of a bayesian network is often decomposed into problems related to its subgraphs although many approaches without decomposition were proposed in 2006 xie geng and zhao proposed using a d separation tree to improve the power of conditional independence tests and the efficiency of structural learning in our research note we study a minimal d separation tree under a partial ordering by which the maximal efficiency can be obtained our results demonstrate that a minimal d separation tree of a directed acyclic graph dag can be constructed by searching for the clique tree of a minimal triangulation of the moral graph for the dag

matrices with prescribed submatrices and number of invariant polynomials abstract we give necessary and sufficient conditions for the existence of a matrix a b over a field where a is a square matrix such that a b has a prescribed submatrix and the pencil xi a b has a prescribed number of invariant factors different from 1 as a corollary we give a necessary and sufficient condition for the existence of a completely controllable pair a b such that a b has a prescribed submatrix

the complexity of acyclic conjunctive queries this paper deals with the evaluation of acyclic booleanconjunctive queries in relational databases by well known resultsof yannakakis 1981 this problem is solvable in polynomial time its precise complexity however has not been pinpointed so far weshow that the problem of evaluating acyclic boolean conjunctivequeries is complete for logcfl the class of decision problems thatare logspace reducible to a context free language since logcfl iscontained in ac1 and nc2 the evaluation problem of acyclic booleanconjunctive queries is highly parallelizable we present a paralleldatabase algorithm solving this problem with alogarithmic number ofparallel join operations the algorithm is generalized to computingthe output of relevant classes of non boolean queries we also showthat the acyclic versions of the following well known database andai problems are all logcfl complete the query output tuple problemfor conjunctive queries conjunctive query containment clausesubsumption and constraint satisfaction the logcfl completenessresult is extended to the class of queries of bounded tree widthand to other relevant query classes which are more general than theacyclic queries

connections in acyclic hypergraphs abstract we demonstrate a sense in which the equivalence between blocks subgraphs without articulation points and biconnected components subgraphs in which there are two edge disjoint paths between any pair of nodes that holds in ordinary graph theory can be generalized to hypergraphs the result has an interpretation for relational databases that the universal relations described by acyclic join dependencies are exactly those for which the connections among attributes are defined uniquely we also exhibit a relationship between the process of graham reduction graham 1979 of hypergraphs and the process of tableau reduction aho sagiv and ullman 1979 that holds only for acyclic hypergraphs

unified theorems on completions of matrix pencils abstract we generalize and unify some theorems concerned with the existence of a pencil with a prescribed subpencil and prescribed strict equivalence class

query evaluation via tree decompositions a number of efficient methods for evaluating first order and monadic second order queries on finite relational structures are based on tree decompositions of structures or queries we systematically study these methods in the first part of the article we consider arbitrary formulas on tree like structures we generalize a theorem of courcelle 1990 by showing that on structures of bounded tree width a monadic second order formula with free first and second order variables can be evaluated in time linear in the structure size plus the size of the output in the second part we study tree like formulas on arbitrary structures we generalize the notions of acyclicity and bounded tree width from conjunctive queries to arbitrary first order formulas in a straightforward way and analyze the complexity of evaluating formulas of these fragments moreover we show that the acyclic and bounded tree width fragments have the same expressive power as the well known guarded fragment and the finite variable fragments of first order logic respectively

decomposition of structural learning about directed acyclic graphs in this paper we propose that structural learning of a directed acyclic graph can be decomposed into problems related to its decomposed subgraphs the decomposition of structural learning requires conditional independencies but it does not require that separators are complete undirected subgraphs domain or prior knowledge of conditional independencies can be utilized to facilitate the decomposition of structural learning by decomposition search for d separators in a large network is localized to small subnetworks thus both the efficiency of structural learning and the power of conditional independence tests can be improved

fast minimal triangulation algorithm using minimum degree criterion we propose an algorithm for minimal triangulation which using simple and efficient strategy subdivides the input graph in different almost non overlapping subgraphs using the technique of matrix multiplication for saturating the minimal separators we show that the partition of the graph can be computed in time o n a where n a is the time required by the binary matrix multiplication after saturating the minimal separators the same procedure is recursively applied on each subgraphs we also present a variant of the algorithm in which the minimum degree criterion is used in this way we obtain an algorithm that uses minimum degree criterion and at the same time produces a minimal triangulation thus shedding new light on the effectiveness of the minimum degree heuristics

on the desirability of acyclic database schemes a class of database schemes called acychc was recently introduced it is shown that this class has a number of desirable properties in particular several desirable properties that have been studied by other researchers m very different terms are all shown to be eqmvalent to acydicity in addition several equivalent charactenzauons of the class m terms of graphs and hypergraphs are given and a smaple algorithm for determining acychclty is presented also given are several eqmvalent characterizations of those sets m of multivalued dependencies such that m is the set of murlvalued dependencies that are the consequences of a given join dependency several characterizations for a conflict free in the sense of lien set of muluvalued dependencies are provided

on the number of invariant polynomials of matrix commutators we study the possible numbers of noneonstant invariant polynomials of the matrix commutator xa ax when x varies elsevier science inc 1997