genetic algorithm approach on multi criteria minimum spanning tree problem minimum spanning tree mst problem is of high importance in network optimization the multi criteria mst mc mst is a more realistic representation of the practical problem in the real world but it is difficult for the traditional network optimization technique to deal with in this paper a genetic algorithm ga approach is developed to deal with this problem without neglecting its network topology the proposed method adopts the prufer number as the tree encoding and applies the multiple criteria decision making mcdm technique and nondominated sorting technique to make the ga search give out all pareto optimal solutions either focused on the region near the ideal point or distributed all along the pareto frontier compared with the enumeration method of pareto optimal solution the numerical analysis shows the efficiency and effectiveness of the ga approach on the mc mst problem

an improved version of the augmented e constraint method augmecon2 for finding the exact pareto set in multi objective integer programming problems generation or a posteriori methods in multi objective mathematical programming momp is the most computationally demanding category among the momp approaches due to the dramatic increase in computational speed and the improvement of mathematical programming algorithms the generation methods become all the more attractive among today s decision makers in the current paper we present the generation method augmecon2 which is an improvement of our development augmecon although augmecon2 is a general purpose method we will demonstrate that augmecon2 is especially suitable for multi objective integer programming moip problems specifically augmecon2 is capable of producing the exact pareto set in moip problems by appropriately tuning its running parameters in this context we compare the previous and the new version in a series of new and old benchmarks found in the literature we also compare augmecon2 s performance in the generation of the exact pareto sets with established methods and algorithms based on specific moip problems knapsack set packing and on published results except from other mathematical programming methods augmecon2 is found to be competitive also with multi objective meta heuristics momh in producing adequate approximations of the pareto set in multi objective combinatorial optimization moco problems

non finite specifiability of projections of functional dependency families abstract a functional dependency fd implicational dependency id family is a family of relational database instances consisting of all instances which satisfy a given set of fds ids an id family is finitely specifiable if it can be defined using a finite set of ids it is known that all fd families are finitely specifiable id families and also that each projection of each fd family while not necessarily an fd family is an id family it is shown here that if the projection of an fd family is not an fd family then it is not a finitely specifiable id family

a genetic based framework for solving multi criteria weighted matching problems abstract the purpose of this paper is to present a flexible genetic based framework for solving the multi criteria weighted matching problem mc wmp in the first part of this paper we design a genetic based framework for solving the ordinary weighted matching problem we present an extensive analysis of the quality of the results and introduce a methodology for tuning its parameters in the second part we develop a modified genetic based algorithm for solving the mc wmp the algorithm generates a significant and representative portion of the pareto optimal set to assess the performance of the algorithm we conduct computational experiments with two and three criteria the potential of the proposed algorithm is demonstrated by comparing to a multi objective simulated annealing algorithm

evolutionary constructive and hybrid procedures for the bi objective set packing problem abstract the bi objective set packing problem is a multi objective combinatorial optimization problem similar to the well known set covering partitioning problems to our knowledge and surprise this problem has not yet been studied whereas several applications have been reported unfortunately solving the problem exactly in a reasonable time using a generic solver is only possible for small instances we designed three alternative procedures for approximating solutions to this problem the first is derived from the original strength pareto evolutionary algorithm which is a population based metaheuristic the second is an adaptation of the greedy randomized adaptative search procedure which is a constructive metaheuristic as underlined in the overview of the literature summarized here almost all the recent effective procedures designed for approximating optimal solutions to multi objective combinatorial optimization problems are based on a blend of techniques called hybrid metaheuristics thus the third alternative which is the primary subject of this paper is an original hybridization of the previous two metaheuristics the algorithmic aspects which differ from the original definition of these metaheuristics are described so that our results can be reproduced the performance of our procedures is reported and the computational results for 120 numerical instances are discussed

schur complements and statistics abstract in this paper we discuss various properties of matrices of the type s h ge 1 f which we call the schur complement of e in a e f g h the matrix e is assumed to be nonsingular when e is singular or rectangular we consider the generalized schur complements s h ge f where e is a generalized inverse of e a comprehensive account of results pertaining to the determinant the rank the inverse and generalized inverses of partitioned matrices and the inertia of a matrix is given both for schur complements and for generalized schur complements we survey the known results in a historical perspective and obtain several extensions numerous applications in numerical analysis and statistics are included the paper ends with an exhaustive bibliography of books and articles related to schur complements

a new approach for modeling and solving set packing problems in recent years the unconstrained quadratic binary program has emerged as a unified modeling and solution framework for a variety of combinatorial optimization problems experience reported in the literature with several problem classes has demonstrated that this unified approach works surprisingly well in terms of solution quality and computational times often rivaling and sometimes surpassing special purpose methods in this paper we report on the application of this unified framework to set packing problems with a variety of coefficient distributions computational experience is reported illustrating the attractiveness of the approach in particular our results highlight both the effectiveness and robustness of our approach across a wide array of test problems

the potentially stable tree sign patterns for dimensions less than five the problem of characterizing potentially stable sign pattern matrices remains unsolved even for tree patterns we present here in tabular form all irreducible n by n potentially stable sign patterns whose undirected graphs are trees for n 5 in hopes that this may be useful to others in formulating and evaluating conjectures on the subject some background and a descriptio of methodology which includes some new theoretical ideas are also included

further properties of the star left star right star and minus partial orderings certain classes of matrices are indicated for which the star left star right star and minus partial orderings or some of them are equivalent characterizations of the left star and right star orderings similar to those devised by hartwig and styan linear algebra appl 82 1986 145 for the star and minus orderings are established along with other auxiliary results which are of independent interest as well some inheritance type properties of matrices are also given the class of ep matrices appears to be essential in several points of our considerations

two local operators and the blue abstract for a given matrix a a matrix p such that pa a is said to be a local identity and such that p2a pa is said to be a local idempotent in the paper some simple properties of such operators are presented their relation to the best linear unbiased estimation in the general gauss markov model is also demonstrated

symmetrizers of matrices abstract a symmetrizer of a given pair of matrices a and b is defined as a matrix x for which the product axb is symmetric right and left symmetrizers of a given matrix a are defined accordingly the main results of the paper are general representations of all three types of symmetrizers the problem considered arose in connection with certain questions pertaining to admissible linear estimation in a gauss markoff model

on some characterizations of the star partial ordering for matrices and rank subtractivity abstract the main result is that drazin s star partial ordering a b holds if and only if a b and b a b a where a b is defined by a a a b and aa ba and where a b denotes rank subtractivity here a and b are m n complex matrices and the superscript denotes the moore penrose inverse several other characterizations of a b are given with particular emphasis on what extra condition must be added in order that rank subtractivity becomes the stronger star order a key tool is a new canonical form for rank subtractivity connections with simultaneous singular value decompositions schur complements and idempotent matrices are also mentioned

an equivalence between relational database dependencies and a fragment of propositional logic it is known that there is an eqmvalence between functional dependencies m a relatmonal database and a certain fragment of proposmonal logic thins eqmvalence is extended to include both functional and multivalued dependencmes thus for each dependency there is a corresponding statement m proposmonal logic it ms then shown that a dependency funcuonal or multivalued is a consequence of a set of dependencies ff and only ff the corresponding proposiuonal statement s a consequence of the corresponding set of proposmonal statements examples are given to show that these techniques are valuable mn provmdmg much shorter proofs of theorems about dependencies than have been obtained by more tradmonal means it is shown that this eqmvalence cannot be extended to include either join dependencies or embedded multmvalued dependencies

on some linear combinations of hypergeneralized projectors abstract the concept of a hypergeneralized projector as a matrix h satisfying h2 h where h denotes the moore penrose inverse of h was introduced by gros and trenkler generalized and hypergeneralized projectors linear algebra appl 264 1997 463 474 in the present paper the problem of when a linear combination c1h1 c2h2 of two hypergeneralized projectors h1 h2 is also a hypergeneralized projector is considered although a complete solution to this problem remains unknown this article provides characterizations of situations in which c1h1 c2h2 2 c1h1 c2h2 derived under certain commutativity property imposed on matrices h1 and h2 the results obtained substantially generalize those given in the above mentioned paper by gros and trenkler

characterizations of the best linear unbiased estimator in the general gauss markov model with the use of matrix partial orderings abstract under the general gauss markov model y x 2 v two new characterizations of blue x are derived involving the lowner and rank subtractivity partial orderings between the dispersion matrix of blue x and the dispersion matrix of y as particular cases of these characterizations three new criteria for the equality between olse x and blue x are given

the minus partial order and the shorted matrix the minus partial order was defined by hartwig 6 weakening the conditions of the star partial order of drazin 5 several new properties of the minus partial order are established the minus partial order is used to redefine the shorted matrix 15 17 and to define the infimum a b and the supremum a b of a pair a b of matrices of the same order the definition of the shorted matrix given here is similar to the krein anderson trapp definition of the shorted positive operator

automorphisms of mn partially ordered by rank subtractivity ordering let mn be the space of all n n matrices with coefficients in r or c where n 3 the rank subtractivity or minus partial order on mn is defined by a rsb iff rankb ranka rank b a we characterize bijective mappings on mn such that a rsb iff a rs b a b mn

sort sets in the relational model the notion of sort set is introduced here to formalize the fact that certain database relations can be sorted so that two or more columns are simultaneously listed in order this notion is shown to be applicable in several ways to enhance the efficiency of an implemented database a characterization of when order dependency implies the existence of sort sets in a database is presented along with several corollaries concerning complexity armstrong relations and cliques of certain graphs sort set dependencies are then introduced a finite sound and complete set of inference rules for sort set dependencies is presented as well as a proof that there is no such set for functional and sort set dependencies taken together deciding logical implication for sort set dependencies is proved to be polynomial but if functional dependencies are included the problem is co np complete each set of sort set and functional dependencies is shown to have an armstrong relation a natural generalization of armstrong relation here called separator is given and then used to study the relationship between order and sort set dependencies

characterizations and dispersion matrix robustness of efficiently estimable parametric functionals in linear models with nuisance parameters abstract algebraic representations dimensional expressions and characterizations are given for the subspace of functionals of the main parameters which can be estimated with full efficiency under a linear model y w z 2 v containing nuisance parameters subspaces of functionals of for which the ordinary least squares estimator is robust against an alternative dispersion matrix v are obtained and a particular subspace of such functionals is found wherein the ordinary least squares estimator is both dispersion matrix robust and robust against the presence of nuisance parameters

partial orderings of matrices referring to singular values or eigenvalues abstract a new partial ordering in the set of complex matrices is defined which is weaker than the star ordering introduced by drazin in 1978 and stronger than the minus ordering introduced by hartwig in 1980 this ordering refers to singular values of matrices and the interest in it was generated by canonical interpretations of the minus and star orderings given by hartwig and styan in 1986 for hermitian matrices a similar ordering referring to eigenvalues is also considered and its connection with a problem concerning distributions of quadratic forms in normal variables is pointed out

order dependency in the relational model abstract the relational model is formally extended to include fixed orderings on attribute domains a new constraint called order dependency is then introduced to incorporate semantic information involving these orderings it is shown that this constraint can be applied to enhance the efficiency of an implemented database the thrust of the paper is to study logical implication for order dependency the main theoretical results consist in i introducing a formalism analogous to propositional calculus for analyzing order dependency ii exhibiting a sound and complete set of inference rules for order dependency and iii demonstrating that determining logical implication for order dependency is co np complete it is also shown that there are sets of order dependencies for which no armstrong relations exist

grasp for set packing problems abstract the principles of the greedy randomized adaptative search procedure grasp metaheuristic are instantiated for the set packing problem we investigated several construction phases and evaluated improvements based on advanced strategies these improvements include a self tuning procedure using reactive grasp an intensification procedure using path relinking and a procedure involving the diversification of the selection using a learning process two sets of various numerical instances were used to perform the computational experiments the first set contains randomly generated instances while the second includes instances relating to real problems in railway planning no metaheuristic has previously been applied to this combinatorial problem consequently we have discussed grasp s performances both in relation to lower upper bounds and to the results obtained with cplex when such results are available our analysis based on the average performances observed shows the impact of the suggested strategies and indicates the configuration that produces the best results

a relationship between the star and minus orderings abstract the result of hartwig and styan 1986 stating that matrices a and b are star ordered if and only if they are minus ordered and b a b a 1 3 or b a b a 1 4 is extended by considering generalized inverses of a linear combination b a with any 0

characterizations for functional dependency and boyce codd normal form families abstract a functional dependency fd family was recently defined 20 as the set of all instances satisfying some set of functional dependencies a boyce codd normal form abbreviated bcnf family is defined here as an fd family specified by some bcnf set of functional dependencies the purpose of this paper is to present set theoretic algebraic characterizations relating to both types of families two characterizations of f i the smallest fd family containing the family i of instances are established the first involves the notion of agreement a concept related to that of a closed set of attributes the second describes f i as the smallest family of instances containing i and closed under four specific operations on instances companion results are also given for bcnf families the remaining results concern characterizations involving the well known operations of projection join and union two characterizations for when the projection of an fd family is again an fd family are given several corollaries are obtained including the effective decidability of whether a projection of an fd family is an fd family the problem for bcnf families disappears since it is shown that the projection of a bcnf family is always a bcnf family analogous to results for fd families presented in 20 characterizations of when the join and union of bcnf families are bcnf families are given finally the collections of all fd families and all bcnf families are characterized in terms of inverse projection operations and intersection

some properties of matrix partial orderings abstract the matrix partial orderings considered are 1 the star ordering and 2 the minus ordering or rank subtractivity both in the set of m n complex matrices and 3 the lowner ordering in the set of m m matrices the problems discussed are 1 inheriting certain properties under a given ordering 2 preserving an ordering under some matrix multiplications 3 relationships between an ordering among direct or kronecker and hadamard products and the corresponding orderings between the factors involved 4 orderings between generalized inverses of a given matrix and 5 preserving or reversing a given ordering under generalized inversions several generalizations of results known in the literature and a number of new results are derived

multiobjective grasp with path relinking in this paper we review and propose different adaptations of the grasp metaheuristic to solve multiobjective combinatorial optimization problems in particular we describe several alternatives to specialize the construction and improvement components of grasp when two or more objectives are considered grasp has been successfully coupled with path relinking for single objective optimization moreover we propose different hybridizations of grasp and path relinking for multiobjective optimization we apply the proposed grasp with path relinking variants to two combinatorial optimization problems the biobjective orienteering problem and the biobjective path dissimilarity problem we report on empirical tests with 70 instances and 30 algorithms that show that the proposed heuristics are competitive with the state of the art methods for these problems

on the structure of armstrong relations for functional dependencies an armstrong relation for a set of functional dependencies fds is a relation that satisfies each fd implied by the set but no fd that is not implied by it the structure and size number of tuples of armstrong relatsons are investigated upper and lower bounds on the size of minimal sized armstrong relations are derived and upper and lower bounds on the number of distinct entries that must appear m an armstrong relation are given it is shown that the time complexity of finding an armstrong relation gwen a set of functional dependencies is precisely exponential in the number of attributes also shown s the falsity of a natural conjecture which says that almost all relations obeying a given set of fds are armstrong relations for that set of fds finally armstrong relations are used to generahze a result obtained by demetrovics using quite complicated methods about the possible sets of keys for a relauon

the matrix inequality m b mb abstract the loewner partial ordering between nonnegative definite matrices m and b mb is considered the main result obtained generalizes a number of results known in the literature the rank subtractivity partial ordering and the drazin partial ordering between m and b mb are also discussed

left star and right star partial orderings abstract two partial orderings in the set of complex matrices are introduced by combining each of the conditions a a a b and aa ba which define the star partial ordering with one of the conditions m a m b and m a m b which define the space preordering several properties of these orderings are examined with main emphasis on comparing the new orderings with the star ordering the minus ordering and other related partial orderings moreover some further characterizations of partial orderings in terms of inclusions of appropriate classes of generalized inverses are derived with the main emphasis on characterizations involving reflexive generalized inverses