on prediction properties of kriging uniform error bounds and robustness kriging based on gaussian random fields is widely used in reconstructing unknown functions the kriging method has pointwise predictive distributions which are computationally simple however in m

approximation algorithms for the feedback vertex set problem with applications to constraint satisfaction and bayesian inference a feedback vertex set of an undirected graph is a subset of vertices that intersects with the vertex set of each cycle in the graph given an undirected graph g with n vertices and weights on its vertices polynomial time algorithms are provided for approximating the problem of finding a feedback vertex set of g with smallest weight when the weights of all vertices in g are equal the performance ratio attained by these algorithms is 4 2 n this improves a previous algorithm which achieved an approximation factor of o sqrt log n for this case for general vertex weights the performance ratio becomes min 2 delta 2 4 log 2 n where delta denotes the maximum degree in g for the special case of planar graphs this ratio is reduced to 10 an interesting special case of weighted graphs where a performance ratio of 4 2 n is achieved is the one where a prescribed subset of the vertices so called blackout vertices is not allowed to participate in any feedback vertex set it is shown how these algorithms can improve the search performance for constraint satisfaction problems an application in the area of bayesian inference of graphs with blackout vertices is also presented

w symmetries of ito stochastic differential equations we discuss w symmetries of ito stochastic differential equations introduced in a recent paper by gaeta and spadaro j math phys 58 053503 2017 in particular we discuss the general form of acceptable generators for continuous lie point w symmetry arguing that they are related to the linear conformal group and how w symmetries can be used in the integration of ito stochastic equations along kozlov theory for standard deterministic or random symmetries it turns out that this requires in general considering more general classes of stochastic equations than just ito ones we discuss w symmetries of ito stochastic differential equations introduced in a recent paper by gaeta and spadaro j math phys 58 053503 2017 in particular we discuss the general form of acceptable generators for continuous lie point w symmetry arguing that they are related to the linear conformal group and how w symmetries can be used in the integration of ito stochastic equations along kozlov theory for standard deterministic or random symmetries it turns out that this requires in general considering more general classes of stochastic equations than just ito ones

characterization of aquifer properties using artificial neural networks neural kriging a method for pattern completion based on the application of artificial neural networks and possessing many operational objectives of the ordinary kriging approach neural kriging is developed a neural kriging nk network is described implemented in a parallelizing algorithm and applied to develop maps of discrete spatially distributed fields e g log hydraulic conductivity nk is in the case of two discrete field values similar to indicator kriging it uses a feed forward counterpropagation training approach because field observations are available and because fast yet reliable results are obtained nk is data driven and requires no estimate of a covariance function the optimal design of the nk network is found to depend on the number of hidden units in a more complex way than expected the quality of the estimate of each pixel of the nk maps can be presented as well as in kriging to help identify areas in which additional information will be most beneficial a comparison with a reference field shows that the nk network produces unbiased errors relative to sample bias and reproduces the variogram of a quantized random field with reasonable accuracy ordinary kriging ok followed by quantization can also perform well however estimation errors in the variogram selected for use in ok in this case the range cofficient in particular must be carefully examined and treated the nk method can provide multiple realizations of the estimated field all of which respect observations hence conditional simulation is demonstrably possible the combination of simplicity interpolation reasonably accurate prediction statistics ability to provide conditional simulations and computational speed suggest that artificial neural networks can be useful tools in geohydrology when applied to specific well defined problems for which they are well suited such as aquifer characterization

on point covers of c oriented polygons let s be any family of n c oriented polygons of the two dimensional euclidean plane e2 i e bounded intersection of halfplanes whose normal directions of edges belong to a fixed collection of c distinct directions let s denote the packing number of s that is the maximum number of pairwise disjoint objects of s let s be the transversal number of s that is the minimum number of points required so that each object contains at least one of those points we provide linear time algorithms t n c t nc for a fat c oriented polytopes translates or homothets of ed proving that g 2 c o a d g 2 c dd and g 2 c 3d3 2 d respectively

random lie point symmetries of stochastic differential equations we study the invariance of stochastic differential equations under random diffeomorphisms and establish the determining equations for random lie point symmetries of stochastic differential equations both in ito and in stratonovich forms we also discuss relations with previous results in the literature

approximation algorithms for submodular vertex cover problems with linear submodular penalties using primal dual technique the notion of penalty has been introduced into many combinatorial optimization models in this paper we consider the submodular vertex cover problems with linear and submodular penalties which are two variants of the submodular vertex cover problem where not all the edges are required to be covered by a vertex cover and the uncovered edges are penalized the problem is to determine a vertex subset to cover some edges and penalize the uncovered edges such that the total cost including covering and penalty is minimized to overcome the difficulty of implementing the primal dual framework directly we relax the two dual programs to slightly weaker versions we then present two primal dual approximation algorithms with approximation ratios of 2 and 4 respectively

the group classification of a scalar stochastic differential equation lie point group classification of a scalar stochastic differential equation sde with one dimensional brownian motion is presented the admitted symmetry group can be zero one two or three dimensional if an equation admits a three dimensional symmetry group it can be transformed into the equation of brownian motion by a change of variables with non random time transformation such equations can be integrated by quadratures there are described drift coefficients for which sdes with constant diffusion coefficients admit two and three dimensional symmetry groups

reference points and approximation algorithms in multicriteria discrete optimization abstract mathematical research on multicriteria optimization problems predominantly revolves around the set of pareto optimal solutions in practice on the other hand methods that output a single solution are more widespread reference point methods are a successful example of this approach and are widely used in real world multicriteria optimization a reference point solution is the solution closest to a given reference point in the objective space we study the connection between reference point methods and approximation algorithms for multicriteria optimization problems over discrete sets in particular we establish that in terms of computational complexity computing approximate reference point solutions is polynomially equivalent to approximating the pareto set complementing these results we show for a number of general algorithmic techniques in single criteria optimization how they can be lifted to reference point optimization in particular we lift the link between dynamic programming and fptas as well as certain lp rounding techniques the latter applies e g to set cover and several machine scheduling problems

surrogate constraint normalization for the set covering problem the set covering problem scp is central in a wide variety of practical applications for which finding good feasible solutions quickly often in real time is crucial surrogate constraint normalization is a classical technique used to derive appropriate weights for surrogate constraint relaxations in mathematical programming this framework remains the core of the most effective constructive heuristics for the solution of the scp chiefly represented by the widely used chvatal method this paper introduces a number of normalization rules and demonstrates their superiority to the classical chvatal rule especially when solving large scale and real world instances directions for new advances on the creation of more elaborate normalization rules for surrogate heuristics are also provided

minimax designs for finite design regions the problem of choosing a design that is representative of a finite candidate set is an important problem in computer experiments the minimax criterion measures the degree of representativeness because it is the maximum distance of a candidate point to the design this article proposes a method for finding minimax designs for finite design regions we establish the relationship between minimax designs and the classical set covering location problem sclp in operations research which is a binary linear program in particular we prove that the set of minimax distances is the set of discontinuities of the function that maps the covering radius to the optimal objective function value we show that solving the sclp at the points of discontinuities which can be determined gives minimax designs these results are employed to design an efficient procedure for finding minimax designs for small sized candidate sets a heuristic procedure is proposed to generate near minimax designs for large candidate sets s

a remark on symmetry of stochastic dynamical systems and their conserved quantities the symmetry properties of stochastic dynamical systems described by a stochastic differential equation of stratonovich type and related conserved quantities are discussed extending previous results by misawa new conserved quantities are given by applying symmetry operators to known conserved quantities some detailed examples are presented

the turning bands method for simulation of random fields using line generation by a spectral method the turning bands method tbm for the simulation of multidimensional random fields is presented these fields commonly occur in the monte carlo simulation of hydrologic processes particularly groundwater flow and mass transport the general tbm equations for two and three dimensional fields are derived with particular emphasis on the more complicated two dimensional case for stationary two dimensional fields the unidimensional line process is generated by a simple spectral method a technique which can be generally applied to any two dimensional covariance function and which is easily extended to anisotropic and areal averaged processes theoretically and by example the tbm is shown to be ergodic even for a finite number of lines and it is demonstrated that it rapidly converges to the true statistics of the field guidelines are presented for the selection of model parameters which will be helpful in the design of simulation experiments the tbm is compared to other methods in terms of cost and accuracy demonstrating that the tbm is as accurate as and much less expensive than multidimensional spectral techniques and more accurate than the most expensive approaches which use matrix inversion such as the nearest neighbor approach the unidimensional spectral technique presented here permits for the first time the inexpensive and accurate tbm simulation of any proper two dimensional covariance function and should be of some help in the stochastic analysis of hydrologic processes

lie point symmetries and stochastic differential equations we discuss lie point symmetries of stochastic ordinary differential equations and the interrelations between these and analogous symmetries of the associated fokker planck equation for the probability measure

semidefinite programming in combinatorial optimization we discuss the use of semidefinite programming for combinatorial optimization problems the main topics covered include i the lovasz theta function and its applications to stable sets perfect graphs and coding theory ii the automatic generation of strong valid inequalities iii the maximum cut problem and related problems and iv the embedding of finite metric spaces and its relationship to the sparsest cut problem

the relationship between attribute reducts in rough sets and minimal vertex covers of graphs the problems to find attribute reduction in rough sets and to obtain the minimal vertex cover for graphs are both np hard problems this paper studies the relationship between the two problems the vertex cover problem for graphs from the perspective of rough sets is first investigated the attribute reduction of an information system is then studied in the framework of graph theory the results in this paper show that finding the minimal vertex cover of a graph is equivalent to finding the attribute reduction of an information system induced from the graph conversely the attribute reduction computation can be translated into the calculation of the minimal vertex cover of a derivative graph finally a new algorithm for the vertex cover problem based on rough sets is presented furthermore experiments are conducted to verify the effectiveness of the proposed method

a tale of two methods we describe two widely used methods for the design and analysis of approximation algorithms the primal dual schema and the local ratio technique we focus on the creation of both methods by revisiting two results by bar yehuda and even the linear time primal dual approximation algorithm for set cover and its local ratio interpretation we also follow the evolution of the two methods by discussing more recent studies

from valid inequalities to heuristics a unified view of primal dual approximation algorithms in covering problems in recent years approximation algorithms based on primal dual methods have been successfully applied to a broad class of discrete optimization problems in this paper we propose a generic primal dual framework to design and analyze approximation algorithms for integer programming problems of the covering type that uses valid inequalities in its design the worst case bound of the proposed algorithm is related to a fundamental relationship called strength between the set of valid inequalities and the set of minimal solutions to the covering problems in this way we can construct an approximation algorithm simply by constructing the required valid inequalities we apply the proposed algorithm to several problems such as covering problems related to totally balanced matrices cyclic scheduling vertex cover general set covering intersections of polymatroids and several network design problems attaining in most cases the best worst case bound known in the literature

on the complexity of learning lexicographic strategies fast and frugal heuristics are well studied models of bounded rationality psychological research has proposed the take the best heuristic as a successful strategy in decision making with limited resources take the best searches for a sufficiently good ordering of cues or features in a task where objects are to be compared lexicographically we investigate the computational complexity of finding optimal cue permutations for lexicographic strategies and prove that the problem is np complete it follows that no efficient that is polynomial time algorithm computes optimal solutions unless p np we further analyze the complexity of approximating optimal cue permutations for lexicographic strategies we show that there is no efficient algorithm that approximates the optimum to within any constant factor unless p np the results have implications for the complexity of learning lexicographic strategies from examples they show that learning them in polynomial time within the model of agnostic probably approximately correct pac learning is impossible unless rp np we further consider greedy approaches for building lexicographic strategies and determine upper and lower bounds for the performance ratio of simple algorithms moreover we present a greedy algorithm that performs provably better than take the best tight bounds on the sample complexity for learning lexicographic strategies are also given in this article

integration of stochastic ordinary differential equations from a symmetry standpoint we derive an algorithm for calculating lie point symmetries of systems of stochastic ordinary differential equations sodes of any order from this algorithm the following facts emerge symmetries of a sode do not in general form a lie algebra the determining equations for ito s equation are in general stochastic linear partial differential equations whereas for sodes of order n 2 the determining equations are linear deterministic partial differential equations that form an overdetemined system which is solvable by classical methods for scalar second order sodes we provide a complete classification of equations admitting finite dimensional symmetry lie algebras this classification is applied to the integration of scalar second order sodes in general a sode admitting a two dimensional symmetry algebra is not integrable by quadratures although it is reducible to a homogeneous ito equation in particular a scalar second order sode admitting a two dimensional symmetry algebra with connected operators is linearizable we also characterize integrable scalar second order sodes admitting three dimensional symmetry algebras finally we show that a sode can admit maximally a zero one two three or four dimensional lie algebra

new complexity results for the k covers problem the k covers problem kcp asks us to compute a minimum cardinality set of strings of given length k 1 that covers a given string it was shown in a recent paper by reduction to 3 sat that the k covers problem is np complete in this paper we introduce a new problem that we call the k bounded relaxed vertex cover problem rvcp k which we show is equivalent to k bounded set cover scp k we show further that kcp is a special case of rvcp k restricted to certain classes g x k of graphs that represent all strings x thus a minimum k cover can be approximated to within a factor k in polynomial time we discuss approximate solutions of kcp and we state a number of conjectures and open problems related to kcp and g x k

approximation of set multi cover via hypergraph matching abstract for a given b n n and a hypergraph h v e with maximum degree the b set multicover problem which we are concerned with in this paper may be stated as follows find a minimum cardinality subset c e such that no vertex v i v is contained in less than b i hyperedges of c peleg schechtman and wool 1997 conjectured that for any fixed and b min i b i the problem cannot be approximated with a ratio strictly smaller than b 1 unless p np in this paper we show that the conjecture of peleg et al is not true on l uniform hypergraphs by presenting a polynomial time approximation algorithm based on a matching covering duality for hypergraphs due to ray chaudhuri 1960 which we convert into an approximative form the given algorithm yields a ratio smaller than 1 b l 1 b moreover we prove that the lower bound conjectured by peleg et al holds for regular hypergraphs under the unique games conjecture

implementation of the three dimensional turning bands random field generator numerical techniques to generate replicates of spatially correlated random fields are often used to synthesize sets of highly variable physical quantities in stochastic models of naturally heterogeneous systems within the realm of hydrologic research for example such tools are widely used to develop hypothetical rainfall distributions hydraulic conductivity fields fracture set properties and other surface or subsurface flow parameters the turning bands method is one such algorithm which generates two and three dimensional fields by combining values found from a series of one dimensional simulations along lines radiating outward from a coordinate origin previous work with two dimensional algorithms indicates that radial lines evenly spaced about the unit circle lead to enhanced convergence properties the same can be said for the three dimensional models but it is more difficult to choose an arbitrary number of evenly spaced lines about the unit sphere the current investigation shows that the use of larger numbers of randomly oriented lines 100 can enhance the performance of the three dimensional algorithm this improved performance is needed to effectively simulate problems characterized by full three dimensionality and or anisotropy in either monte carlo or single realization applications use of a large number of lines will also reduce the presence of a distortion effect manifested as linelike patterns in the field increased computational costs can be reduced by employing a fast fourier transform technique to generate the line processes

new conserved quantities derived from symmetry for stochastic dynamical systems recently the author has proposed an elementary theory of conserved quantities and symmetry for stochastic dynamical systems described by stochastic differential equations of stratonovich type within the framework a new method for deriving conserved quantities from symmetry is developed in a similar manner to that of hojman s work 1992 which gives a new conservation law constructed without using either lagrangians or hamiltonians for deterministic dynamical systems some examples of the conserved quantities obtained through the new method are given

fixed domain asymptotic properties of tapered maximum likelihood estimators when the spatial sample size is extremely large which occurs in many environmental and ecological studies operations on the large covariance matrix are a numerical challenge covariance tapering is a technique to alleviate the numerical challenges under the assumption that data are collected along a line in a bounded region we investigate how the tapering affects the asymptotic efficiency of the maximum likelihood estimator mle for the microergodic parameter in the matern covariance function by establishing the fixed domain asymptotic distribution of the exact mle and that of the tapered mle our results imply that under some conditions on the taper the tapered mle is asymptotically as efficient as the true mle for the microergodic parameter in the matern model

stochastic analysis of nonstationary subsurface solute transport 1 unconditional moments this paper applies stochastic methods to the analysis and prediction of solute transport in heterogeneous saturated porous media partial differential equations for three unconditional ensemble moments the concentration mean concentration covariance and velocity concentration cross covariance are derived by applying perturbation techniques to the governing transport equation for a conservative solute concentration uncertainty is assumed to be the result of unmodeled small scale fluctuations in a steady state velocity field the moment expressions which describe how each moment evolves over time and space resemble the classic deterministic advection dispersion equation and can be solved using similar methods a solution procedure based on a galerkin finite element algorithm is illustrated with a hypothetical two dimensional example for this example the required steady state velocity statistics are obtained from an infinite domain spectral solution of the stochastic groundwater flow equation the perturbation solution is shown to reproduce the statistics obtained from a monte carlo simulation quite well for a natural log conductivity standard deviation of 0 5 and moderately well for a natural log conductivity standard deviation of 1 0 the computational effort required for a perturbation solution is significantly less than that required for a monte carlo solution of acceptable accuracy sensitivity analyses conducted with the perturbation approach provide qualitative confirmation of a number of results obtained by other investigators for more restrictive special cases

aquifer operator scaling and the effect on solute mixing and dispersion 1 since aquifer parameters may have statistical dependence structures that are present across a huge range of scales the concepts of fractional brownian motion fbm have been used in both analytic and numerical stochastic settings most previous models have used isotropic scaling characterized by a single scalar hurst coefficient any real world anisotropy has been handled by an elliptical stretching random k field we define a d dimensional extension of fbm in which the fractional order integration may take on different orders in the d primary possibly nonorthogonal scaling directions and the degree of connectivity and long range dependence is freely assigned via a probability measure on the unit sphere this approach accounts for the different scaling found in the vertical versus horizontal directions in sedimentary aquifers and allows very general degrees of continuity of k in certain directions it also allows for the representation of fracture networks in a continuum setting the eigenvectors of the scaling matrix describe the primary fracture scaling directions and discrete weights of fractional integration represent fracture continuity that may be limited to a small number of directions in a numerical experiment the motion of solutes through 2 d operator fractional gaussian fields depends very much on transverse hurst coefficients transverse scaling in the range of fractional gaussian noise engenders greater plume mixing and a transition to a fickian regime higher orders of integration in the range of fractional brownian motion are associated with thicker layering of aquifer sediments and more preferential unmixed transport therefore direct representation of the unique directional scaling properties of an aquifer is important for realistic simulation of transport

probabilistic analysis of a combined aggregation and math programming heuristic for a general class of vehicle routing and scheduling problems we propose and analyze a heuristic that uses region partitioning and an aggregation scheme for customer attributes load size time windows etc to create a finite number of customer types a math program is solved based on these aggregated customer types to generate a feasible solution to the original problem the problem class we address is quite general and defined by a number of general consistency properties problems in this class include vrps with general distance norms capacitated problems time window vrps pick up and delivery problems combined inventory control and routing problems and arc routing we provide a probabilistic analysis of this heuristic under very general probabilistic assumptions in particular we do not require independence between customer locations and their various attributes the heuristic is a s optimal as the number of customers n tends to infinity further it runs in o n log n time for a fixed relative error and can be designed to be asymptotically optimal w

effects of kriging and inverse modeling on conditional simulation of the avra valley aquifer in southern arizona the avra valley aquifer in southern arizona is modeled stochastically at three levels of uncertainty the highest level of uncertainty occurs when log transmissivity estimates are based on measured values of this parameter but without regard to the geographic location of each measurement point the resulting steady state hydraulic heads in the aquifer computed by unconditional simulation with the aid of a multivariate normal random number generator coupled with a finite element model have a relatively large variance this variance can be reduced by conditioning the log transmissivity estimates on the spatial arrangement of the data by means of kriging when conditional simulation of the aquifer is performed at this intermediate level of uncertainty with the aid of the same technique as in the unconditional case the variance of the predicted head values is reduced by a factor of 3 2 the lowest level of uncertainty is achieved when the log transmissivity estimates are further conditioned on data relating to the flow regime such as flow rates and water levels in observation wells by means of a statistical inverse procedure conditional simulation at this level of uncertainty is a novel concept it results in a hydraulic head prediction variance that is 14 3 times lower than the corresponding variance based on kriged log transmissivities the net effect of conditioning by means of kriging and inverse modeling is to reduce the prediction variance by a factor of 46 0 a similar study performed by binsarti 1980 on the cortaro aquifer in southern arizona showed insignificant variance reduction due to kriging but a factor of four reduction after inverse modeling these results indicate that the conditioning effect of the inverse method may be much greater than that of kriging the fact that conditioning can reduce the hydraulic head prediction variance by as much as a factor of 46 implies that one should be cautious when dealing with the results of stochastic aquifer models in which conditioning effects are disregarded

solving integer programs over monotone inequalities in three variables a framework for half integrality and good approximations abstract we define a class of monotone integer programs with constraints that involve up to three variables each a generic constraint in such integer program is of the form ax by z c where a and b are nonnegative and the variable z appears only in that constraint we devise an algorithm solving such problems in time polynomial in the length of the input and the range of variables u the solution is obtained from a minimum cut on a graph with o nu nodes and o mu arcs where n is the number of variables of the types x and y and m is the number of constraints our algorithm is also valid for nonlinear objective functions nonmonotone integer programs are optimization problems with constraints of the type ax by z c without restriction on the signs of a and b such problems are in general np hard we devise here an algorithm relying on a transformation to the monotone case that delivers half integral superoptimal solutions in polynomial time such solutions provide bounds on the optimum value that can only be superior to bounds provided by linear programming relaxation when the half integral solution can be rounded to an integer feasible solution this is a 2 approximate solution in that the technique is a unified 2 approximation technique for a large class of problems the results apply also for general integer programming problems with worse approximation factors that depend on a quantifier measuring how far the problem is from the class of problems we describe the algorithm described here has a wide array of problem applications an additional important consequence of our results is that nonmonotone problems in the framework are max snp hard and at least as hard to approximate as vertex cover problems that are amenable to the analysis provided here are easily recognized the analysis itself is entirely technical and involves manipulating the constraints and transforming them to a totally unimodular system while losing no more than a factor of 2 in the integrality

computational experience with approximation algorithms for the set covering problem the set covering problem scp is a well known combinatorial optimization problem which is np hard we conducted a comparative study of eight different approximation algorithms for the scp including several greedy variants fractional relaxations randomized algorithms and a neural network algorithm the algorithms were tested on a set of random generated problems with up to 500 rows and 5000 columns and on two sets of problems originating in combinatorial questions with up to 28160 rows and 11264 columns on the random problems and on one set of combinatorial problems the best algorithm among those we tested was the neural network algorithm with greedy variants very close in second and third place on the other set of combinatorial problems the best algorithm was a greedy variant and the neural network performed quite poorly the other algorithms we tested were always inferior to the ones mentioned above

a graph approximation heuristic for the vertex cover problem on planar graphs abstract the vertex cover problem is hard in general and remains so on planar graphs on the other hand it is always solvable on bipartite graphs in this paper we introduce a strategy which creates from an arbitrary planar graph g a minimum size bipartite homeomorph of g say g h we solve the vertex cover problem on g h and convert the solution to a cover on g tests on a set of sample graphs suggest that the resulting heuristic performs quite well

storage capacity as an information theoretic vertex cover and the index coding rate motivated by applications in distributed storage the storage capacity of a graph was recently defined to be the maximum amount of information that can be stored across the vertices of a graph such that the information at any vertex can be recovered from the information stored at the neighboring vertices computing the storage capacity is a fundamental problem in network coding and is related or equivalent to some well studied problems such as index coding with side information and generalized guessing games in this paper we consider storage capacity as a natural information theoretic analogue of the minimum vertex cover of a graph indeed while it was known that storage capacity is upper bounded by minimum vertex cover we show that by treating it as such we can get a 3 2 approximation for planar graphs and a 4 3 approximation for triangle free planar graphs since the storage capacity is intimately related to the index coding rate we get a 2 approximation of index coding rate for planar graphs and 3 2 approximation for triangle free planar graphs previously only a trivial 4 approximation of the index coding rate was known for planar graphs we also show a polynomial time approximation scheme for the index coding rate when the alphabet size is constant we then develop a general method of gadget covering to upper bound the storage capacity in terms of the average of a set of vertex covers this method is intuitive and leads to the exact characterization of storage capacity for various families of graphs as an illustrative example we use this approach to derive the exact storage capacity of cycles with chords a family of graphs related to outerplanar graphs finally we generalize the storage capacity notion to include recovery from partial node failures in distributed storage we show tight upper and lower bounds on this partial recovery capacity that scales nicely with the fraction of failures in a vertex

bounds on the efficiency of linear predictions using an incorrect covariance function suppose z is a random process defined on a bounded set r r d with finite second moments consider the behavior of linear predictions based on z t 1 z t n where t 1 t 2 is a dense sequence of points in r a general method is described for obtaining rates of convergence when the covariance function is misspecified but compatible with the correct covariance function when z is gaussian these bounds are related to the entropy distance between the measures for the random field under the actual and presumed covariance functions explicit bounds are given when r 0 1 and z is stationary with spectral density of the form f a 2 2 p where is a known positive integer and a is the parameter that is misspecified

a distributed 2 e approximation for vertex cover in o log δ e log log δ rounds we present a simple deterministic distributed 2 e approximation algorithm for minimum weight vertex cover which completes in o log elog log rounds where is the maximum degree in the graph for any e 0 that is at most o 1 for a constant e this implies a constant approximation in o log log log rounds which contradicts the lower bound of kmw10

the primal dual method for approximation algorithms in this survey we give an overview of a technique used to design and analyze algorithms that provide approximate solutions to np hard problems in combinatorial optimization because of parallels with the primal dual method commonly used in combinatorial optimization we call it the primal dual method for approximation algorithms we show how this technique can be used to derive approximation algorithms for a number of different problems including network design problems feedback vertex set problems and facility location problems

guaranteed convergence in a class of hopfield networks a class of symmetric hopfield networks with nonpositive synapses and zero threshold is analyzed in detail it is shown that all stationary points have a one to one correspondence with the minimal vertex covers of certain undirected graphs that the sequential hopfield algorithm as applied to this class of networks converges in at most 2n steps n being the number of neurons and that the parallel hopfield algorithm either converges in one step or enters a two cycle in one step the necessary and sufficient condition on the initial iterate for the parallel algorithm to converge in one step are given a modified parallel algorithm which is guaranteed to converge in 3n 2 steps x being the integer part of x for an n neuron network of this particular class is also given by way of application it is shown that this class naturally solves the vertex cover problem simulations confirm that the solution provided by this method is better than those provided by other known methods

approximation algorithms for the set covering and vertex cover problems we propose a heuristic that delivers in o n 3 steps a solution for the set covering problem the value of which does not exceed the maximum number of sets covering an element times the optimal value

lie point symmetries of stratonovich stochastic differential equations this paper considers lie point symmetries of stochastic differential equations sdes in stratonovich form first we derive the determining equations of the random symmetries which correspond to lie group transformations acting in the space of the independent variable time the dependent variables and the brownian motion then the determining equations of the random symmetries are used to obtain the determining equations of the deterministic symmetries which correspond to lie group transformations acting in the space of the independent variable and dependent variables in the case of the deterministic symmetries the transformation of the brownian motion is induced by the random time change it is proved that the corresponding ito and stratonovich sdes admit the same random symmetries and the same deterministic symmetries

symmetries of systems of stochastic differential equations with diffusion matrices of full rank lie point symmetries of a system of stochastic differential equations sdes with diffusion matrices of full rank are considered it is proved that the maximal dimension of a symmetry group admitted by a system of n sdes is n 2 in addition such systems cannot admit symmetry operators whose coefficients are proportional to a nonconstant coefficient of proportionality these results are applied to compute the lie group classification of a system of two sdes the classification is obtained with the help of non equivalent realizations of real lie algebras by fiber preserving vector fields in 1 2 variables possibilities of using symmetries for integration of sdes by quadratures are discussed

prediction of spatial cumulative distribution functions using subsampling abstract the spatial cumulative distribution function scdf is a random function that provides a statistical summary of a random field over a spatial domain of interest in this article we develop a spatial subsampling method for predicting an scdf based on observations made on a hexagonal grid similar to the one used in the environmental monitoring and assessment program of the u s environmental protection agency we show that under quite general conditions the proposed subsampling method provides accurate data based approximations to the sampling distributions of various functionals of the scdf predictor in particular it produces estimators of different population characteristics such as the quantiles and weighted mean integrated squared errors of the empirical predictor as an illustration we apply the subsampling method to construct large sample prediction bands for the scdf of an ecological index for foliage condition of red maple trees in the state of maine

combinatorics for smaller kernels the differential of a graph let g v e be a graph of order n and let b d be the set of vertices in v d that have a neighbor in the vertex set d the differential of d is defined as d b d d and the differential of a graph g written g is equal to max d d v if g is connected and n 3 g n 5 is known this immediately leads to a linear vertex kernel result in the terminology of parameterized complexity for the problem of deciding whether g k taking k as the parameter we then establish a new combinatorial result which establishes that g n 4 if g is a connected graph of order n 6 and if g contains no induced path of five vertices whose midpoint is a cut vertex and whose endpoints have degree one this technical combinatorial theorem can be used to derive an even smaller linear vertex kernel for general graphs also we show that the related maximization problem allows for a polynomial time factor 14 approximation algorithm

a randomised approximation algorithm for the hitting set problem let h v e be a hypergraph with vertex set v and edge set e where n v and m e let l be the maximum size of an edge and be the maximum vertex degree a hitting set or vertex cover in h is a subset of v in which all edges are incident the hitting set problem is to find a hitting set of minimum cardinality it is known that an approximation ratio of l can be achieved easily on the other hand for constant l an approximation ratio better than l cannot be achieved in polynomial time under the unique games conjecture khot and regev 2008 17 thus breaking the l barrier for significant classes of hypergraphs is a complexity theoretically and algorithmically interesting problem which has been studied by several authors krivelevich 1997 18 halperin 2000 12 okun 2005 23 we propose a randomised algorithm of hybrid type for the hitting set problem which combines lp based randomised rounding graphs sparsening and greedy repairing and analyse it for different classes of hypergraphs for hypergraphs with o n 1 4 and l o n we achieve an approximation ratio of l 1 c for some constant c 0 with constant probability for the case of hypergraphs where l and are constants we prove a ratio of l 1 l 1 8 the latter is done by analysing the expected size of the hitting set and using concentration inequalities moreover for quasi regularisable hypergraphs we achieve an approximation ratio of l 1 n 8 m we show how and when our results improve over the results of krivelevich halperin and okun

fast stabbing of boxes in high dimensions abstract we present in this paper a simple yet efficient algorithm for stabbing a set s of n axis parallel boxes in d dimensional space with c s points in output sensitive time o dn log c s and linear space let c s and b s be respectively the minimum number of points required to stab s and the maximum number of pairwise disjoint boxes of s we prove that b s c s c s b s 1 log 2 b s d 1 since finding a minimal set of c s points is np complete as soon as d 1 we obtain a fast precision sensitive heuristic for stabbing s whose quality does not depend on the input size in the case of congruent or constrained isothetic boxes our algorithm reports respectively c s 2 d 1 b s and c s o d b s stabbing points moreover we show that the bounds we get on c s are asymptotically tight and corroborate our results with some experiments we also describe an optimal output sensitive algorithm for finding a minimal size optimal stabbing point set of intervals finally we conclude with insights for further research

efficient approximation algorithms for maximum coverage with group budget constraints abstract given a ground set u with a non negative weight w i for each i u a positive integer k and a collection of sets s which is partitioned into a family of disjoint groups g the goal of the maximum coverage problem with group budget constraints mcg is to select k sets from s such that the total weight of the union of the k sets is maximized and at most one set is selected from each group g g we first present an approximation algorithm with a factor 1 1 e and an exponential time via randomized linear programming rounding technique then we improve the time complexity of the algorithm to o m n 3 5 l k 3 5 q 7 l for s m u n and l being the length of the input by the key idea of modeling the selection of groups as computing a constrained flow in a corresponding auxiliary graph the algorithm is later shown can be extended to solve two generalizations of mcg last but not the least we present another algorithm with a time complexity o m n 3 5 l k 10 5 l and a slightly increased approximation ratio 1 e 1 1 mainly based on the idea of partition where 2 is a parameter tuning which can balance the time complexity and the ratio

random lie symmetries of ito stochastic differential equations lie point symmetries of stochastic differential equations sdes which include transformations depending on the brownian motion are considered the corresponding lie group transformations acting in the space of the independent variable time the dependent variables and the brownian motion preserve the differential form of sdes and the properties of the brownian motion such symmetries are called random symmetries the focus of the paper is a definition of the random symmetries and their properties it is shown that the random symmetries form a lie algebra we also investigate relations between the random symmetries and the conserved quantities of sdes for several stochastic differential equations we find their random symmetries and compare them with the deterministic symmetries which correspond to lie group transformations in the space of the independent and dependent variables

the multicovering problem abstract the multicovering problem is min cx subject to ax b x 0 where a is a matrix whose elements are all zero or one and b is a vector of positive integers this problem has many applications in scheduling location and business areas a variety of industrial commercial and military contexts give rise to large problems of this kind frequently at or beyond the limits of current computational tractability our research goal is the development of a special purpose algorithm capable of solving large problems of this kind conceptual and computational inputs to the algorithm arise from three sources from the generalization of computational tools developed for more specific problems in particular the set covering problem from the adaptation of general integer programming and combinatorial optimization methods and from the development within this paper of special techniques designed to take advantage of the structure of the problem studied following a description of the algorithm we give some computational results for randomly generated problems

asymptotically efficient prediction of a random field with a misspecified covariance function on peut obtenir les meilleurs predicteurs lineaires sans biais d un champ aleatoire si la fonction de covariance du champ aleatoire est specifiee correctement

partial multicuts in trees let t v e be an undirected tree in which each edge is associated with a non negative cost and let s1 t1 sk tk be a collection of k distinct pairs of vertices given a requirement parameter t k the partial multicut on a tree problem asks to find a minimum cost set of edges whose removal from t disconnects at least t out of these k pairs this problem generalizes the well known multicut on a tree problem in which we are required to disconnect all given pairs the main contribution of this paper is an 8 3 e approximation algorithm for partial multicut on a tree whose run time is strongly polynomial for any fixed e 0 this result is achieved by introducing problem specific insight to the general framework of using the lagrangian relaxation technique in approximation algorithms our algorithm utilizes a heuristic for the closely related prize collecting variant in which we are not required to disconnect all pairs but rather incur penalties for failing to do so we provide a lagrangian multiplier preserving algorithm for the latter problem with an approximation factor of 2 finally we present a new 2 approximation algorithm for multicut on a tree based on lp rounding

a multiscale finite element linearization scheme for the unsaturated flow problems in heterogeneous porous media 1 numerical solution of unsaturated water flow in heterogeneous porous media is difficult not only because of the spatial variability in the parameters used to characterize the relevant physical properties of the natural porous media but also because of the mathematical problems that arise in dealing with the nonlinearities in this study a multiscale finite element linearization scheme is presented for effectively simulating unsaturated flow in heterogeneous porous media spanning over many scales the central goal of this method is to obtain the large scale solution of richards equation with heterogeneous coefficients accurately and efficiently on a coarse grid without resolving all the small scale details the underlying idea is to employ the slodicka linear relaxation approximation scheme to address the nonlinear characteristics of the equation and to use the multiscale finite element base functions to account for the spatial variability in the equation coefficients we describe the principle for constructing such a method and give an algorithm for implementing it numerical experiments are carried out for the unsaturated flow equation with both periodic and randomly generated lognormal hydraulic conductivity to demonstrate the efficiency and accuracy of the proposed method

spatial averaging of transmissivity in heterogeneous fields with flow toward a well steady state nonuniform flow in a heterogeneous medium is investigated using a combined numerical empirical approach the objective of the study is to determine the effective transmissivity of a square field with a constant head external boundary and a constant rate well at its center transmissivity at the point scale is modeled as an isotropic and multivariate lognormal spatial random function block transmissivities are defined using an empirical scaling up of point support values within the field the scaling process is a weighted spatial geometric averaging where log transmissivities are weighted by the inverse square of their distance from the well block transmissivities thus obtained and true effective values calculated using a numerical flow model are compared for realizations a finely discretized transmissivity field the principal finding of this study is that effective transmissivities obtained from the numerical flow model and block values obtained by spatial averaging are in excellent agreement for low to moderate variances of log transmissivity and moderately eccentric field geometries the geostatistical model for point scale transmissivity and the deterministic spatial averaging law are combined in order to create a geostatistical model for transmissivity at the block scale exact expressions are derived for the ensemble moments of block averaged transmissivities and well drawdowns in both the nonconditional case and the conditional case when transmissivity at the well bore is known the ensemble mean of block averaged transmissivity is found to decrease from the ensemble arithmetic mean toward the ensemble geometric mean as the field size becomes large compared to the integral range of spatial correlation this result contradicts earlier published work numerical results presented in this study also disagree with the analytical results of a recent stochastic analysis of specific discharge in radial systems

rounding algorithms for covering problems in the last 25 years approximation algorithms for discrete optimization problems have been in the center of research in the fields of mathematical programming and computer science recent results from computer science have identified barriers to the degree of approximability of discrete optimization problems unless p np as a result as far as negative results are concerned a unifying picture is emerging on the other hand as far as particular approximation algorithms for different problems are concerned the picture is not very clear different algorithms work for different problems and the insights gained from a successful analysis of a particular problem rarely transfer to another our goal in this paper is to present a framework for the approximation of a class of integer programming problems covering problems through generic heuristics all based on rounding deterministic using primal and dual information or randomized but with nonlinear rounding functions of the optimal solution of a linear programming lp relaxation we apply these generic heuristics to obtain in a systematic way many known as well as new results for the set covering facility location general covering network design and cut covering problems 1998 the mathematical programming society inc published by elsevier science b v

clustering through continuous facility location problems we consider the continuous facility location problem conflp given a finite set of clients c r d and a number f r conflp consists in opening a set f r d of facilities each at cost f and connecting each client to an open facility the objective is to minimize the costs of opening facilities and connecting clients we reduce conflp to the standard facility location problem flp by using the so called approximate center sets this reduction preserves the approximation except for an error e and implies 1 488 e and 2 04 e approximations when the connection cost is given by the euclidean distance and the squared euclidean distance respectively moreover we obtain approximate center sets for the case that the connection cost is the th power of the euclidean distance achieving approximations for the corresponding problems for any 1 as a byproduct we also obtain a polynomial time approximation scheme for the k median problem with this cost function for any fixed k

fixed domain asymptotics for spatial periodograms abstract the periodogram for a spatial process observed on a lattice is often used to estimate the spectral density the bases for such estimators are two asymptotic properties that periodograms commonly possess 1 the periodogram at a particular frequency is approximately unbiased for the spectral density and 2 the correlation of the periodogram at distinct frequencies is approximately zero for spatial data it is often appropriate to use fixed domain asymptotics in which the observations get increasingly dense in some fixed region as their number increases using fixed domain asymptotics this article shows that standard asymptotic results for periodograms do not apply and that using the periodogram of the raw data can yield highly misleading results but by appropriately filtering the data before computing the periodogram it is possible to obtain results similar to the standard asymptotic results for spatial periodograms

a bicriteria algorithm for the minimum submodular cost partial set multi cover problem abstract this paper presents a bicriteria approximation algorithm for the minimum submodular cost partial set multi cover problem scpsmc the goal of which is to find a minimum cost sub collection of sets to fully cover q percentage of total profit of all elements where the cost on sub collections is a submodular function and an element e with covering requirement r e is fully covered if it belongs to at least r e picked sets assuming that the maximum covering requirement r max max e e r e is a constant and the cost function is nonnegative and submodular we give a deterministic b q e 1 e bicriteria algorithm for scpsmc the output of which fully covers at least 1 e q percentage of the total profit and the performance ratio is b q e where b max e f e r e and f e is the number of sets containing element e

simple lagrangian heuristic for the set covering problem abstract in this paper a simple lagrangian heuristic is proposed for the set covering problem it is based on simple and classical ideas lagrangian duality greedy heuristic for the set covering problem subgradient optimization and redundant covers the main interesting point of the paper lies in the fact that the proposed heuristic turns out to be efficient for low density set covering problems with a large number of variables

efficient approximation of min set cover by moderately exponential algorithms we study the approximation of min set cover combining ideas and results from polynomial approximation and from exact computation with non trivial worst case complexity upper bounds for np hard problems we design approximation algorithms for min set cover achieving ratios that cannot be achieved in polynomial time unless problems in np could be solved by slightly super polynomial algorithms with worst case complexity much lower though super polynomial than those of an exact computation

on maximal lie point symmetry groups admitted by scalar stochastic differential equations it is proved that the lie point symmetry group admitted by a scalar stochastic differential equation sde of order n 3 is at most n 2 dimensional this result supplements those for first and second order sdes maximal dimensionality can be used to carry out lie group classifications lie point group classification of the third order sdes is provided