numerical approximation of the schrodinger equation with the electromagnetic field by the hagedorn wave packets in this paper we approximate the semi classical schrodinger equation in the presence of electromagnetic field by the hagedorn wave packets approach by operator splitting the hamiltonian is divided into the modified part and the residual part the modified hamiltonian which is the main new idea of this paper is chosen by the fact that hagedorn wave packets are localized both in space and momentum so that a crucial correction term is added to the truncated hamiltonian and is treated by evolving the parameters associated with the hagedorn wave packets the residual part is treated by a galerkin approximation we prove that with the modified hamiltonian only the hagedorn wave packets dynamics give the asymptotic solution with error o e1 2 where e is the scaled planck constant we also prove that the galerkin approximation for the residual hamiltonian can reduce the approximation error to o ek 2 where k depends on the number of hagedorn wave packets added to the dynamics this approach is easy to implement and can be naturally extended to the multidimensional cases unlike the high order gaussian beam method in which the non constant cut off function is necessary and some extra error is introduced the hagedorn wave packets approach gives a practical way to improve accuracy even when e is not very small

a wavefront based gaussian beam method for computing high frequency wave propagation problems we present a novel wavefront method based on gaussian beams for computing high frequency wave propagation problems unlike standard geometrical optics gaussian beams compute the correct solution of the wave field also at caustics the method tracks a front of two canonical beams with two particular initial values for width and curvature in a fast post processing step from the canonical solutions we recreate any other gaussian beam with arbitrary initial data on the initial front this provides a simple mechanism to include a variety of optimization processes including error minimization and beam width minimization for a posteriori selection of optimal beam initial parameters the performance of the method is illustrated with two numerical examples

computing network reliability in time polynomial in the number of cuts we present a new algorithm that computes the probability that there is an operating path from a node s to a node t in a stochastic network the computation time of this algorithm is bounded by a polynomial in the number of s t cuts in the network we also examine the complexity of other connectedness reliability problems with respect to the number of cutsets and pathsets in the network these problems are distinguished as either having algorithms that are polynomial in the number of such sets or having no such algorithms unless p np

the complexity of counting cuts and of computing the probability that a graph is connected several enumeration and reliability problems are shown to be p complete and hence at least as hard as np complete problems included are important problems in network reliability analysis namely computing the probability that a graph is connected and counting the number of minimum cardinality s t cuts or directed network cuts also shown to be p complete are counting vertex covers in a bipartite graph counting antichains in a partial order and approximating the probability that a graph is connected and the probability that a pair of vertices is connected

on the complexity of nonoverlapping multivariate marginal bounds for probabilistic combinatorial optimization problems given a combinatorial optimization problem with an arbitrary partition of the set of random objective coefficients we evaluate the tightest possible bound on the expected optimal value for joint distributions consistent with the given multivariate marginals of the subsets in the partition for univariate marginals this bound was first proposed by meilijson and nadas meilijson i a nadas 1979 convex majorization with an application to the length of critical path j appl probab 16 3 671 677 we generalize the bound to nonoverlapping multivariate marginals using multiple choice integer programming new instances of polynomial time computable bounds are identified for discrete distributions for the problem of selecting up to m items out of a set of n items of maximum total weight the multivariate marginal bound is shown to be computable in polynomial time when the size of each subset in the partition is o log n for an activity on arc pert network the partition is naturally defined by subsets of incoming arcs into nodes the multivariate marginal bound on expected project duration is shown to be computable in time polynomial in the maximum number of scenarios for any subset and the size of the network as an application a polynomial time solvable two stage stochastic program for project crashing is identified an important feature of the bound developed in this paper is that it is exactly achievable by a joint distribution unlike many of the existing bounds

computing the numbers of independent sets and matchings of all sizes for graphs with bounded treewidth abstract in the theory and applications of graphs it is a basic problem to compute the numbers of independent sets and matchings of given sizes since the problem of computing the total number of independent sets and that of matchings of graphs is p complete it is unlikely to give efficient algorithms to find the numbers of independent sets and matchings of given sizes in this paper for graphs with order n and treewidth at most p we present two dynamic algorithms to compute the numbers of independent sets of all sizes with runtime o 2p pn3 and the numbers of matchings of all sizes with runtime o 22p pn3 respectively by the algorithms presented in this paper for graphs with small treewidths the numbers of independent sets and matchings of all possible sizes can be computed efficiently

convergence of a semiclassical wavepacket based time splitting for the schrodinger equation we propose a new algorithm for solving the semiclassical time dependent schrodinger equation the algorithm is based on semiclassical wavepackets the focus of the analysis is only on the time discretization convergence is proved to be quadratic in the time step and linear in the semiclassical parameter varepsilon

gabor representations of evolution operators we perform a time frequency analysis of fourier multipliers and more generally pseudodierential operators with symbols of gevrey analytic and ultra analytic type as an application we show that gabor frames which provide optimally sparse decompositions for schrodinger type propagators 14 surprisingly reveal to be an equally ecient tool for representing solutions to hyperbolic and parabolic type dierential equations with constant coecients in fact the gabor matrix represen tation of the corresponding propagator displays super exponential decay away from the diagonal

fast gaussian wavepacket transforms and gaussian beams for the schrodinger equation this paper introduces a wavepacket transform based gaussian beam method for solving the schrodinger equation we focus on addressing two computational issues of the gaussian beam method how to generate a gaussian beam representation for general initial conditions and how to perform long time propagation for any finite period of time to address the first question we introduce fast gaussian wavepacket transforms and develop on top of them an efficient initialization algorithm for general initial conditions based on this new initialization algorithm we address the second question by reinitializing the beam representation when the beams become too wide numerical examples in one two and three dimensions demonstrate the efficiency and accuracy of the proposed algorithms the methodology can be readily generalized to deal with other semi classical quantum mechanical problems

pseudo spectral solution of nonlinear schro dinger equations abstract we compare four pseudo spectral split step methods for solving a class of nonlinear schrodinger nls equations the importance of observing the l 2 invariance of the continuous problem is demonstrated through numerical experiments the best performance is obtained by transforming the given equation to an nls equation where two of the coefficients satisfy a simple algebraic relationship the problem can be solved efficiently in terms of the new variables and the l 2 norm of the computed solution is time invariant

gaussian beam decomposition of high frequency wave fields in this paper we present a method of decomposing a highly oscillatory wave field into a sparse superposition of gaussian beams the goal is to extract the necessary parameters for a gaussian beam superposition from this wave field so that further evolution of the high frequency waves can be computed by the method of gaussian beams the methodology is described for r d with numerical examples for d 2 in the first example a field generated by an interface reflection of gaussian beams is decomposed into a superposition of gaussian beams the beam parameters are reconstructed to a very high accuracy the data in the second example is not a superposition of a finite number of gaussian beams the wave field to be approximated is generated by a finite difference method for a geometry with two slits the accuracy in the decomposition increases monotonically with the number of beams

gaussian beam methods for the schrodinger equation with discontinuous potentials we propose eulerian and lagrangian gaussian beam methods for the schrodinger equation with discontinuous potentials at the quantum barriers where the potential is discontinuous we derive suitable interface conditions to account for quantum scattering information these scattering interface conditions are then built into the numerical fluxes in the eulerian level set formulation of the gaussian beam methods and are also used in the lagrangian formulation including an interface condition for the hessian matrix we carry both 1d and 2d numerical examples to verify the accuracy of the method

the gaussian wave packet transform efficient computation of the semi classical limit of the schrodinger equation part 1 formulation and the one dimensional case in this paper an efficient and accurate method for simulating the propagation of a localized solution of the schrodinger equation with a smooth potential near the semiclassical limit is presented we are interested computing arbitrarily accurate solutions when the non dimensional planck s constant e is small but not negligible the method is based on a time dependent transformation of the independent variables closely related to gaussian wave packets a rescaled wave function w satisfies a new schrodinger equation with a time dependent potential which is a perturbation of the harmonic oscillator the perturbation being o e so that all stiffness in space and time is greatly reduced in fact for integration in a fixed time interval the number of modes required to fully resolve the problem decreases when e is decreased the original wave function may be reconstructed by fourier interpolation although expectation values of the observables can be computed directly from the function w itself if the initial condition is a gaussian wave packet very few modes are necessary to fully resolve the w variable so for short time very accurate solutions can be obtained at low computational cost initial conditions other than gaussians wave packets can also be used in this paper the gaussian wave packet transform is carefully outlined and applied to the schrodinger equation in one dimension detailed numerical tests show the efficiency and accuracy of the approach in the sequel of this paper this approach has been extended to the multidimensional case

a note on on the existence of solutions of quasilinear wave equations with viscosity abstract in the note we study the existence and uniqueness of the global generalized solution for the following initial boundary value problem 1 u t t t x div u t x 2 u t x u t x u t t x u t t x p 1 u t t x u t x q 1 u t x x t 0 t 2 u t x 0 t 0 t 3 u 0 x u 0 x u t 0 x u 1 x x where is a bounded domain in r n with smooth boundary p 1 q 1 0 0 denotes the laplacian operator in r n denotes the gradient operator and s is a given nonlinear function

fast multiscale gaussian beam methods for wave equations in bounded convex domains motivated by fast multiscale gaussian wavepacket transforms and multiscale gaussian beam methods which were originally designed for pure initial value problems of wave equations we develop fast multiscale gaussian beam methods for initial boundary value problems of wave equations in bounded convex domains in the high frequency regime to compute the wave propagation in bounded convex domains we have to take into account reflecting multiscale gaussian beams which are accomplished by enforcing reflecting boundary conditions during beam propagation and carrying out suitable reflecting beam summation to propagate multiscale beams efficiently we prove that the ratio of the squared magnitude of beam amplitude and the beam width is roughly conserved and accordingly we propose an effective indicator to identify significant beams we also prove that the resulting multiscale gaussian beam methods converge asymptotically numerical examples demonstrate the accuracy and efficiency of the method

gradient estimates for solutions of parabolic equations and systems 1 statement of the problem let 52 c r n 2 be a bounded domain with smooth boundary r 2 in this note we study scalar quasilinear parabolic differential equations u div y g ivul vu f u 0 1 u a u f u o 2 and systems u da u f u 0 3 on q x 0 co with initial conditions u 0 u0 resp u 0 u0 and homogeneous dirichlet boundary conditions u i x o co o for 1 and 2 resp 4 u1r x 0 00 0 for 3 5

a deterministic optimal design problem for the heat equation for the heat equation on a bounded subdomain omega of mathbb r d we investigate the optimal shape and location of the observation domain in observability inequalities a new decomposition of l 2 mathbb r d into heat packets allows us to remove the randomization procedure and assumptions on the geometry of omega in previous works the explicit nature of the heat packets gives new information about the observability constant in the inverse problem

semitotal domination new hardness results and a polynomial time algorithm for graphs of bounded mim width abstract a semitotal dominating set of a graph g with no isolated vertex is a dominating set d of g such that every vertex in d is within distance two of another vertex in d the minimum size t 2 g of a semitotal dominating set of g is squeezed between the domination number g and the total domination number t g semitotal dominating set is the problem of finding given a graph g a semitotal dominating set of g of size t 2 g in this paper we continue the systematic study on the computational complexity of this problem when restricted to special graph classes in particular we show that it is solvable in polynomial time for the class of graphs of bounded mim width by a reduction to total dominating set and we provide several approximation lower bounds for subclasses of subcubic graphs moreover we obtain complexity dichotomies in monogenic classes for the decision versions of semitotal dominating set and total dominating set finally we show that it is np complete to recognise the graphs such that t 2 g t g and those such that g t 2 g even if restricted to be planar and with maximum degree at most 4 and we provide forbidden induced subgraph characterisations for the graphs hereditarily satisfying either of these two equalities

the computational complexity of probabilistic inference using bayesian belief networks research note abstract bayesian belief networks provide a natural efficient method for representing probabilistic dependencies among a set of variables for these reasons numerous researchers are exploring the use of belief networks as a knowledge representation in artificial intelligence algorithms have been developed previously for efficient probabilistic inference using special classes of belief networks more general classes of belief networks however have eluded efforts to develop efficient inference algorithms we show that probabilistic inference using belief networks is np hard therefore it seems unlikely that an exact algorithm can be developed to perform probabilistic inference efficiently over all classes of belief networks this result suggests that research should be directed away from the search for a general efficient probabilistic inference algorithm and toward the design of efficient special case average case and approximation algorithms

the gaussian wave packet transform a gaussian wave packet transform is developed for the efficient computation of the semi classical limit of the multidimensional schrodinger equation with smooth potential this transformation based on gaussian wave packets yields a schrodinger type equation that is very amenable to numerical solution in the semi classical limit the transformed schrodinger equation is solved with a 4th order splitting method the wave function can be reconstructed from the transformed wave function whereas some expectation values can easily be evaluated directly the number of grid points needed per degree of freedom is small enough that computations in dimension of up to 4 or 5 are feasible without the use of any basis thinning procedures

algorithmic combinatorics based on slicing posets a combinatorial problem usually requires enumerating counting or ascertaining existence of structures that satisfy a given property b in a set of structures l this paper describes a technique based on a generalization of birkhoff s theorem of representation of finite distributive lattices that can be used for solving such problems mechanically and efficiently specifically we give an efficient polynomial time algorithm to enumerate count or detect structures that satisfy b when the total set of structures is large but the set of structures satisfying b is small we illustrate our techniques by analyzing problems in integer partitions set families and set of permutations

calculating k connectedness reliability using steiner bounds the k connectedness reliability problem considered in this paper has as input undirected graph g subset k of terminal vertices and common edge failure probability p and as output the probability rg k p that when edges fail independently each with probability p the set of operating edges connect every pair of vertices of k we show how the steiner property held by the underlying simplicial complex associated with this problem can lead to an extension of the ball provan shellability bounds to this more general problem we show in computational studies that this bound performs quite favorably in comparison with known approximation techniques

the complexity of counting locally maximal satisfying assignments of boolean csps we investigate the computational complexity of the problem of counting the locally maximal satisfying assignments of a constraint satisfaction problem csp over the boolean domain 0 1 a satisfying assignment is locally maximal if any new assignment which is obtained from it by changing a 0 to a 1 is unsatisfying for each constraint language localmaxcsp denotes the problem of counting the locally maximal satisfying assignments given an input csp with constraints in we give a complexity dichotomy for the problem of exactly counting the locally maximal satisfying assignments and a complexity trichotomy for the problem of approximately counting them relative to the problem csp which is the problem of counting all satisfying assignments the locally maximal version can sometimes be easier but never harder this finding contrasts with the recent discovery that approximately counting locally maximal independent sets in a bipartite graph is harder under the usual complexity theoretic assumptions than counting all independent sets

eulerian gaussian beams for high frequency wave propagation we design an eulerian gaussian beam summation method for solving helmholtz equations in the high frequency regime the traditional gaussian beam summation method is based on lagrangian ray tracing and local ray centered coordinates we propose a new eulerian formulation of gaussian beam theory which adopts global cartesian coordinates level sets and liouville equations yielding uniformly distributed eulerian traveltimes and amplitudes in phase space simultaneously for multiple sources the time harmonic wavefield can be constructed by summing up gaussian beams with ingredients provided by the new eulerian formulation the conventional gaussian beam summation method can be derived from the proposed method there are three advantages of the new method 1 we have uniform resolution of ray distribution 2 we can obtain wavefields excited at different sources by varying only source locations in the summation formula 3 we can obtain wavefields excited at different frequencies by varying only frequencies in the summation formula numerical experiments indicate that the gaussian beam summation method yields accurate asymptotic wavefields even at caustics the new method may be used for seismic modeling and migration

a ray based ipdg method for high frequency time domain acoustic wave propagation in inhomogeneous media abstract the numerical approximation of high frequency wave propagation in inhomogeneous media is a challenging problem in particular computing high frequency solutions by direct simulations requires several points per wavelength for stability and usually requires many points per wavelength for a satisfactory accuracy in this paper we propose a new method for the acoustic wave equation in inhomogeneous media in the time domain to achieve superior accuracy and stability without using a large number of unknowns the method is based on a discontinuous galerkin discretization together with carefully chosen basis functions to obtain the basis functions we use the idea from geometrical optics and construct the basis functions by using the leading order term in the asymptotic expansion also we use a wavefront tracking method and a dimension reduction procedure to obtain dominant rays in each cell we show numerically that the accuracy of the numerical solutions computed by our method is significantly higher than that computed by the ipdg method using polynomials moreover the relative errors of our method grow only moderately as the frequency increases

global existence and exponential stability of small solutions to nonlinear viscoelasticity the global existence of smooth solutions to the equations of nonlinear hyperbolic system of 2nd order with third order viscosity is shown for small and smooth initial data in a bounded domain ofn dimensional euclidean space with smooth boundary dirichlet boundary condition is studied and the asymptotic behaviour of exponential decay type of solutions ast tending to is described time periodic solutions are also studied as an application of our main theorem nonlinear viscoelasticity strongly damped nonlinear wave equation and acoustic wave equation in viscous conducting fluid are treated

on the hardness of approximate reasoning abstract many ai problems when formalized reduce to evaluating the probability that a propositional expression is true in this paper we show that this problem is computationally intractable even in surprisingly restricted cases and even if we settle for an approximation to this probability we consider various methods used in approximate reasoning such as computing degree of belief and bayesian belief networks as well as reasoning techniques such as constraint satisfaction and knowledge compilation that use approximation to avoid computational difficulties and reduce them to model counting problems over a propositional domain we prove that counting satisfying assignments of propositional languages is intractable even for horn and monotone formulae and even when the size of clauses and number of occurrences of the variables are extremely limited this should be contrasted with the case of deductive reasoning where horn theories and theories with binary clauses are distinguished by the existence of linear time satisfiability algorithms what is even more surprising is that as we show even approximating the number of satisfying assignments i e approximating approximate reasoning is intractable for most of these restricted theories we also identify some restricted classes of propositional formulae for which efficient algorithms for counting satisfying assignments can be given

complexity of generalized satisfiability counting problems abstract the class of generalized satisfiability problems introduced in 1978 by schaefer presents a uniform way of studying the complexity of satisfiability problems with special conditions the complexity of each decision and counting problem in this class depends on the involved logical relations in 1979 valiant defined the class p the class of functions definable as the number of accepting computations of a polynomial time nondeterministic turing machine clearly all satisfiability counting problems belong to this class p we prove a dichotomy theorem for generalized satisfiability counting problems that is if all logical relations involved in a generalized satisfiability counting problem are affine then the number of satisfying assignments of this problem can be computed in polynomial time otherwise this function is p complete this gives us a comparison between decision and counting generalized satisfiability problems we can determine exactly the polynomial satisfiability decision problems whose number of solutions can be computed in polynomial time and also the polynomial satisfiability decision problems whose counting counterparts are already p complete moreover taking advantage of a similar dichotomy result proved in 1978 by schaefer for generalized satisfiability decision problems we get as a corollary the implication that the counting counterpart of each np complete generalized satisfiability decision problem is p complete

computational aspects of mining maximal frequent patterns in this paper we study the complexity theoretic aspects of mining maximal frequent patterns from the perspective of counting the number of all distinct solutions we present the first formal proof that the problem of counting the number of maximal frequent itemsets in a database of transactions given an arbitrary support threshold is p complete thereby providing theoretical evidence that the problem of mining maximal frequent itemsets is np hard we also extend our complexity analysis to other similar data mining problems that deal with complex data structures such as sequences trees and graphs we investigate several variants of these mining problems in which the patterns of interest are subsequences subtrees or subgraphs and show that the associated problems of counting the number of maximal frequent patterns are all either p complete or p hard

transducing markov sequences a markov sequence is a basic statistical model representing uncertain sequential data and it is used within a plethora of applications including speech recognition image processing computational biology radio frequency identification rfid and information extraction the problem of querying a markov sequence is studied under the conventional semantics of querying a probabilistic database where queries are formulated as finite state transducers specifically the complexity of two main problems is analyzed the first problem is that of computing the confidence probability of an answer the second is the enumeration of the answers in the order of decreasing confidence with the generation of the top k answers as a special case or in an approximate order thereof in particular it is shown that enumeration in any subexponential approximate order is generally intractable even for some fixed transducers and a matching upper bound is obtained through a proposed heuristic due to this hardness a special consideration is given to restricted yet common classes of transducers that extract matches of a regular expression subject to prefix and suffix constraints and it is shown that these classes are indeed significantly more tractable

network reliability and inner four cycle free graphs a planar graph g v e is an inner four cycle free graph ifcf graph if after all series and parallel replacements it has a planar embedding g with a face f such that every cycle of g having four or more edges contains a vertex of f it is shown that an ifcf graph can be reduced to an edge by a recursive application of series parallel and y replacements this property forms the basis of an o v 2 algorithm to compute the probability that a probabilistic ifcf graph is connected this result is extended to graphs whose triconnected components are ifcf such graphs are not necessarily ifcf the class of ifcf graphs includes the series parallel graphs wheel graphs ladder networks and graphs in normal form for which polynomial network reliability algorithms are already known

the backward phase flow and fbi transform based eulerian gaussian beams for the schrodinger equation we propose the backward phase flow method to implement the fourier bros iagolnitzer fbi transform based eulerian gaussian beam method for solving the schrodinger equation in the semi classical regime the idea of eulerian gaussian beams has been first proposed in 12 in this paper we aim at two crucial computational issues of the eulerian gaussian beam method how to carry out long time beam propagation and how to compute beam ingredients rapidly in phase space by virtue of the fbi transform we address the first issue by introducing the reinitialization strategy into the eulerian gaussian beam framework essentially we reinitialize beam propagation by applying the fbi transform to wavefields at intermediate time steps when the beams become too wide to address the second issue inspired by the original phase flow method we propose the backward phase flow method which allows us to compute beam ingredients rapidly numerical examples demonstrate the efficiency and accuracy of the proposed algorithms

computational complexity of loss networks abstract in this paper we examine the theoretical limits on developing algorithms to find blocking probabilities in a general loss network we demonstrate that exactly computing the blocking probabilities of a loss network is a p complete problem we also show that a general algorithm for approximating the blocking probabilities is also intractable unless rp np which seems unlikely according to current common notions in complexity theory given these results we examine implications for designing practical algorithms for finding blocking probabilities in special cases

least reliable networks and the reliability domination a well known model in communication network reliability consists of an undirected graph g whose edges operate independently with the same probability p then the reliability r g p of g is the probability that g is connected it is known that r g p is a polynomial in p and its coefficient of the least order term is the number of spanning trees t g while the coefficient of the highest order term is the reliability domination d g of g presented is a complete characterization of graphs that achieve the minimum absolute value mod d g mod over the class of n node e edge connected graphs furthermore the class of graphs that yield minimum t g is shown to minimize mod d g mod the results have applications in the synthesis of least reliable networks

hardness of computing width parameters based on branch decompositions over the vertex set many width parameters of graphs are defined using branch decompositions over the vertex set of the graph and a corresponding cut function in this paper we give a general framework for showing hardness of many width parameters defined in such a way by reducing from the problem of deciding the exact value of the cut function we show that this implies np hardness for deciding both boolean width and mim width and that mim width is w1 hard and not in apx unless np zpp

on time splitting spectral approximations for the schrodinger equation in the semiclassical regime in this paper we study time splitting spectral approximations for the linear schrodinger equation in the semiclassical regime where the planck constant e is small in this regime the equation propagates oscillations with a wavelength of o e and finite difference approximations require the spatial mesh size h o e and the time step k o e in order to obtain physically correct observables much sharper mesh size constraints are necessary for a uniform l2 approximation of the wave function the spectral time splitting approximation under study will be proved to be unconditionally stable time reversible and gauge invariant it conserves the position density and gives uniform l2 approximation of the wave function for k o e and h o e extensive numerical examples in both one and two space dimensions and analytical considerations based on the wigner transform even show that weaker constraints e g k independent of e and h o e are admissible for obtaining correct observables finally we address the application to nonlinear schrodinger equations and conduct some numerical experiments to predict the corresponding admissible meshing strategies

error estimates for gaussian beam superpositions gaussian beams are asymptotically valid high frequency solutions to hyperbolic partial differential equations concentrated on a single curve through the physical domain they can also be extended to some dispersive wave equations such as the schrodinger equation superpositions of gaussian beams provide a powerful tool to generate more general high frequency solutions that are not necessarily concentrated on a single curve this work is concerned with the accuracy of gaussian beam superpositions in terms of the wavelength epsilon we present a systematic construction of gaussian beam superpositions for all strictly hyperbolic and schrodinger equations subject to highly oscillatory initial data of the form ae i phi epsilon through a careful estimate of an oscillatory integral operator we prove that the k th order gaussian beam superposition converges to the original wave field at a rate proportional to epsilon k 2 in the appropriate norm dictated by the well posedness estimate in particular we prove that the gaussian beam superposition converges at this rate for the acoustic wave equation in the standard epsilon scaled energy norm and for the schrodinger equation in the l 2 norm the obtained results are valid for any number of spatial dimensions and are unaffected by the presence of caustics we present a numerical study of convergence for the constant coefficient acoustic wave equation in r 2 to analyze the sharpness of the theoretical results

an o e time algorithm for computing the reliability of a class of directed networks this paper studies the problem of computing the source to terminal reliability the probability that a source s can communicate with a terminal t in a probabilistic directed network it introduces a set of reliability preserving reductions called 2 neighbor node reductions we show that for a class of directed networks called bsp digraphs such reductions are always admissible and the source to terminal reliability can be computed in time linear in the size of the network a directed network d v e is a bsp digraph if its underlying undirected graph is series parallel the bsp digraphs considered can be cyclic or acyclic and both vertices as well as edges can fail any two vertices can be chosen as s and t previously no polynomial time algorithms were known for this class of digraphs the proposed method is also applicable to mixed graphs containing directed and undirected edges

the complexity of counting in sparse regular and planar graphs we show that a number of graph theoretic counting problems remain cal np hard indeed cal p complete in very restricted classes of graphs in particular we prove that the problems of counting matchings vertex covers independent sets and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree we obtain corollaries about counting cliques in restricted classes of graphs and counting satisfying assignments to restricted classes of monotone 2 cnf formulae to achieve these results a new interpolation based reduction technique which preserves properties such as constant degree is introduced

fast computation of bounds for two terminal network reliability in this paper an algorithm for the fast computation of network reliability bounds is proposed the evaluation of the network reliability is an intractable problem for very large networks and hence approximate solutions based on reliability bounds have assumed importance the proposed bounds computation algorithm is based on an efficient bdd representation of the reliability graph model and a novel search technique to find important minpaths mincuts to quickly reduce the gap between the reliability upper and lower bounds furthermore our algorithm allows the control of the gap between the two bounds by controlling the overall execution time therefore a trade off between prediction accuracy and computational resources can be easily made in our approach the numerical results are presented for large real example reliability graphs to show the efficacy of our approach

a survey of network reliability and domination theory we present a brief survey of the current state of the art in network reliability we survey only exact methods and do not consider monte carlo methods most network reliability problems are in the worst case np hard and are in a sense more difficult than many standard combinatorial optimization problems nevertheless there are in fact linear and polynomial time algorithms for network reliability problems of special structure we review general methods for network reliability computation and discuss the central role played by domination theory in network reliability computational complexity we also point out the connection with the more general problem of computing the reliability of coherent structures the class of coherent structures contains both directed and undirected networks as well as logic or fault trees without not gates this topic is a rich area for further research

disjoint products and efficient computation of reliability in this paper we analyze the problem of computing the probability of the union of a set of events where each event is given as the product of a set of boolean variables each boolean variable represents the operation or failure of a particular component the problem has direct applications to the reliability analysis of complex systems as well as more general applications after showing that the problem is np hard in general we define simple computationally efficient procedures for generating upper and lower bounds on the required probability we show that these procedures produce the exact answer i e the upper bound equals the lower bounds for two classes of systems matroids and threshold systems these results draw on the relationship between this problem and the notion of shellability studied in the context of simplicial polytopes shellability is shown to have a very interesting and useful interpretation in this problem setting

on randomized approximation for finding a level ideal of a poset and the generalized median stable matchings this study is concerned with finding a level ideal li of a partially ordered set poset given a finite poset p the level of each element p p is defined as the number of ideals that do not include p then the problem is to find the ith li the ideal consisting of elements whose levels are less than a given integer i the concept of a level ideal is naturally derived from the generalized median stable matchings introduced by teo and sethuraman teo c p j sethuraman 1998 the geometry of fractional stable matchings and its applications math oper res 23 4 874 891 in the context of fairness of matchings in a stable marriage problem cheng cheng c t 2010 understanding the generalized median stable matchings algorithmica58 1 34 51 showed that finding the ith li is p hard when i n where n is the total number of ideals of p this paper shows that finding the ith li is p hard even if i n1 c where c is an arbitrary constant at least one meanwhile we present a polynomial time exact algorithm when i o log n c where c is an arbitrary positive constant we also devise two randomized approximation schemes for the ideals of a poset by using an oracle of an almost uniform sampler

finding optimal bayesian networks using precedence constraints we consider the problem of finding a directed acyclic graph dag that optimizes a decomposable bayesian network score while in a favorable case an optimal dag can be found in polynomial time in the worst case the fastest known algorithms rely on dynamic programming across the node subsets taking time and space 2n to within a factor polynomial in the number of nodes n in practice these algorithms are feasible to networks of at most around 30 nodes mainly due to the large space requirement here we generalize the dynamic programming approach to enhance its feasibility in three dimensions first the user may trade space against time second the proposed algorithms easily and efficiently parallelize onto thousands of processors third the algorithms can exploit any prior knowledge about the precedence relation on the nodes underlying all these results is the key observation that given a partial order p on the nodes an optimal dag compatible with p can be found in time and space roughly proportional to the number of ideals of p which can be significantly less than 2n considering sufficiently many carefully chosen partial orders guarantees that a globally optimal dag will be found aside from the generic scheme we present and analyze concrete tradeoff schemes based on parallel bucket orders

complexity of model checking for reaction systems reaction systems are a new mathematical formalism inspired by the living cell and driven by only two basic mechanisms facilitation and inhibition as a modeling framework they differ from the traditional approaches based on odes and ctmcs in two fundamental aspects their qualitative character and the non permanency of resources in this article we introduce to reaction systems several notions of central interest in biomodeling mass conservation invariants steady states stationary processes elementary fluxes and periodicity we prove that the decision problems related to these properties span a number of complexity classes from p to np and conp complete to pspace complete

the complexity of planar counting problems we prove the p hardness of the counting problems associated with various satisfiability graph and combinatorial problems when restricted to planar instances these problems include 3sat 1 3sat 1 ex3sat minimum vertex cover minimum dominating set minimum feedback vertex set x3c partition into triangles and clique cover we also prove the np completeness of the ambiguous satisfiability problems j b saxe two papers on graph embedding problems tech report cmu cs 80 102 dept of computer science carnegie mellon univ pittsburgh pa 1980 and the dp completeness with respect to random polynomial reducibility of the unique satisfiability problems l g valiant and v v vazirani np is as easy as detecting unique solutions in proc 17th acm symp on theory of computing 1985 pp 458 463 associated with several of the above problems when restricted to planar instances previously very few p hardness results no sf np hardness results and no dp completeness results were known for counting problems ambiguous satisfiability problems and unique satisfiability problems respectively when restricted to planar instances assuming sf p neq np one corollary of the above results is that there are no epsilon approximation algorithms for the problems of maximizing or minimizing a linear objective function subject to a planar system of linear inequality constraints over the integers

eulerian gaussian beams for schrodinger equations in the semi classical regime we propose gaussian beam based eulerian methods to compute semi classical solutions of the schrodinger equation traditional gaussian beam type methods for the schrodinger equation are based on the lagrangian ray tracing based on the first eulerian gaussian beam framework proposed in leung et al s leung j qian r burridge eulerian gaussian beams for high frequency wave propagation geophysics 72 2007 sm61 sm76 we develop a new eulerian gaussian beam method which uses global cartesian coordinates level set based implicit representation and liouville equations the resulting method gives uniformly distributed phases and amplitudes in phase space simultaneously to obtain semi classical solutions to the schrodinger equation with different initial wave functions we only need to slightly modify the summation formula this yields a very efficient method for computing semi classical solutions to the schrodinger equation for instance in the one dimensional case the proposed algorithm requires only o snm2 operations to compute s different solutions with s different initial wave functions under the influence of the same potential where n o 1 is the planck constant and m n is the number of computed beams which depends on weakly numerical experiments indicate that this eulerian gaussian beam approach yields accurate semi classical solutions even at caustics

dichotomy results for fixed point counting in boolean dynamical systems we present dichotomy theorems regarding the computational complexity of counting fixed points in boolean discrete dynamical systems i e finite discrete dynamical systems over the domain 0 1 for a class f of boolean functions and a class g of graphs an f g system is a boolean dynamical system with local transitions functions lying in f and graphs in g we show that if local transition functions are given by lookup tables then the following complexity classification holds let f be a class of boolean functions closed under superposition and let g be a graph class closed under taking minors if f contains all min functions all max functions or all self dual and monotone functions and g contains all planar graphs then it is p complete to compute the number of fixed points in an f g system otherwise it is computable in polynomial time we also prove a dichotomy theorem for the case that local transition functions are given by formulas over logical bases this theorem has a significantly more complicated structure than the theorem for lookup tables a corresponding theorem for boolean circuits coincides with the theorem for formulas

computing on binary strings many problems in computer science can be abstracted to the following question given a set of objects and rules respectively which new objects can be produced in the paper we consider a succinct version of the question given a set of binary strings and several operations like conjunction and disjunction which new binary strings can be generated although it is a fundamental problem to the best of our knowledge the problem hasn t been studied yet in this paper an o m 2 n algorithm is presented to determine whether a string s is representable by a set w where n is the number of strings in w and each string has the same length m however looking for the minimum subset to represent a given string is shown to be np hard also finding the smallest subset to represent each string in the original set is np hard we establish tight inapproximability results and approximation algorithms for these np hard problems in addition we prove that counting the number of representable strings is p complete we then explore how the problems change when the operator negation is available for example if the operator negation can be used the counting problem is as simple as some power of 2 this difference may help us to better understand the problem

fast huygens sweeping methods for multiarrival green s functions of helmholtz equations in the high frequency regime abstractmultiarrival green s functions are essential in seismic modeling migration and inversion huygens kirchhoff hk integrals provide a bridge to integrate locally valid first arrival green s functions into a globally valid multiarrival green s function we have designed robust and accurate finite difference methods to compute first arrival traveltimes and amplitudes so that first arrival green s functions can be constructed rapidly we adapted a fast butterfly algorithm to evaluate discretized hk integrals the resulting fast huygens sweeping method has the following unique features 1 it precomputes a set of local traveltime and amplitude tables 2 it automatically takes care of caustics 3 it constructs green s functions of the helmholtz equation for arbitrary frequencies and for many point sources and 4 for a fixed number of points per wavelength it constructs each green s function in nearly optimal complexity o n log n in terms of the total number of mesh points n where the prefa

mim width iii graph powers and generalized distance domination problems abstract we generalize the family of problems and locally checkable vertex partition problems to their distance versions which naturally captures well known problems such as distance r dominating set and distance r independent set we show that these distance problems are in xp parameterized by the structural parameter mim width and hence polynomial time solvable on graph classes where mim width is bounded and quickly computable such as k trapezoid graphs dilworth k graphs circular permutation graphs interval graphs and their complements convex graphs and their complements k polygon graphs circular arc graphs complements of d degenerate graphs and h graphs if given an h representation we obtain these results by showing that taking any power of a graph never increases its mim width by more than a factor of two to supplement these findings we show that many classes of problems are w 1 hard parameterized by mim width solution size we show that powers of graphs of tree width w 1 or path width w and powers of graphs of clique width w have mim width at most w these results provide new classes of bounded mim width we prove a slight strengthening of the first statement which implies that surprisingly leaf power graphs which are of importance in the field of phylogenetic studies have mim width at most 1

holographic algorithm with matchgates is universal for planar csp over boolean domain we prove a complexity classification theorem that classifies all counting constraint satisfaction problems csp over boolean variables into exactly three classes 1 polynomial time solvable 2