technical note a langrangian algorithm for the multiple choice integer program in the multiple choice integer program the binary variables are partitioned into generalized upper bounding sets or special ordered sets we solve this problem using a branch and bound scheme designed to take advantage of this structure the basic procedure is enhanced by solving a lagrangean problem as an approximation further we use a variable reduction technique to eliminate many of the variables prior to problem solution computational experience is promising for randomly generated problems scheduling problems and budgeting problems tested

strong mixed integer programming formulations for trained neural networks we present strong mixed integer programming mip formulations for high dimensional piecewise linear functions that correspond to trained neural networks these formulations can be used for a number of important tasks such as verifying that an image classification network is robust to adversarial inputs or solving decision problems where the objective function is a machine learning model we present a generic framework which may be of independent interest that provides a way to construct sharp or ideal formulations for the maximum of d affine functions over arbitrary polyhedral input domains we apply this result to derive mip formulations for a number of the most popular nonlinear operations e g relu and max pooling that are strictly stronger than other approaches from the literature we corroborate this computationally showing that our formulations are able to offer substantial improvements in solve time on verification tasks for image classification networks

frequency analysis of corner point supported mindlin plates by a finite difference energy method abstract the first known solution of the title problem is presented a method based on the variational procedure in conjunction with a finite difference method is used to examine the free vibration characteristics numerical studies show that the convergence is relatively slow requiring a finer mesh model in view of the symmetry of the geometry of the plate and the symmetry in the distribution of the point supports only one quarter of the plate is examined three mode types are developed to account for the full plate detailed results are given for the frequencies and mode shapes of the first six modes of each symmetry class and the effect of transverse shear deformation and rotary inertia is analyzed the validity of the results is partially demonstrated by comparing them with existing solutions based on the classical thin plate theory

the dichotomy of probabilistic inference for unions of conjunctive queries we study the complexity of computing a query on a probabilistic database we consider unions of conjunctive queries ucq which are equivalent to positive existential first order logic sentences and also to nonrecursive datalog programs the tuples in the database are independent random events we prove the following dichotomy theorem for every ucq query either its probability can be computed in polynomial time in the size of the database or is p hard our result also has applications to the problem of computing the probability of positive boolean expressions and establishes a dichotomy for such classes based on their structure for the tractable case we give a very simple algorithm that alternates between two steps applying the inclusion exclusion formula and removing one existential variable a key and novel feature of this algorithm is that it avoids computing terms that cancel out in the inclusion exclusion formula in other words it only computes those terms whose mobius function in an appropriate lattice is nonzero we show that this simple feature is a key ingredient needed to ensure completeness for the hardness proof we give a reduction from the counting problem for positive partitioned 2cnf which is known to be p complete the hardness proof is nontrivial and combines techniques from logic classical algebra and analysis

vibrations of axially moving viscoelastic plate with parabolically varying thickness based on the two dimensional viscoelastic differential constitutive relation the differential equation of motion of the axially moving viscoelastic rectangular plate constituted by the kelvin voigt model with parabolically varying thickness in the y direction is derived the dimensionless complex frequencies of axially moving viscoelastic plate with different boundary conditions versus the dimensionless moving speed for various aspect ratio thickness parameter and the dimensionless delay time are analyzed by the differential quadrature method the effects of various parameters such as aspect ratio thickness parameter the dimensionless moving speed and the dimensionless delay time on the vibration characteristics of the axially moving viscoelastic rectangular plate with parabolically varying thickness are discussed

on new symplectic approach for exact bending solutions of moderately thick rectangular plates with two opposite edges simply supported the symplectic geometry method is introduced for exact bending solutions of moderately thick rectangular plates with two opposite edges simply supported the basic equations for the plates are first transferred into hamilton canonical equations the whole state variables are then separated using the method of eigenfunction expansion in the symplectic geometry typical examples for plates with selected boundary conditions are solved and exact bending solutions obtained since only the basic elasticity equations of the plates are used this method eliminates the need to pre determine the deformation function and is hence more reasonable than conventional methods numerical results were presented to demonstrate the validity and accuracy of this approach as compared to those reported in other literatures

asymptotic conditional probabilities the unary case motivated by problems that arise in computing degrees of belief we consider the problem of computing asymptotic conditional probabilities for first order sentences given first order sentences phi and theta we consider the structures with domain 1 ldots n that satisfy theta and compute the fraction of them in which phi is true we then consider what happens to this fraction as n gets large this extends the work on 0 1 laws that considers the limiting probability of first order sentences by considering asymptotic conditional probabilities as shown by liogon kii math notes acad ussr 6 1969 pp 856 861 and by grove halpern and koller res rep rj 9564 ibm almaden research center san jose ca 1993 in the general case asymptotic conditional probabilities do not always exist and most questions relating to this issue are highly undecidable these results however all depend on the assumption that theta can use a nonunary predicate symbol liogonkii math notes acad ussr 6 1969 pp 856 861 shows that if we condition on formulas theta involving unary predicate symbols only but no equality or constant symbols then the asymptotic conditional probability does exist and can be effectively computed this is the case even if we place no corresponding restrictions on phi we extend this result here to the case where theta involves equality and constants we show that the complexity of computing the limit depends on various factors such as the depth of quantifier nesting or whether the vocabulary is finite or infinite we completely characterize the complexity of the problem in the different cases and show related results for the associated approximation problem

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

probabilistic combinatorial optimization moments semidefinite programming and asymptotic bounds we address the problem of evaluating the expected optimal objective value of a 0 1 optimization problem under uncertainty in the objective coefficients the probabilistic model we consider prescribes limited marginal distribution information for the objective coefficients in the form of moments we show that for a fairly general class of marginal information a tight upper lower bound on the expected optimal objective value of a 0 1 maximization minimization problem can be computed in polynomial time if the corresponding deterministic problem is solvable in polynomial time we provide an efficiently solvable semidefinite programming formulation to compute this tight bound we also analyze the asymptotic behavior of a general class of combinatorial problems that includes the linear assignment spanning tree and traveling salesman problems under knowledge of complete marginal distributions with and without independence we calculate the limiting constants exactly

the value of side information in network flow optimization abstract consider a group of agents who seek to simultaneously traverse a graph each edge of the graph has an associated weight e g a delay and the agents seek to minimize the cumulative weight incurred by all agents as each traverses a path of the graph an edge s weight is a function of the number of agents that use that edge as well as an inherent random weight if the agents have no side information about the random component they will deterministically organize themselves so as to optimize their average performance we consider a generalization of this framework whereby the agents have access to a limited amount of shared side information about the edge weights and we study the relationship between information quantity and performance

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

persistence in discrete optimization under data uncertainty an important question in discrete optimization under uncertainty is to understand the persistency of a decision variable i e the probability that it is part of an optimal solution for instance in project management when the task activity times are random the challenge is to determine a set of critical activities that will potentially lie on the longest path in the spanning tree and shortest path network problems when the arc lengths are random the challenge is to pre process the network and determine a smaller set of arcs that will most probably be a part of the optimal solution under different realizations of the arc lengths building on a characterization of moment cones for single variate problems and its associated semidefinite constraint representation we develop a limited marginal moment model to compute the persistency of a decision variable under this model we show that finding the persistency is tractable for zero one optimization problems with a polynomial sized representation of the convex hull of the feasible region through extensive experiments we show that the persistency computed under the limited marginal moment model is often close to the simulated persistency value under various distributions that satisfy the prescribed marginal moments and are generated independently

on theoretical and empirical aspects of marginal distribution choice models in this paper we study the properties of a recently proposed class of semiparametric discrete choice models referred to as the marginal distribution model mdm by optimizing over a family of joint error distributions with prescribed marginal distributions surprisingly the choice probabilities arising from the family of generalized extreme value models of which the multinomial logit model is a special case can be obtained from this approach despite the difference in assumptions on the underlying probability distributions we use this connection to develop flexible and general choice models to incorporate consumer and product level heterogeneity in both partworths and scale parameters in the choice model furthermore the extremal distributions obtained from the mdm can be used to approximate the fisher s information matrix to obtain reliable standard error estimates of the partworth parameters without having to bootstrap the method we use simulated and empirical data sets to test the performance of this approach we evaluate the performance against the classical multinomial logit mixed logit and a machine learning approach that accounts for partworth heterogeneity our numerical results indicate that mdm provides a practical semiparametric alternative to choice modeling this paper was accepted by eric bradlow special issue on business analytics

on the mautner phenomenon and groups with property t this paper is inspired by moore s work on the mautner phenomenon 18 in 18 the best possible result of this kind has been established for a one parameter subgroup of a lie group our purpose here is to extend moore s result to a cyclic subgroup of a lie group or of an algebraic group over a local field we make various applications of this result to study groups with property t first we show in theorem 1 2 that moore s result can be extended to a cyclic subgroup of a lie group as applications we have the following theorems

extension and separation properties of positive definite functions on locally compact groups continuing earlier work we investigate two related aspects of the set p g of continuous positive definite functions on a locally compact group g the first one is the problem of when for a closed subgroup h of g every function in p h extends to some function in p g the second one is the question whether elements in g h can be separated from h by functions in p g which are identically one on h

exact methods for the knapsack problem and its generalizations abstract a unified approach and a summary of the most important results concerned with exact methods for solving the binary knapsack problem and its generalizations are given we stress the importance of dual methods for solving linear programming relaxations of the considered problems two ways of generalization of the knapsack problem are described if the special ordered sets are added then the multiple choice knapsack problem is obtained if the constraints have the nested structure then we get the nested knapsack problem also the multiple choice nested knapsack problem is discussed

a separation property of positive definite functions on locally compact groups and applications to fourier algebras abstract for a closed subgroup h of a locally compact group g consider the property that the continuous positive definite functions on g which are identically one on h separate points in g h from points in h we prove a structure theorem for almost connected groups having this separation property for every closed subgroup also when a pair g h has this separation property there are interesting consequences in the ideal theory of the fourier algebra of g

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

a distributionally robust perspective on uncertainty quantification and chance constrained programming the objective of uncertainty quantification is to certify that a given physical engineering or economic system satisfies multiple safety conditions with high probability a more ambitious goal is to actively influence the system so as to guarantee and maintain its safety a scenario which can be modeled through a chance constrained program in this paper we assume that the parameters of the system are governed by an ambiguous distribution that is only known to belong to an ambiguity set characterized through generalized moment bounds and structural properties such as symmetry unimodality or independence patterns we delineate the watershed between tractability and intractability in ambiguity averse uncertainty quantification and chance constrained programming using tools from distributionally robust optimization we derive explicit conic reformulations for tractable problem classes and suggest efficiently computable conservative approximations for intractable ones

the complexity of bayesian networks specified by propositional and relational languages abstract we examine the complexity of inference in bayesian networks specified by logical languages we consider representations that range from fragments of propositional logic to function free first order logic with equality in doing so we cover a variety of plate models and of probabilistic relational models we study the complexity of inferences when network query and domain are the input the inferential and the combined complexity when the network is fixed and query and domain are the input the query data complexity and when the network and query are fixed and the domain is the input the domain complexity we draw connections with probabilistic databases and liftability results and obtain complexity classes that range from polynomial to exponential levels we identify new languages with tractable inference and we relate our results to languages based on plates and probabilistic relational models

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

distributionally robust optimization with polynomial densities theory models and algorithms in distributionally robust optimization the probability distribution of the uncertain problem parameters is itself uncertain and a fictitious adversary e g nature chooses the worst distribution from within a known ambiguity set a common shortcoming of most existing distributionally robust optimization models is that their ambiguity sets contain pathological discrete distributions that give nature too much freedom to inflict damage we thus introduce a new class of ambiguity sets that contain only distributions with sum of squares sos polynomial density functions of known degrees we show that these ambiguity sets are highly expressive as they conveniently accommodate distributional information about higher order moments conditional probabilities conditional moments or marginal distributions exploiting the theoretical properties of a measure based hierarchy for polynomial optimization due to lasserre siam j optim 21 3 864 885 2011 we prove that certain worst case expectation constraints are polynomial time solvable under these new ambiguity sets we also show how sos densities can be used to approximately solve the general problem of moments we showcase the applicability of the proposed approach in the context of a stylized portfolio optimization problem and a risk aggregation problem of an insurance company

the complexity of the counting constraint satisfaction problem the counting constraint satisfaction problem csp h over a finite relational structure h can be expressed as follows given a relational structure g over the same vocabulary determine the number of homomorphisms from g to h in this article we characterize relational structures h for which csp h can be solved in polynomial time and prove that for all other structures the problem is p complete

polyhedral combinatorics and network reliability this paper studies the reliability of systems comprised of variables which must satisfy a set of linear equalities and nonnegativity constraints and which are subject to random failure a major example which will be given special emphasis is the reachability source to all connectedness reliability problem for stochastic networks we show how properties of convex polyhedra and their network counterparts are useful in determining properties of the reliability polynomial for these systems in particular we show how recent results by stanley and by billera and lee concerning the face numbers of convex polytopes and polyhedra can be applied to provide good upper and lower bounds on reliability in linear systems and we apply these bounds in the network examples

a probabilistic model for minmax regret in combinatorial optimization in this paper we propose a new probabilistic model for minimizing the anticipated regret in combinatorial optimization problems with distributional uncertainty in the objective coefficients the interval uncertainty representation of data is supplemented with information on the marginal distributions as a decision criterion we minimize the worst case conditional value at risk of regret the proposed model includes the interval data minmax regret model as a special case for the class of combinatorial optimization problems with a compact convex hull representation a polynomial sized mixed integer linear program is formulated when a the range and mean are known and b the range mean and mean absolute deviation are known and a mixed integer second order cone program is formulated when c the range mean and standard deviation are known for the subset selection problem of choosing k elements of maximum total weight out of a set of n elements the probabilistic regret model is shown to be solvable in polynomial time in the instances a and b above this extends the current known polynomial complexity result for minmax regret subset selection with range information only

counting and sampling minimum s t cuts in weighted planar graphs in polynomial time we give an o nd nlogn algorithm computing the number of minimum s t cuts in weighted planar graphs where n is the number of vertices and d is the length of the shortest s t path in the corresponding unweighted graph previously ball and provan gave a polynomial time algorithm for unweighted planar graphs with both s and t lying on the outer face our results hold for all locations of s and t and weighted graphs and have direct applications in computer vision

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

stochastic bounds on distributions of optimal value functions with applications to pert network flows and reliability in many classical combinatorial optimization problems including critical and shortest paths maximum flow and network reliability the introduction of uncertainty considerably complicates the calculation of system performance in fact in these contexts computing system performance exactly can often be an impossible task therefore obtaining stochastic bounds on the system s performance becomes an attractive and useful alternative this paper studies several stochastic bounds that are applicable to these contexts and to a broader set of problems that can be described by the general combinatorial concepts of clutters and blocking clutters we begin our discussion by defining these unifying concepts and illustrating their specialization in several problem contexts

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

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

exact vibration solutions for circular mindlin plates with multiple concentric ring supports this paper presents an analytical approach to investigate the vibration behaviour of circular mindlin plates with multiple concentric ring supports a circular plate is first divided into multiple annular segments and a circular segment at the locations of the ring supports the governing differential equations and the solutions of these equations for the annular and circular segments based on the mindlin plate theory are presented a homogenous equation system governing the vibration of circular mindlin plates with ring supports is derived by imposing the essential and natural boundary and segment interface conditions the first known exact vibration frequencies for circular mindlin plates with multiple concentric ring supports are obtained and the modal shapes of displacement fields and stress resultants for several selected cases are presented the influence of the ring support locations plate boundary conditions and plate thickness ratios on the vibration behaviour of circular plates is discussed

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

robustness to dependency in portfolio optimization using overlapping marginals in this paper we develop a distributionally robust portfolio optimization model where the robustness is across different dependency structures among the random losses for a fr echet class of discrete distributions with overlapping marginals we show that the distributionally robust portfolio optimization problem is efficiently solvable with linear programming to guarantee the existence of a joint multivariate distribution consistent with the overlapping marginal information we make use of a graph theoretic property known as the running intersection property building on this property we develop a tight linear programming formulation to find the optimal portfolio that minimizes the worst case conditional value at risk measure lastly we use a data driven approach with financial return data to identify the fr echet class of distributions satisfying the running intersection property and then optimize the portfolio over this class of distributions numerical results in two different datasets show that the distributionally robust portfolio optimization model improves on the sample based approach

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

persistency model and its applications in choice modeling given a discrete maximization problem with a linear objective function where the coefficients are chosen randomly from a distribution we would like to evaluate the expected optimal value and the marginal distribution of the optimal solution we call this the persistency problem for a discrete optimization problem under uncertain objective and the marginal probability mass function of the optimal solution is named the persistence value in general this is a difficult problem to solve even if the distribution of the objective coefficient is well specified in this paper we solve a subclass of this problem when the distribution is assumed to belong to the class of distributions defined by given marginal distributions or given marginal moment conditions under this model we show that the persistency problem maximizing the expected objective value over the set of distributions can be solved via a concave maximization model the persistency model solved using this formulation can be used to obtain important qualitative insights to the behavior of stochastic discrete optimization problems we demonstrate how the approach can be used to obtain insights to problems in discrete choice modeling using a set of survey data from a transport choice modeling study we calibrate the random utility model with choice probabilities obtained from the persistency model numerical results suggest that our persistency model is capable of obtaining estimates that perform as well if not better than classical methods such as logit and cross nested logit models we can also use the persistency model to obtain choice probability estimates for more complex choice problems we illustrate this on a stochastic knapsack problem which is essentially a discrete choice problem under budget constraint

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

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

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

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

extending positive definite functions from subgroups of nilpotent locally compact groups continuing research begun in 15 and 16 we investigate the problem of when a closed subgroup h of a nilpotent locally compact group g has the property that every continuous positive definite function on h extends to some such function on g the main result is an identification of all subgroups of the heisenberg groups which share this property

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

risk taking in stochastic pert networks abstract this paper studies a pert problem which incorporates project managers risk taking behavior in stochastic environments in this stochastic pert network which we refer to as riskpert activity durations are nonnegative random variables and a project manager uses a disutility function to evaluate each path and chooses that path which maximizes his expected disutility as an optimal or critical path this path as opposed to the classical critical path is used as a basis to analyze the pert network and to estimate the probability of completion time the general riskpert is difficult to solve however special cases when activity durations are statistically independent and disutility functions are linear exponential quadratic or linear exponential are solvable exactly the illustrative examples demonstrate that riskpert captures managers risk taking behavior and provides results that are more realistic than those of the classical pert models furthermore this paper extends some of the riskpert models to include multi dimensional pert networks

the value of side information in shortest path optimization consider an agent who seeks to traverse the shortest path in a graph having random edge weights if the agent has no side information about the realizations of the edge weights it should simply take the path of least average length we consider a generalization of this framework whereby the agent has access to a limited amount of side information about the edge weights ahead of choosing a path we define a measure for information quantity provide bounds on the agent s performance relative to the amount of side information it receives and present algorithms for optimizing side information the results are based on a new graph characterization tied to shortest path optimization

bounding the project completion time distribution in pert networks we consider the pert model of a project composed of activities whose durations are random variables with known distributions for the situations in which the activity durations are completely independent we present a new method for obtaining a probability distribution function that bounds the exact probability distribution of the project completion time from below the bounding distribution can be used to obtain an upper bound on the mean completion time of the project we also prove and illustrate that this bounding distribution is better tighter than any of the existing lower bounds implying that the corresponding upper bound on the mean completion time is tighter than any of the existing upper bounds

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

constant access systems a general framework for greedy optimization on stochastic networks we consider network optimization problems in which the weights of the edges are random variables we develop conditions on the combinatorial structure of the problem which guarantee that the objective function value is a first passage time in an appropriately constructed continuous time markov chain the arc weights must be distributed exponentially the method of solution of the deterministic problem must be greedy in a general sense and the accumulation of objective function value during the greedy procedure must occur at a constant rate we call these structures constant access systems after the third property examples of constant access systems include the shortest path system the longest path system the time until disconnection in a network of failing components and some bottleneck optimization problems for each system we give the distribution of the objective function the distribution of the solution of the problem and the probability that a given arc is a member of the optimal solution we

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

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

the complexity of computing the sign of the tutte polynomial we study the complexity of computing the sign of the tutte polynomial of a graph as there are only three possible outcomes positive negative and zero this seems at first sight more like a decision problem than a counting problem surprisingly however there are large regions of the parameter space for which computing the sign of the tutte polynomial is actually p hard as a trivial consequence approximating the polynomial is also p hard in this case thus approximately evaluating the tutte polynomial in these regions is as hard as exactly counting the satisfying assignments to a cnf boolean formula for most other points in the parameter space we show that computing the sign of the polynomial is in fp whereas approximating the polynomial can be done in polynomial time with an np oracle as a special case we completely resolve the complexity of computing the sign of the chromatic polynomial this is easily computable at q 2 and when q leq 32 27 and is np hard to compute for all other va

vibration of tapered mindlin plates in terms of static timoshenko beam functions abstract in this paper the free vibrations of rectangular mindlin plates with variable thickness in one or two directions are investigated the thickness variation of the plate is continuous and can be represented by a power function of the rectangular co ordinates a wide range of tapered rectangular plates can be described by giving various index values to the power function two sets of new admissible functions are developed respectively to approximate the flexural displacement and the angle of rotation due to bending of the plate the eigenfrequency equation is obtained by using the rayleigh ritz method the complete solutions of displacement and angle of rotation due to bending for a tapered timoshenko beam a strip taken from the tapered mindlin plate in some direction under a taylor series of static load have been derived which are used as the admissible functions of the rectangular mindlin plates with taper thickness in one or two directions unlike conventional admissible functions which are independent of the thickness variation of the plate the static timoshenko beam functions presented in this paper are closely connected with the thickness variation of the plate so that higher accuracy and more rapid convergence can be expected some numerical results are furnished for both truncated mindlin plates and sharp ended mindlin plates on the basis of convergence study and comparison with available results in literature it is shown that the first few eigenfrequencies can be obtained with quite satisfactory accuracy by using only a small number of terms of the static timoshenko beam functions

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

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

flexural vibration of shear deformable circular and annular plates on ring supports this paper presents an analysis of free flexural vibration of circular and annular mindlin plates lying on multiple internal concentric ring supports the rayleigh ritz method with an admissible displacement function expressed in terms of a set of simple polynomials is used to obtain the governing eigenvalue equation the natural frequencies of circular and annular mindlin plates with one or two ring supports having inner and outer peripheries with different combinations of free simply supported and clamped conditions are determined a comprehensive set of vibration results is presented and where possible these have been verified with existing solutions published in the open literature no previous theoretical results for thick annular plates supported with internal rings are known to exist thus the study should be useful and valuable to engineers and researchers

reliability polynomials and their asymptotic limits for families of graphs we present exact calculations of reliability polynomials r g p for lattice strips g of fixed widths l y 4 and arbitrarily great length l x with various boundary conditions we introduce the notion of a reliability per vertex r g p lim v r g p 1 v where v denotes the number of vertices in g and g denotes the formal limit lim v g we calculate this exactly for various families of graphs we also study the zeros of r g p in the complex p plane and determine exactly the asymptotic accumulation set of these zeros b across which r g is nonanalytic

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

vibration of shear deformable plates with variable thickness first order and higher order analyses this work presents accurate numerical calculations of the natural frequencies for elastic rectangular plates of variable thickness with various combinations of boundary conditions the thickness variation in one or two directions of the plate is taken in polynomial form the first order shear deformation plate theory of mindlin and the higher order shear deformation plate theory of reddy have been applied to the plate analysis the governing equations and the boundary conditions are derived using the dynamic version of the principle of minimum of the lagrangian function the solution is obtained by the extended kantorovich method this approach is combined with the exact element method for the vibration analysis of members with variable flexural rigidity which provides for the derivation of the exact dynamic stiffness matrix of varying cross sections strips the large number of numerical examples demonstrates the applicability and versatility of the present method the results obtained by both shear deformation theories are compared with those obtained by the classical thin plate theory and with published results

free vibration analysis of mindlin plates with linearly varying thickness abstract a method based on the variational principles in conjunction with the finite difference technique is applied to examine the free vibration characteristics of isotropic rectangular plates of linearly varying thickness by including the effects of transverse shear deformation and rotary inertia the validity of the present approach is demonstrated by comparing the results with other solutions proposed for plates with uniform and linearly varying thickness natural frequencies and mode shapes of mindlin plates with simply supported and clamped edges are determined for various values of relative thickness ratio and the taper thickness constant

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

counting almost minimum cutsets with reliability applications the numbern1 ofi edge network cutsets in a graph is an important parameter in reliability analysis using a theorem of lomonosov and polesskii ball and provan have shown that the numberni of minimum cardinality network cutsets can be determined in polynomial time this paper gives a polynomial time algorithm for determiningnr k for any fixedk as a special case of counting cuts of specified maximum weight in an edge weighted graph the resulting improvement in existing reliability bounds is shown to be substantial

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

solving a class of stochastic minimization problems this work gives a methodology for analyzing a class of discrete minimization problems with random element weights the minimum weight solution is shown to be an absorbing state in a markov chain while the distribution of weight of the minimum weight element is shown to be of phase type we then present two sided bounds for matroids with nbue distributed weights as well as for weights with bounded positive hazard rates we illustrate our method using a realistic military communications problem