bayesian and non bayesian evidential updating abstract four main results are arrived at in this paper 1 closed convex sets of classical probability functions provide a representation of belief that includes the representations provided by shafer probability mass functions as a special case 2 the impact of uncertain evidence can be formally represented by dempster conditioning in shafer s framework 3 the impact of uncertain evidence can be formally represented in the framework of convex sets of classical probabilities by classical conditionalization 4 the probability intervals that result from dempster shafer updating on uncertain evidence are included in and may be properly included in the intervals that result from bayesian updating on uncertain evidence

on multi index assignment polytopes we investigate an integer programming model for multi dimensional assignment problems this model enables us to establish the dimension for entire families of assignment polytopes thus unifying and generalising previous results in particular we establish the dimension of the linear assignment polytope as well as that of every axial and planar assignment polytope further for the axial polytopes we identify a family of clique facets we also give a necessary condition for the existence of a solution for assignment problems

large sample learning of bayesian networks is np hard in this paper we provide new complexity results for algorithms that learn discrete variable bayesian networks from data our results apply whenever the learning algorithm uses a scoring criterion that favors the simplest structure for which the model is able to represent the generative distribution exactly our results therefore hold whenever the learning algorithm uses a consistent scoring criterion and is applied to a sufficiently large dataset we show that identifying high scoring structures is np hard even when any combination of one or more of the following hold the generative distribution is perfect with respect to some dag containing hidden variables we are given an independence oracle we are given an inference oracle we are given an information oracle we restrict potential solutions to structures in which each node has at most k parents for all k 3 our proof relies on a new technical result that we establish in the appendices in particular we provide a method for constructing the local distributions in a bayesian network such that the resulting joint distribution is provably perfect with respect to the structure of the network

the algebra of bounded linear operators on lp lq has infinitely many closed ideals the first author s research was supported by nsf grant dms 1160633 the second author was supported by the 2014 workshop in analysis and probability at texas a m university

fundamental concepts of qualitative probabilistic networks abstract graphical representations for probabilistic relationships have recently received considerable attention in ai qualitative probabilistic networks abstract from the usual numeric representations by encoding only qualitative relationships which are inequality constraints on the joint probability distribution over the variables although these constraints are insufficient to determine probabilities uniquely they are designed to justify the deduction of a class of relative likelihood conclusions that imply useful decision making properties two types of qualitative relationship are defined each a probabilistic form of monotonicity constraint over a group of variables qualitative influences describe the direction of the relationship between two variables qualitative synergies describe interactions among influences the probabilistic definitions chosen justify sound and efficient inference procedures based on graphical manipulations of the network these procedures answer queries about qualitative relationships among variables separated in the network and determine structural properties of optimal assignments to decision variables

complexity of a 3 dimensional assignment problem abstract we show that a certain 3 dimensional assignment problem is np complete to do this we show that the following problem is np complete given bipartite graphs g 1 g 2 with the same sets of vertices do there exist perfect matchings m 1 m 2 of g 1 g 2 respectively such that m 1 m 2 o

context specific sign propagation in qualitative probabilistic networks qualitative probabilistic networks are qualitative abstractions of probabilistic networks summarising probabilistic influences by qualitative signs as qualitative networks model influences at the level of variables knowledge about probabilistic influences that hold only for specific values cannot be expressed the results computed from a qualitative network as a consequence can be weaker than strictly necessary and may in fact be rather uninformative we extend the basic formalism of qualitative probabilistic networks by providing for the inclusion of context specific information about influences and show that exploiting this information upon reasoning has the ability to forestall unnecessarily weak results

sequential decision making with partially ordered preferences this paper presents new insights and novel algorithms for strategy selection in sequential decision making with partially ordered preferences that is where some strategies may be incomparable with respect to expected utility we assume that incomparability amongst strategies is caused by indeterminacy imprecision in probability values we investigate six criteria for consequentialist strategy selection c maximin c maximax c maximix interval dominance maximality and e admissibility we focus on the popular decision tree and influence diagram representations algorithms resort to linear multilinear programming we describe implementation and experiments

spaceability and operator ideals abstract let i 1 and i 2 be arbitrary operator ideals in the sense of pietsch and e and f be banach spaces such that the set i 1 e f i 2 e f is non empty we give a quite general sufficient condition on the banach spaces in order to obtain the spaceability of i 1 e f i 2 e f some consequences are provided when considering the ideals of compact operators strictly singular operators finitely strictly singular operators and schatten operator classes

a logic to reason about likelihood abstract we present a logic ll which uses a modal operator l to help capture the notion of being likely despite the fact that likelihood is not assigned quantitative values through probabilities ll captures many of the properties of likelihood in an intuitively appealing way we give a possible worlds style semantics to ll and using standard techniques of modal logic we give a complete axiomatization for ll and show that satisfiability of ll formulas can be decided in exponential time we discuss how the logic might be used in areas such as medical diagnosis where decision making in the face of uncertainties is crucial we conclude by using ll to give a formal proof of correctness of some aspects of a protocol for exchanging secrets

sequence singular operators abstract in this paper we study l p related collections of operators introduced by beanland and freeman in 6 on the subject of forming operator ideals we show that these collections are not always closed under addition and hence do not form operator ideals nevertheless they allow us to construct an uncountable chain of closed ideals in each of the operator algebras l l 1 l q 1 q and l l 1 c 0 this finishes answering a longstanding question of pietsch

partially observable markov decision processes with imprecise parameters this study extends the framework of partially observable markov decision processes pomdps to allow their parameters i e the probability values in the state transition functions and the observation functions to be imprecisely specified it is shown that this extension can reduce the computational costs associated with the solution of these problems first the new framework pomdps with imprecise parameters pomdpips is formulated we consider 1 the interval case in which each parameter is imprecisely specified by an interval that indicates possible values of the parameter and 2 the point set case in which each probability distribution is imprecisely specified by a set of possible distributions second a new optimality criterion for pomdpips is introduced as in pomdps the criterion is to regard a policy i e an action selection rule as optimal if it maximizes the expected total reward the expected total reward however cannot be calculated precisely in pomdpips because of the parameter imprecision instead we estimate the total reward by adopting arbitrary second order beliefs i e beliefs in the imprecisely specified state transition functions and observation functions although there are many possible choices for these second order beliefs we regard a policy as optimal as long as there is at least one of such choices with which the policy maximizes the total reward thus there can be multiple optimal policies for a pomdpip we regard these policies as equally optimal and aim at obtaining one of them by appropriately choosing which second order beliefs to use in estimating the total reward computational costs incurred in obtaining such an optimal policy can be reduced significantly we provide an exact solution algorithm for pomdpips that does this efficiently third the performance of such an optimal policy as well as the computational complexity of the algorithm are analyzed theoretically last empirical studies show that our algorithm quickly obtains satisfactory policies to many pomdpips

closed ideals of operators on the tsirelson and schreier spaces abstract let b x denote the banach algebra of bounded operators on x where x is either tsirelson s banach space or the schreier space of order n for some n n we show that the lattice of closed ideals of b x has a very rich structure in particular b x contains at least continuum many maximal ideals our approach is to study the closed ideals generated by the basis projections indeed the unit vector basis is an unconditional basis for each of the above spaces so there is a basis projection p n b x corresponding to each non empty subset n of n a closed ideal of b x is spatial if it is generated by p n for some n we can now state our main conclusions as follows the family of spatial ideals lying strictly between the ideal of compact operators and b x is non empty and has no minimal or maximal elements for each pair i j of spatial ideals there is a family l l where the index set has the cardinality of the continuum such that l is an uncountable chain of spatial ideals l is a closed ideal that is not spatial and i l j and l m j whenever l m are distinct and l l m m

signed directed acyclic graphs for causal inference formal rules governing signed edges on causal directed acyclic graphs are described in this paper and it is shown how these rules can be useful in reasoning about causality specifically the notions of a monotonic effect a weak monotonic effect and a signed edge are introduced results are developed relating these monotonic effects and signed edges to the sign of the causal effect of an intervention in the presence of intermediate variables the incorporation of signed edges into the directed acyclic graph causal framework furthermore allows for the development of rules governing the relationship between monotonic effects and the sign of the covariance between two variables it is shown that when certain assumptions about monotonic effects can be made then these results can be used to draw conclusions about the presence of causal effects even when data is missing on confounding variables

a reduction approach to the repeated assignment problem abstract we consider the repeated assignment problem rap which is a k fold repetition of the n n linear assignment problem lap with the additional requirement that no assignment can be repeated more than once in actual applications k is typically much smaller than n first we derive upper and lower bounds respectively by a heuristic together with local search and an efficient method solving the continuous relaxation the latter also solves a lagrangian relaxation such that the related pegging test to fix variables at zero or one decomposes into k independent pegging tests to laps these can be solved exactly by transforming them into all pairs shortest path problems together with these procedures we also employ a virtual pegging test and reduce rap in size numerical experiments show that the reduced instances with k n can be solved exactly using standard mip solvers

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

an algorithm for the planar three index assignment problem abstract a branch and bound algorithm for the planar three index assignment problem is presented the branching operation is performed on sets of variables belonging to a common constraint rather than on a single variable the algorithm comprises an upper and lower bound procedure the upper bound procedure consists of two parts one for finding an initial solution and one for improving it the local improvement rule takes advantage of the latin square structure of the solution the lower bound procedure combines a new dual heuristic and a lagrangean relaxation scheme solved by subgradient optimization the use of reduced costs in the evaluation of bounds is another feature of the algorithm computational experience both on the exact algorithm and on the primal heuristics as standalone procedures is reported

minimal sufficient causation and directed acyclic graphs notions of minimal sufficient causation are incorporated within the directed acyclic graph causal framework doing so allows for the graphical representation of sufficient causes and minimal sufficient causes on causal directed acyclic graphs while maintaining all of the properties of causal directed acyclic graphs this in turn provides a clear theoretical link between two major conceptualizations of causality one counterfactual based and the other based on a more mechanistic understanding of causation the theory developed can be used to draw conclusions about the sign of the conditional covariances among variables

a survey for the quadratic assignment problem the quadratic assignment problem qap one of the most difficult problems in the np hard class models many real life problems in several areas such as facilities location parallel and distributed computing and combinatorial data analysis combinatorial optimization problems such as the traveling salesman problem maximal clique and graph partitioning can be formulated as a qap in this paper we present some of the most important qap formulations and classify them according to their mathematical sources we also present a discussion on the theoretical resources used to define lower bounds for exact and heuristic algorithms we then give a detailed discussion of the progress made in both exact and heuristic solution methods including those formulated according to metaheuristic strategies finally we analyze the contributions brought about by the study of different approaches

the bilinear assignment problem complexity and polynomially solvable special cases in this paper we study the bilinear assignment problem bap with size parameters m and n m le n bap is a generalization of the well known quadratic assignment problem and the three dimensional assignment problem and hence np hard we show that bap cannot be approximated within a constant factor unless p np even if the associated quadratic cost matrix q is diagonal further we show that bap remains np hard if m o root r of n for some fixed r but is solvable in polynomial time if m o left frac log n log log n right when the rank of q is fixed bap is observed to admit fptas and when this rank is one it is solvable in polynomial time under some additional restrictions we then provide a necessary and sufficient condition for bap to be equivalent to two linear assignment problems a closed form expression to compute the average of the objective function values of all solutions is presented whereas the median of the solution values cannot be identified in polynomial time unless p np we then provide polynomial time heuristic algorithms that find a solution with objective function value no worse than that of m 1 n 1 solutions however computing a solution whose objective function value is no worse than that of m n lceil frac m beta rceil lceil frac n beta rceil solutions is np hard for any fixed rational number beta 1

efficient algorithms with performance guarantees for some problems of finding several discrete disjoint subgraphs in complete weighted graph some hard to solve combinatorial problems of finding several disjoint discrete structures in complete weighted graph are considered efficient algorithms with performance guarantees are constructed for the euclidean m peripatetic salesman problem m weighted clique problem and m layer planar three index assignment problem

a variant of time minimizing assignment problem a time minimizing assignment problem tmap dealing with the allocation of n jobs to m n persons is considered in this paper one job is to be allocated to exactly one person and each person does at least one job all the persons start working on the jobs simultaneously if a person is to do more than one job he does them one after the other in any order the aim of the present study is to find that feasible assignment which minimizes the total time for completing all the jobs a lexi search approach is proposed to find an optimal feasible assignment

probability intervals over influence diagrams a mechanism for performing probabilistic reasoning in influence diagrams using interval rather than point valued probabilities is described procedures for operations corresponding to conditional expectation and bayesian conditioning in influence diagrams are derived where lower bounds on probabilities are stored at each node the resulting bounds for the transformed diagram are shown to be the tightest possible within the class of constraints on probability distributions that can be expressed exclusively as lower bounds on the component probabilities of the diagram sequences of these operations can be performed to answer probabilistic queries with indeterminacies in the input and for performing sensitivity analysis on an influence diagram the storage requirements and computational complexity of this approach are comparable to those for point valued probabilistic inference mechanisms

polyhedral combinatorics of multi index axial transportation problems abstract this paper studies integer points ip and integer vertices iv of the p index axial transportation polytope p atp of order n 1 n 2 n p n 1 n 2 n p 2 p 2 defined by integer vectors as well as noninteger vertices of the 3 atp in particular for the p atp we establish criteria for the minimum and maximum number of ips and describe the class of polytopes for which the number of ips coincides with the number of ivs for the 3 atp of order n n n we prove the theorem on the exponential growth of denominators of fractional components of the polytope vertices three conjectures are stated regarding the maximum number of vertices of the p atp the maximum number of ivs and the structure of the nondegenerate polytopes with the maximum number of ips

lower bounds for the axial three index assignment problem this paper describes new bounding methods for the axial three index assignment problem 3ap for calculating 3ap lower bounds we use a projection method followed by a hungarian algorithm based on a new lagrangian relaxation we also use a cost transformation scheme which iteratively transforms 3ap costs in a series of equivalent 3aps which provides the possibility of improving the 3ap lower bound these methods produce efficiently computed relatively tight lower bound

on linear programs with random costs we consider linear programs in which the objective function cost coefficients are independent non negative random variables and give upper bounds for the random minimum cost one application shows that for quadratic assignment problems with such costs certain branch and bound algorithms usually take more than exponential time

rigidity of composition operators on the hardy space h p abstract let be an analytic map taking the unit disk d into itself we establish that the class of composition operators f c f f exhibits a rather strong rigidity of non compact behaviour on the hardy space h p for 1 p and p 2 our main result is the following trichotomy which states that exactly one of the following alternatives holds i c is a compact operator h p h p ii c fixes a linearly isomorphic copy of l p in h p but c does not fix any copies of l 2 in h p iii c fixes a copy of l 2 in h p moreover in case iii the operator c actually fixes a copy of l p 0 1 in h p provided p 1 we reinterpret these results in terms of norm closed ideals of the bounded linear operators on h p which contain the compact operators k h p in particular the class of composition operators on h p does not reflect the quite complicated lattice structure of such ideals

the multiperiod assignment problem a multicommodity network flow model and specialized branch and bound algorithm abstract the multiperiod assignment problem a specialization of the three dimensional assignment problem is an optimization model that describes the situation of assigning people to activities or jobs over time we consider the most general case which has both a cost of assigning a person to an activity in each time period and a cost of transferring the person from one activity in each period to another activity in the next period in general the number of time periods need not equal the number of persons and activities we present a new formulation of the multiperiod assignment problem that of an integer multicommodity network flow model the special structure of the model allows us to find a good feasible solution relatively quickly by a shortest path heuristic algorithm we discuss a new branch and bound algorithm for solving this problem to optimality the subproblems of the branch and bound tree are evaluated by solving a set of special structured shortest path problems all of which can be solved in order n 2 t time where n is the number of persons and activities and t is the number of time periods we present computational tests of the algorithm on moderately sized problems