a class of joint models for multivariate longitudinal measurements and a binary event predicting binary events such as newborns with large birthweight is important for obstetricians in their attempt to reduce both maternal and fetal morbidity and mortality such predictions have been a challenge in obstetric practice where longitudinal ultrasound measurements taken at multiple gestational times during pregnancy may be useful for predicting various poor pregnancy outcomes the focus of this article is on developing a flexible class of joint models for the multivariate longitudinal ultrasound measurements that can be used for predicting a binary event at birth a skewed multivariate random effects model is proposed for the ultrasound measurements and the skewed generalized t link is assumed for the link function relating the binary event and the underlying longitudinal processes we consider a shared random effect to link the two processes together markov chain monte carlo sampling is used to carry out bayesian posterior computation several variations of the proposed model are considered and compared via the deviance information criterion the logarithm of pseudomarginal likelihood and with a training test set prediction paradigm the proposed methodology is illustrated with data from the nichd successive small for gestational age births study a large prospective fetal growth cohort conducted in norway and sweden

lagrangean relaxation based heuristics for lot sizing with setup times we consider a lot sizing problem with setup times where the objective is to minimize the total inventory carrying cost only the demand is dynamic over time and there is a single resource of limited capacity we show that the approaches implemented in the literature for more general versions of the problem do not perform well in this case we examine the lagrangean relaxation lr of demand constraints in a strong reformulation of the problem we then design a primal heuristic to generate upper bounds and combine it with the lr problem within a subgradient optimization procedure we also develop a simple branch and bound heuristic to solve the problem computational results on test problems taken from the literature show that our relaxation procedure produces consistently better solutions than the previously developed heuristics in the literature

solving the degree concentrated fault tolerant spanning subgraph problem by dc programming in this paper we consider the maximum and minimum versions of degree concentrated fault tolerant spanning subgraph problem which has many applications in network communications we prove that both this two problems are np hard for the maximum version we use dc programming relaxation to propose a heuristic algorithm numerical tests indicate that the proposed algorithm is efficient and effective for the minimum version we also formulate it as a dc program and show that the dc algorithm does not perform well for this problem

multilevel capacitated lotsizing complexity and lp based heuristics abstract this paper presents the first heuristics capable of solving multilevel lotsizing problems with capacity constraints on more than one level moreover the form of the heuristics is quite general so that they can easily be extended to solve a variety of problems if one wants to solve these problems on a routine basis in a real environment one needs to find fast and easy algorithms however we show that for certain problem classes this is a very difficult task far more difficult than has been suggested in the literature for problems with setup times we show that finding a feasible solution is np complete even without setup times testing for feasibility can be very difficult just how time consuming such heuristics must be is demonstrated this leaves little chance to build fast and easy heuristics except for the most simple cases our exploration of the complexity issues points to mathematical programming as a potential source of heuristics for these problems this paper presents a new and general approach based on rounding an lp solution for the problem without setup times the methods use different information and patterns evident in the lp solution are explored the approach is tested on a large set of problems the main contributions of this paper are the way in which we distinguish between the easy and hard lotsizing problems the lp based heuristics and the test set of capacitated multilevel lotsizing problems

on asymptotic normality of limiting density functions with bayesian implications walker 1969 presented a straightforward approach to asymptotic normality of posterior distributions based on independent and identically distributed i i d observations dawid 1970 extended this work by providing a weaker set of conditions to cover some distributions whose ranges depend on the parameters of interest more recently heyde and johnstone 1979 simplified walker s conditions for stochastic processes in this paper we introduce three basic normality conditions for limiting density functions in the bayesian context our approach is data determined and is nonprobabilistic in nature nevertheless the results thus obtained can be easily converted to probabilistic ones when compared with those given by other investigators it is shown that our conditions as a whole have a greater generality and flexibility than those previously assumed either by dawid or heyde and johnstone in particular for the i i d case the commonly assumed wald type uniform boundedness condition for tail behaviours is shown to be unnecessary for asymptotic normality section 3 3 the advantage of our approach is further exemplified by a simple proof of the asymptotic normality of distributions conjugate to the exponential family

bayesian analysis of covariance matrices and dynamic models for longitudinal data summary parsimonious modelling of the within subject covariance structure while heeding its positive definiteness is of great importance in the analysis of longitudinal data using the cholesky decomposition and the ensuing unconstrained and statistically meaningful reparameterisation we provide a convenient and intuitive framework for developing conditionally conjugate prior distributions for covariance matrices and show their connections with generalised inverse wishart priors our priors offer many advantages with regard to elicitation positive definiteness computations using gibbs sampling shrinking covariances toward a particular structure with considerable flexibility and modelling covariances using covariates bayesian estimation methods are developed and the results are compared using two simulation studies these simulations suggest simpler and more suitable priors for the covariance structure of longitudinal data

improved combinatorial algorithms for facility location problems we present improved combinatorial approximation algorithms for the uncapacitated facility location problem two central ideas in most of our results are cost scaling and greedy improvement we present a simple greedy local search algorithm which achieves an approximation ratio of 2 414 epsilon in tilde o n 2 epsilon time this also yields a bicriteria approximation tradeoff of 1 gamma 1 2 gamma for facility cost versus service cost which is better than previously known tradeoffs and close to the best possible combining greedy improvement and cost scaling with a recent primal dual algorithm for facility location due to jain and vazirani we get an approximation ratio of 1 853 in tilde o n 3 time this is very close to the approximation guarantee of the best known algorithm which is linear programming lp based further combined with the best known lp based algorithm for facility location we get a very slight improvement in the approximation factor for facility location achieving 1 728 we also consider a variant of the capacitated facility location problem and present improved approximation algorithms for this

bayesian measures of model complexity and fit summary we consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined using an information theoretic argument we derive a measure pd for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest in general pd approximately corresponds to the trace of the product of fisher s information and the posterior covariance which in normal models is the trace of the hat matrix projecting observations onto fitted values its properties in exponential families are explored the posterior mean deviance is suggested as a bayesian measure of fit or adequacy and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages adding pd to the posterior mean deviance gives a deviance information criterion for comparing models which is related to other information criteria and has an approximate decision theoretic justification the procedure is illustrated in some examples and comparisons are drawn with alternative bayesian and classical proposals throughout it is emphasized that the quantities required are trivial to compute in a markov chain monte carlo analysis

a hybrid adaptive large neighborhood search heuristic for lot sizing with setup times abstract this paper presents a hybrid of a general heuristic framework and a general purpose mixed integer programming mip solver the framework is based on local search and an adaptive procedure which chooses between a set of large neighborhoods to be searched a mixed integer programming solver and its built in feasibility heuristics is used to search a neighborhood for improving solutions the general reoptimization approach used for repairing solutions is specifically suited for combinatorial problems where it may be hard to otherwise design suitable repair neighborhoods the hybrid heuristic framework is applied to the multi item capacitated lot sizing problem with setup times where experiments have been conducted on a series of instances from the literature and a newly generated extension of these on average the presented heuristic outperforms the best heuristics from the literature and the upper bounds found by the commercial mip solver ilog cplex using state of the art mip formulations furthermore we improve the best known solutions on 60 out of 100 and improve the lower bound on all 100 instances from the literature

computing a tree having a small vertex cover abstract we consider a new steiner tree problem called vertex cover weighted steiner tree problem this problem defines the weight of a steiner tree as the minimum weight of vertex covers in the tree and seeks a minimum weight steiner tree in a given vertex weighted undirected graph the problem admits an o log n approximation algorithm in general graphs with n vertices and this approximation factor is tight up to a constant because it is np hard to achieve an o log n approximation for the vertex cover weighted steiner tree problem in general graphs even if the given vertex weights are uniform and a spanning tree is required in this paper we present constant factor approximation algorithms for the problem in unit disk graphs and in graphs excluding a fixed minor for the latter graph class our algorithm can be also applied for the steiner tree activation problem

monte carlo em for missing covariates in parametric regression models we propose a method for estimating parameters for general parametric regression models with an arbitrary number of missing covariates we allow any pattern of missing data and assume that the missing data mechanism is ignorable throughout when the missing covariates are categorical a useful technique for obtaining parameter estimates is the em algorithm by the method of weights proposed in ibrahim 1990 journal of the american statistical association85 765 769 we extend this method t o continuous or mixed categorical and continuous covariates and for arbitrary parametric regression models by adapting a monte carlo version of the em algorithm as discussed by wei and tanner 1990 journal of the american statistical association85 699 704 in addition we discuss the gibbs sampler for sampling from the conditional distribution of the missing covariates given the observed data and show that the appropriate complete conditionals are log concave the log concavity property of the conditional distributions will facilitate a straightforward implementation of the gibbs sampler via the adaptive rejection algorithm of gilks and wild 1992 applied statistics41 337 348 we assume the model for the response given the covariates is an arbitrary parametric regression model such as a generalized linear model a parametric survival model or a nonlinear model we model the marginal distribution of the covariates as a product of one dimensional conditional distributions this allows us a great deal of flexibility in modeling the distribution of the covariates and reduces the number of nuisance parameters that are introduced in the e step we present examples involving both simulated and real data

the k steiner ratio in graphs a steiner minimum tree smt is the shortest length tree in a metric space interconnecting a set of points called the regular points possibly using additional vertices a k size steiner minimum tree ksmt is one that can be split into components where all regular points are leaves and all components have at most k leaves the k steiner ratio rho k is the infimum of the ratios smt ksmt over all finite sets of regular points in all possible metric spaces where the distances are given by a complete graph previously only rho 2 and rho 3 were known exactly in graphs and some bounds were known for other values of k in this paper we determine rho k exactly for all k from this we prove a better approximation ratio for the steiner tree problem in graphs

approximation via cost sharing simpler and better approximation algorithms for network design we present constant factor approximation algorithms for several widely studied np hard optimization problems in network design including the multicommodity rent or buy virtual private network design and single sink buy at bulk problems our algorithms are simple and their approximation ratios improve over those previously known in some cases by orders of magnitude we develop a general analysis framework to bound the approximation ratios of our algorithms this framework is based on a novel connection between random sampling and game theoretic cost sharing

a bayesian proportional hazards regression model with non ignorably missing time varying covariates missing covariate data are common in observational studies of time to an event especially when covariates are repeatedly measured over time failure to account for the missing data can lead to bias or loss of efficiency especially when the data are non ignorably missing previous work has focused on the case of fixed covariates rather than those that are repeatedly measured over the follow up period hence here we present a selection model that allows for proportional hazards regression with time varying covariates when some covariates may be non ignorably missing we develop a fully bayesian model and obtain posterior estimates of the parameters via the gibbs sampler in winbugs we illustrate our model with an analysis of post diagnosis weight change and survival after breast cancer diagnosis in the long island breast cancer study project follow up study our results indicate that post diagnosis weight gain is associated with lower all cause and breast cancer specific survival among women diagnosed with new primary breast cancer our sensitivity analysis showed only slight differences between models with different assumptions on the missing data mechanism yet the complete case analysis yielded markedly different results

latent variable poisson models for assessing the regularity of circadian patterns over time abstractmany researchers in biology and medicine have focused on trying to understand biological rhythms and their potential impact on disease a common biological rhythm is circadian where the cycle repeats itself every 24 hours however a disturbance of the circadian pattern may be indicative of future disease in this article we develop new statistical methodology for assessing the degree of disturbance or irregularity in a circadian pattern for count sequences that are observed over time in a population of individuals we develop a latent variable poisson modeling approach with both circadian and stochastic short term trend autoregressive latent process components that allow for individual variation in the degree of each component a parameterization is proposed for modeling covariate dependence on the proportion of these two model components across individuals in addition we incorporate covariate dependence in the overall mean the magnitude of the trend and the phase shift of the circadian p

a predictive approach to the analysis of designed experiments abstract viewing the analysis of designed experiments as a model selection problem we introduce the use of a predictive bayesian criterion in this context based on the predictive density of a replicate experiment pdre a calibration of the criterion is provided to assist in the model choice the relationships of the proposed criterion to other prevalent criteria such as aic bic and mallows s c p are given an information theoretic criterion based on the pdre s of two competing models is also introduced and compared with the usual f statistic for two nested models examples are given to illustrate the proposed methodology

a pseudo bayesian shrinkage approach to regression with missing covariates we consider the linear regression of outcome y on regressors w and z with some values of w missing when our main interest is the effect of z on y controlling for w three common approaches to regression with missing covariates are i complete case analysis cc which discards the incomplete cases and ii ignorable likelihood methods which base inference on the likelihood based on the observed data assuming the missing data are missing at random rubin 1976b and iii nonignorable modeling which posits a joint distribution of the variables and missing data indicators another simple practical approach that has not received much theoretical attention is to drop the regressor variables containing missing values from the regression modeling dv for drop variables dv does not lead to bias when either i the regression coefficient of w is zero or ii w and z are uncorrelated we propose a pseudo bayesian approach for regression with missing covariates that compromises between the cc and dv estimates exploiting information in the incomplete cases when the data support dv assumptions we illustrate favorable properties of the method by simulation and apply the proposed method to a liver cancer study extension of the method to more than one missing covariate is also discussed

embedding learning capability in lagrangean relaxation an application to the travelling salesman problem abstract this paper presents an effective procedure that finds lower bounds for the travelling salesman problem based on the 1 tree using a learning based lagrangian relaxation technique the procedure can dynamically alter its step size depending upon its previous iterations along with having the capability of expansion contraction the procedure performs a learning process in which lagrange multipliers are influenced by a weighted cost function of their neighbouring nodes in analogy with simulated annealing paradigm here a learning process is equivalent to escaping local optimality via exploiting the structure of the problem computational results conducted on euclidean benchmarks from the tsplib library show that the procedure is very effective

missing covariates in generalized linear models when the missing data mechanism is non ignorable we propose a method for estimating parameters in generalized linear models with missing covariates and a non ignorable missing data mechanism we use a multinomial model for the missing data indicators and propose a joint distribution for them which can be written as a sequence of one dimensional conditional distributions with each one dimensional conditional distribution consisting of a logistic regression we allow the covariates to be either categorical or continuous the joint covariate distribution is also modelled via a sequence of one dimensional conditional distributions and the response variable is assumed to be completely observed we derive the e and m steps of the em algorithm with non ignorable missing covariate data for categorical covariates we derive a closed form expression for the e and m steps of the em algorithm for obtaining the maximum likelihood estimates mles for continuous covariates we use a monte carlo version of the em algorithm to obtain the mles via the gibbs sampler computational techniques for gibbs sampling are proposed and implemented the parametric form of the assumed missing data mechanism itself is not testable from the data and thus the non ignorable modelling considered here can be viewed as a sensitivity analysis concerning a more complicated model therefore although a model may have passed the tests for a certain missing data mechanism this does not mean that we have captured even approximately the correct missing data mechanism hence model checking for the missing data mechanism and sensitivity analyses play an important role in this problem and are discussed in detail several simulations are given to demonstrate the methodology in addition a real data set from a melanoma cancer clinical trial is presented to illustrate the methods proposed

a partition based relaxation for steiner trees the steiner tree problem is a classical np hard optimization problem with a wide range of practical applications in an instance of this problem we are given an undirected graph g v e a set of terminals r subseteq v and non negative costs c e for all edges e in e any tree that contains all terminals is called a steiner tree the goal is to find a minimum cost steiner tree the vertices v backslash r are called steiner vertices the best approximation algorithm known for the steiner tree problem is a greedy algorithm due to robins and zelikovsky siam j discrete math 19 1 122 134 2005 it achieves a performance guarantee of 1 frac ln 3 2 approx 1 55 the best known linear programming lp based algorithm on the other hand is due to goemans and bertsimas math program 60 145 166 1993 and achieves an approximation ratio of 2 2 r in this paper we establish a link between greedy and lp based approaches by showing that robins and zelikovsky s algorithm can be viewed as an iterated primal dual algorithm with respect to a novel lp relaxation the lp used in the first iteration is stronger than the well known bidirected cut relaxation an instance is b quasi bipartite if each connected component of g backslash r has at most b vertices we show that robins and zelikovsky s algorithm has an approximation ratio better than 1 frac ln 3 2 for such instances and we prove that the integrality gap of our lp is between frac 8 7 and frac 2b 1 b 1

a 3 2 approximation algorithm for some minimum cost graph problems we consider a class of graph problems introduced in a paper of goemans and williamson that involve finding forests of minimum edge cost this class includes a number of location routing problems it also includes a problem in which we are given as input a parameter k k and want to find a forest such that each component has at least k k vertices goemans and williamson gave a 2 approximation algorithm for this class of problems we give an improved 3 2 approximation algorithm

lot sizing in general assembly systems with setup costs setup times and multiple constrained resources we introduce a heuristic method for finding good feasible solutions for multiproduct lot sizing problems with general assembly structures multiple constrained resources and nonzero setup costs and setup times we evaluate the performance of this heuristic by comparing its solutions to optimal solutions of small randomly generated problems and to time truncated optimization subroutine library osl solutions of medium sized randomly generated problems in the first case the heuristic locates solutions averaging 4 percent worse than optimal in less than 1 percent of time required by osl the heuristic solutions to medium sized problems are approximately 26 percent better than solutions osl finds after 10 000 cpu seconds and the heuristic finds these solutions in approximately 10 percent of osl time

predictive model selection we consider the problem of selecting one model from a large class of plausible models a predictive bayesian viewpoint is advocated to avoid the specification of prior probabilities for the candidate models and the detailed interpretation of the parameters in each model using criteria derived from a certain predictive density and a prior specification that emphasizes the observables we implement the proposed methodology for three common problems arising in normal linear models variable subset selection selection of a transformation of predictor variables and estimation of a parametric variance function interpretation of the relative magnitudes of the criterion values for various models is facilitated by a calibration of the criteria relationships between the proposed criteria and other well known criteria are examined

a general approximation technique for constrained forest problems we present a general approximation technique for a large class of graph problems our technique mostly applies to problems of covering at minimum cost the vertices of a graph with trees cycles or paths satisfying certain requirements in particular many basic combinatorial optimization problems fit in this framework including the shortest path minimum cost spanning tree minimum weight perfect matching traveling salesman and steiner tree problems our technique produces approximation algorithms that run in o n 2 log n time and come within a factor of 2 of optimal for most of these problems for instance we obtain a 2 approximation algorithm for the minimum weight perfect matching problem under the triangle inequality our running time of o n 2 log n time compares favorably with the best strongly polynomial exact algorithms running in o n 3 time for dense graphs a similar result is obtained for the 2 matching problem and its variants we also derive the first approximation algorithms for many np complete problems including the non fixed point to point connection problem the exact path partitioning problem and complex location design problems moreover for the prize collecting traveling salesman or steiner tree problems we obtain 2 approximation algorithms therefore improving the previously best known performance guarantees of 2 5 and 3 respectively math programming 59 1993 pp 413 420

improved approximation algorithms for prize collecting steiner tree and tsp we study the prize collecting steiner tree pcst prize collecting traveling salesman pctsp and prize collecting path pc path problems given a graph v e with a cost on each edge and a penalty a k a prize on each node the goal is to find a tree for pcst cycle for pctsp or path for pc path that minimizes the sum of the edge costs in the tree cycle path and the penalties of the nodes not spanned by it in addition to being a useful theoretical tool for helping to solve other optimization problems pcst has been applied fruitfully by att a 2 approximation for pc path appeared in 2003 the natural linear programming relaxation of pcst has an integrality gap of 2 which has been a barrier to further improvements for this problem we present 2 epsilon approximation algorithms for all three problems connected by a unified technique for improving prize collecting algorithms that allows us to circumvent the integrality gap barrier specifically our approximation ratio for prize collecting steiner tree is below 1 9672

bayesian analysis of binary and polychotomous response data abstract a vast literature in statistics biometrics and econometrics is concerned with the analysis of binary and polychotomous response data the classical approach fits a categorical response regression model using maximum likelihood and inferences about the model are based on the associated asymptotic theory the accuracy of classical confidence statements is questionable for small sample sizes in this article exact bayesian methods for modeling categorical response data are developed using the idea of data augmentation the general approach can be summarized as follows the probit regression model for binary outcomes is seen to have an underlying normal regression structure on latent continuous data values of the latent data can be simulated from suitable truncated normal distributions if the latent data are known then the posterior distribution of the parameters can be computed using standard results for normal linear models draws from this posterior are used to sample new latent data and t

capacitated lot sizing and scheduling by lagrangean relaxation abstract development of new models and solution procedures for production planning has been of research interest for several decades implementation of these models has resulted in lower production costs by reducing inventories number of setups and labor costs in this paper we develop several optimal near optimal procedures for the capacitated lot sizing and scheduling problem clsp with setup times limited regular time and limited overtime we formulate a mixed integer linear programming model of the problem and solve it by lagrangean relaxation we experiment with alternative lagrangean relaxations and develop new procedures to solve these relaxations overall the capacity constraints relaxation seems to be superior to the demand constraints relaxation our results show that large problems can be solved in reasonable computer times and within one percent accuracy of the optimal solutions we solved 99 8 i e 99 items and 8 periods 50 12 and 50 8 problems in 30 61 36 25 and 12 65 seconds of cdc cyber 730 computer time respectively our procedures are general enough to be applied directly or with slight modifications in real life production settings

model choice a minimum posterior predictive loss approach summary model choice is a fundamental and much discussed activity in the analysis of datasets nonnested hierarchical models introducing random effects may not be handled by classical methods bayesian approaches using predictive distributions can be used though the formal solution which includes bayes factors as a special case can be criticised we propose a predictive criterion where the goal is good prediction of a replicate of the observed data but tempered by fidelity to the observed values we obtain this criterion by minimising posterior loss for a given model and then for models under consideration selecting the one which minimises this criterion for a broad range of losses the criterion emerges as a form partitioned into a goodness of fit term and a penalty term we illustrate its performance with an application to a large dataset involving residential property transactions

on the equivalence of the bidirected and hypergraphic relaxations for steiner tree the bottleneck of the currently best ln 4 varepsilon ln 4 approximation algorithm for the np hard steiner tree problem is the solution of its large so called hypergraphic linear programming relaxation hyp hypergraphic lps are strongly np hard to solve exactly and it is a formidable computational task to even approximate them sufficiently well we focus on another well studied but poorly understood lp relaxation of the problem the bidirected cut relaxation bcr this lp is compact and can therefore be solved efficiently its integrality gap is known to be greater than 1 16 and while this is widely conjectured to be close to the real answer only a trivial upper bound of 2 is known in this article we give an efficient constructive proof that bcr and hyp are polyhedrally equivalent in instances that do not have an edge induced claw on steiner vertices i e they do not contain a steiner vertex with three steiner neighbours this implies faster ln 4 ln 4 approximations for these graphs and is a significant step forward from the previously known equivalence for so called quasi bipartite instances in which steiner vertices form an independent set we complement our results by showing that even restricting to instances where steiner vertices induce one single star determining whether the two relaxations are equivalent is np hard

bayesian analysis for generalized linear models with nonignorably missing covariates we propose bayesian methods for estimating parameters in generalized linear models glms with nonignorably missing covariate data we show that when improper uniform priors are used for the regression coefficients phi of the multinomial selection model for the missing data mechanism the resulting joint posterior will always be improper if i all missing covariates are discrete and an intercept is included in the selection model for the missing data mechanism or ii at least one of the covariates is continuous and unbounded this impropriety will result regardless of whether proper or improper priors are specified for the regression parameters beta of the glm or the parameters alpha of the covariate distribution to overcome this problem we propose a novel class of proper priors for the regression coefficients phi in the selection model for the missing data mechanism these priors are robust and computationally attractive in the sense that inferences about beta are not sensitive to the choice of the hyperparameters of the prior for phi and they facilitate a gibbs sampling scheme that leads to accelerated convergence in addition we extend the model assessment criterion of chen dey and ibrahim 2004a biometrika 91 45 63 called the weighted l measure to glms and missing data problems as well as extend the deviance information criterion dic of spiegelhalter et al 2002 journal of the royal statistical society b 64 583 639 for assessing whether the missing data mechanism is ignorable or nonignorable a novel markov chain monte carlo sampling algorithm is also developed for carrying out posterior computation several simulations are given to investigate the performance of the proposed bayesian criteria as well as the sensitivity of the prior specification real datasets from a melanoma cancer clinical trial and a liver cancer study are presented to further illustrate the proposed methods

steiner tree approximation via iterative randomized rounding the steiner tree problem is one of the most fundamental np hard problems given a weighted undirected graph and a subset of terminal nodes find a minimum cost tree spanning the terminals in a sequence of papers the approximation ratio for this problem was improved from 2 to 1 55 robins and zelikovsky 2005 all these algorithms are purely combinatorial a long standing open problem is whether there is an lp relaxation of steiner tree with integrality gap smaller than 2 rajagopalan and vazirani 1999 in this article we present an lp based approximation algorithm for steiner tree with an improved approximation factor our algorithm is based on a seemingly novel iterative randomized rounding technique we consider an lp relaxation of the problem which is based on the notion of directed components we sample one component with probability proportional to the value of the associated variable in a fractional solution the sampled component is contracted and the lp is updated consequently we iterate this process until all terminals are connected our algorithm delivers a solution of cost at most ln 4 e as a by product of our analysis we show that the integrality gap of our lp is at most 1 55 hence answering the mentioned open question

improved algorithms for joint optimization of facility locations and network connections abstract this paper studies a k median steiner forest problem that jointly optimizes the opening of at most k facility locations and their connections to the client locations so that each client is connected by a path to an open facility with the total connection cost minimized the problem has wide applications in the telecommunication and transportation industries but is strongly np hard in the literature only a 2 approximation algorithm is known it being based on a lagrangian relaxation of the problem and using a sophisticated primal dual schema in this study we have developed an improved approximation algorithm using a simple transformation from an optimal solution of a minimum spanning tree problem compared with the existing 2 approximation algorithm our new algorithm not only achieves a better approximation ratio that is easier to be proved but also guarantees to produce solutions of equal or better quality up to 50 percent improvement in some cases in addition for two non trivial special cases where either every location contains a client or all the locations are in a tree shaped network we have developed for the first time in the literature new algorithms that can solve the problem to optimality in polynomial time

bayesian criterion based model assessment for categorical data we propose a general bayesian criterion for model assessment for categorical data called the weighted l measure which is constructed from the posterior predictive distribution of the data the measure is based on weighting the observations according to the sampling variance of their future response vector the weight component in the weighted l measure plays the role of a penalty term in the criterion in which a greater weight assigned to covariate values implies a greater penalty term on the dimension of the model a detailed justification is provided for such a weighting procedure and several theoretical properties of the weighted l measure are presented for a wide variety of discrete data models for these models we examine properties of the weighted l measure and show that it can perform better than the unweighted l measure in a variety of settings in addition we show that the weighted quadratic loss l measure is more attractive than the unweighted l measure and the deviance loss l measure for categorical data moreover a calibration for the weighted l measure is motivated and proposed which allows us to compare formally the l measure values of competing models a detailed simulation study is presented to examine the performance of the weighted l measure and it is compared to other established model selection methods finally the method is applied to a real dataset using a bivariate ordinal response model

when trees collide an approximation algorithm for thegeneralized steiner problem on networks we give the first approximation algorithm for the generalized network steiner problem a problem in network design an instance consists of a network with link costs and for each pair i j of nodes an edge connectivity requirement r ij the goal is to find a minimum cost network using the available links and satisfying the requirements our algorithm outputs a solution whose cost is within 2 lceil log 2 r 1 rceil of optimal where r is the highest requirement value in the course of proving the performance guarantee we prove a combinatorial min max approximate equality relating minimum cost networks to maximum packings of certain kinds of cuts as a consequence of the proof of this theorem we obtain an approximation algorithm for optimally packing these cuts we show that this algorithm has application to estimating the reliability of a probabilistic network

new approaches for virtual private network design virtual private network design is the following np hard problem we are given a communication network represented as a weighted graph with thresholds on the nodes which represent the amount of flow that a node can send to and receive from the network the task is to reserve capacities at minimum cost and to specify paths between every ordered pair of nodes such that all valid traffic matrices can be routed along the corresponding paths recently this network design problem has received considerable attention in the literature it is motivated by the fact that the exact amount of flow which is exchanged between terminals is not known in advance and prediction is often elusive the main contributions of this paper are as follows 1 using hu s 2 commodity flow theorem we provide a new and considerably stronger lower bound on the cost of an optimum solution with this lower bound we reanalyze a simple routing scheme which has been described in the literature many times and provide an improved upper bound on its approximation ratio 2 we present a new randomized approximation algorithm in contrast to earlier approaches from the literature the resulting solution does not have tree structure a combination of our new algorithm with the simple routing scheme yields an expected performance ratio of 3 79 for virtual private network design this is a considerable improvement of the previously best known 5 55 approximation result a gupta a kumar and t roughgarden simpler and better approximation algorithms for network design in proceedings of the acm symposium on theory of computing acm new york 2003 pp 365 372 3 our vpnd algorithm uses a steiner tree approximation algorithm as a subroutine it is known that an optimum steiner tree can be computed in polynomial time if the number of terminals is logarithmic replacing the approximate steiner tree computation with an exact one whenever the number of terminals is sufficiently small we finally reduce the approximation ratio to 3 55 to the best of our knowledge this is the first time that a nontrivial result from exact exponential algorithms leads to an improved polynomial time approximation algorithm

improved approximations for two stage min cut and shortest path problems under uncertainty in this paper we study the robust and stochastic versions of the two stage min cut and shortest path problems introduced in dhamdhere et al in how to pay come what may approximation algorithms for demand robust covering problems in focs pp 367 378 2005 and give approximation algorithms with improved approximation factors specifically we give a 2 approximation for the robust min cut problem and a 4 approximation for the stochastic version for the two stage shortest path problem we give a 3 39 3 39 approximation for the robust version and 6 78 6 78 approximation for the stochastic version our results significantly improve the previous best approximation factors for the problems in particular we provide the first constant factor approximation for the stochastic min cut problem our algorithms are based on a guess and prune strategy that crucially exploits the nature of the robust and stochastic objective in particular we guess the worst case second stage cost and based on the guess select a subset of costly scenarios for the first stage solution to address the second stage solution for any scenario is simply the min cut or shortest path problem in the residual graph the key contribution is to show that there is a near optimal first stage solution that completely satisfies the subset of costly scenarios that are selected by our procedure while the guess and prune strategy is not directly applicable for the stochastic versions we show that using a novel lp formulation we can adapt a guess and prune algorithm for the stochastic versions our algorithms based on the guess and prune strategy provide insights about the applicability of this approach for more general robust and stochastic versions of combinatorial problems

from uncertainty to nonlinearity solving virtual private network via single sink buy at bulk the virtual private network problem vpn models scenarios in which traffic is uncertain or rapidly changing the goal is supporting at minimum cost a given family of traffic matrices which are implicitly given by upper bounds on the ingoing and outgoing traffic at each node costs are classically defined by a linear function linear vpn but we consider here also the more general case of concave increasing costs concave vpn in this paper we give the first constant factor approximation for concave vpn and we improve the best known approximation factor for linear vpn our approximation results build on a novel reduction based on konig s theorem which allows us to turn uncertainty of traffic into nonlinearity of the objective function this way we are able to reduce linear vpn and concave vpn to the single sink rent or buy problem srob and the single sink buy at bulk problem ssbb respectively using the machinery developed for the latter two problems plus additional ideas we are able to improve the approximation ratio for vpn along the way we also obtain among other results an improved approximation algorithm for ssbb and a tighter bound on the gap between the costs of arbitrary solutions and tree solutions for vpn furthermore solving an open problem we show that vpn remains np hard even in the balanced case where the sum of ingoing and outgoing traffic bounds is equal

improved approximation algorithms for minimum power covering problems given an undirected graph with edge costs the power of a node is the maximum cost of an edge incident to it and the power of a graph is the sum of the powers of its nodes motivated by applications in wireless networks we consider two network design problems under the power minimization criteria in both problems we are given a graph g v e with edge costs and a set t subseteq v of terminals the goal is to find a minimum power edge subset f subseteq e such that the graph h v f satisfies some prescribed requirements in the min power edge cover problem h should contain an edge incident to every terminal using the iterative randomized rounding irr method we give an algorithm with expected approximation ratio 1 41 the ratio is reduced to 73 60 1 217 when t is an independent set in g in the case of unit costs we also achieve ratio 73 60 and in addition give a simple efficient combinatorial algorithm with ratio 5 4 for all these np hard problems the previous best known ratio was 3 2 in the related min power terminal backup problem h should contain a path from every t in t to some node in t setminus t we obtain ratio 3 2 for this np hard problem improving the trivial ratio of 2

a fully polynomial approximation scheme for single product scheduling in a finite capacity facility this paper considers a version of the economic lot sizing problem for a single product produced in a facility of finite capacity over a finite time horizon with specifiable start and end conditions a set of algorithms is presented that will approximate the optimal production schedule to a given allowable error e algorithms with computation time bounds of o1 e2 are presented which allow for setups of finite length setups with or without direct cash flow quite general cost and demand functions and a wide variety of production policy constraints the procedures make no a priori assumptions about the form of the optimal solution numerical results are included

the steiner tree problem on graphs inapproximability results the steiner tree problem on weighted graphs seeks a minimum weight subtree containing a given subset of the vertices terminals we show that it is np hard to approximate the steiner tree problem within a factor 96 95 our inapproximability results are stated in a parametric way and explicit hardness factors would be improved automatically by providing gadgets and or expanders with better parameters

a lagrangean based heuristic for dynamic multilevel multiitem constrained lotsizing with setup times in this paper a heuristic approach for the dynamic multilevel multiitem lotsizing problem in general product structures with multiple constrained resources and setup times is proposed with the help of lagrangean relaxation the capacitated multilevel multiitem lotsizing problem is decomposed into several uncapacitated single item lotsizing problems from the solutions of these single item problems lower bounds on the minimum objective function value are derived upper bounds are generated by means of a heuristic finite scheduling procedure the quality of the approach is tested with reference to various problem groups of differing sizes

polynomial time approximation schemes for euclidean traveling salesman and other geometric problems we present a polynomial time approximation scheme for euclidean tsp in fixed dimensions for every fixed c 1 and given any n nodes in r 2 a randomized version of the scheme finds a 1 1 c approximation to the optimum traveling salesman tour in o n log n o c time when the nodes are in r d the running time increases to o n log n o d c d 1 for every fixed c d the running time is n poly log n that is nearly linear in n the algorithmm can be derandomized but this increases the running time by a factor o n d the previous best approximation algorithm for the problem due to christofides achieves a 3 2 aproximation in polynomial time we also give similar approximation schemes for some other np hard euclidean problems minimum steiner tree k tsp and k mst the running times of the algorithm for k tsp and k mst involve an additional multiplicative factor k the previous best approximation algorithms for all these problems achieved a constant factor approximation we also give efficient approximation schemes for euclidean min cost matching a problem that can be solved exactly in polynomial time all our algorithms also work with almost no modification when distance is measured using any geometric norm such as e p for p 1 or other minkowski norms they also have simple parallel i e nc implementations

model selection criteria for missing data problems using the em algorithm we consider novel methods for the computation of model selection criteria in missing data problems based on the output of the em algorithm the methodology is very general and can be applied to numerous situations involving incomplete data within an em framework from covariates missing at random in arbitrary regression models to nonignorably missing longitudinal responses and or covariates toward this goal we develop a class of information criteria for missing data problems called ic h q which yields the akaike information criterion and the bayesian information criterion as special cases the computation of ic h q requires an analytic approximation to a complicated function called the h function along with output from the em algorithm used in obtaining maximum likelihood estimates the approximation to the h function leads to a large class of information criteria called ic h k q theoretical properties of ic h k q including consistency are investigated in detail to eliminate the analytic approximation to the h function a computationally simpler approximation to ic h q called ic q is proposed the computation of which depends solely on the q function of the em algorithm advantages and disadvantages of ic h k q and ic q are discussed and examined in detail in the context of missing data problems extensive simulations are given to demonstrate the methodology and examine the small sample and large sample performance of ic h k q and ic q in missing data problems an aids data set also is presented to illustrate the proposed methodology

propriety of the posterior distribution and existence of the mle for regression models with covariates missing at random characterizing model identifiability in the presence of missing covariate data is a very important issue in missing data problems in this article we characterize the propriety of the posterior distribution of the regression coefficients for some general classes of regression models including the class of generalized linear models glm s and parametric survival models with right censored data toward this goal we derive some very general and easy to check conditions for the matrix of covariates we also derive sufficient conditions for the existence of the maximum likelihood estimates and establish novel results for checking propriety of the posterior when the sample size is large several theorems are given to establish propriety of the posterior and the existence of the maximum likelihood estimator the conditions reduce to solving a system of linear equations which can be carried out using software such as maple imsl or sas we assume that the missing covariates are missing at random and assume

primal dual approach to the single level capacitated lot sizing problem abstract the lagrangean relaxation of the single level capacitated dynamic lot sizing problem can be solved using the primal dual method the algorithm has monotone and finite convergence properties it works as a steepest ascent method a variant of this approach is also studied a heuristic routine used to obtain a feasible solution in each iteration is presented computational experiences show that this method usually yields better solutions than the subgradient method although it requires greater cpu times