a polynomial time approximation algorithm for the permanent of a matrix with nonnegative entries we present a polynomial time randomized algorithm for estimating the permanent of an arbitrary n n matrix with nonnegative entries this algorithm technically a fully polynomial randomized approximation scheme computes an approximation that is with high probability within arbitrarily small specified relative error of the true value of the permanent

approximation algorithms for the capacitated multi item lot sizing problem via flow cover inequalities we study the classical capacitated multi item lot sizing problem with hard capacities there are n items each of which has a specified sequence of demands over a finite planning horizon of t discrete periods the demands are known in advance but can vary from period to period all demands must be satisfied on time each order incurs a time dependent fixed ordering cost regardless of the combination of items or the number of units ordered but the total number of units ordered cannot exceed a given capacity c on the other hand carrying inventory from period to period incurs holding costs the goal is to find a feasible solution with minimum overall ordering and holding costs we show that the problem is strongly np hard and then propose a novel facility location type lp relaxation that is based on an exponentially large subset of the well known flow cover inequalities the proposed lp can be solved to optimality in polynomial time via an efficient separation procedure for this subset of inequalities moreover the optimal solution of the lp can be rounded to a feasible integer solution with cost that is at most twice the optimal cost this provides a 2 approximation algorithm which is the first constant approximation algorithm for the problem we also describe an interesting on the fly variant of the algorithm that does not require solving the lp a priori with all the flow cover inequalities as a by product we obtain the first theoretical proof regarding the strength of flow cover inequalities in capacitated inventory models

convergence of min sum message passing for quadratic optimization we establish the convergence of the min sum message passing algorithm for minimization of a quadratic objective function given a convex decomposition our results also apply to the equivalent problem of the convergence of gaussian belief propagation

lost sales inventory systems with a service level criterion competitive retail environments are characterized by service levels and lost sales in case of excess demand we contribute to research on lost sales models with a service level criterion in multiple ways first we study the optimal replenishment policy for this type of inventory system as well as base stock policies and r s s policies furthermore we derive lower and upper bounds on the order up to level and we propose efficient approximation procedures to determine the order up to level the procedures find values of the inventory control variables that are close to the best r s s policy and comply to the service level restriction for most of the instances with an average cost increase of 2 3 and 1 2 for the case without and with fixed order costs respectively

on the number of conjugacy classes of finite nilpotent groups abstract we establish the first super logarithmic lower bound for the number of conjugacy classes of a finite nilpotent group in particular we obtain that for any constant c there are only finitely many finite p groups of order p m with at most c m conjugacy classes this answers a question of l pyber

approximation balancing policies for inventory systems with remanufacturing we consider a single product periodic review inventory system with remanufacturable returned products in which the serviceable product used to fulfill stochastic customer demand can be either manufactured from new parts or remanufactured from returned products demand and returns follow general stochastic processes and may be correlated nonstationary and evolving across different periods the system costs include remanufacturing and manufacturing costs as well as inventory holding and demand backlogging costs the objective is to minimize the expected total discounted cost over a finite planning horizon the optimal policy for this model has a simple structure but its computation can be practically intractable because of the large and high dimensional state space henceforth we propose an efficient approximation algorithm based on cost balancing techniques to compute manufacturing and remanufacturing quantities in each period and show that the expected cost under this remanufacturing balancing policy

hausdorff dimension pro p groups and kac moody algebras every finitely generated profinite group can be given the structure of a metric space and as such it has a well defined hausdorff dimension function in this paper we study hausdorff dimension of closed subgroups of finitely generated pro p groups g we prove that if g is p adic analytic and h c g is a closed subgroup then the hausdorff dimension of h is dimh dimg where the dimensions are of h and g as lie groups letting the spectrum spec g of g denote the set of hausdorff dimensions of closed subgroups of g it follows that the spectrum of p adic analytic groups is finite and consists of rational numbers we then consider some non p adic analytic groups g and study their spectrum in particular we investigate the maximal hausdorff dimension of nonopen subgroups of g and show that it is equal to 1 1 d 1 in the case of g sld fp t where p 2 and to 1 2 if g is the so called nottingham group where p 5 we also determine the spectrum of sl2 fp t p 2 completely showing that it is equal to 0 2 3 1 some of the proofs rely on the description of maximal graded subalgebras of kac moody algebras recently obtained by the authors in joint work with e i zelmanov

on nilpotent multipliers of some verbal products of groups the paper is devoted to finding a homomorphic image for the c nilpotent multiplier of the verbal product of a family of groups with respect to a variety v when v nc or nc v also a structure of the c nilpotent multiplier of a special case of the verbal product the nilpotent product of cyclic groups is given in fact we present an explicit formula for the c nilpotent multiplier of the nth nilpotent product of the group g z n nz nzr1 n nzrt where ri 1 divides ri for all i 1 i t 1 and p r1 1 for any prime p less than or equal to n c for all positive integers n c

stability in two stage stochastic programming we analyze the effect of changes in problem functions and or distributions in certain two stage stochastic programming problems with recourse under reasonable assumptions the locally optimal value of the perturbed problem will be continuous and the corresponding set of local optimizers will be upper semicontinuous with respect to the parameters including the probability distribution in the second stage

open loop routing of n arrivals to m parallel queues we consider the problem of routing n arrivals to m queues in parallel when no information other than the prior statistics are available for the decision process the work we present here differs from results in the literature in two significant ways 1 we address the nonstationary problem of open loop routing for n spl infin arrivals and 2 we do not decompose the problem into an allocation phase and a routing phase but look instead for an overall optimal policy exhaustive enumeration and dynamic programming turn out to be computationally infeasible approaches for realistic values of the parameters n and m examination of the necessary conditions for optimality as given by pontryagin s maximum principle leads to the formulation of a policy iteration algorithm convergence of the algorithm in a finite number of iterations as well as monotonicity results are established while only convergence to a local minimum is guaranteed extensive computational experimentation points to its near optimality some variants of the basic policy iteration algorithm are also discussed

on the relation between upper central quotients and lower central series of a group let h be a group with a normal subgroup n contained in the upper central subgroup zch in this article we study the influence of the quotient group g h n on the lower central subgroup yc lh in particular for any finite group g we give bounds on the order and exponent of yc l h for g equal to a dihedral group or quaternion group or extra special group we list all possible groups that can arise as yc lh our proofs involve i the baer invariants of g ii the schur multiplier m l g of g relative to a normal subgroup l and iii the nonabelian tensor product of groups some results on the nonabelian tensor product may be of independent interest

provably near optimal sampling based policies for stochastic inventory control models in this paper we consider two fundamental inventory models the single period newsvendor problem and its multiperiod extension but under the assumption that the explicit demand distributions are not known and that the only information available is a set of independent samples drawn from the true distributions under the assumption that the demand distributions are given explicitly these models are well studied and relatively straightforward to solve however in most real life scenarios the true demand distributions are not available or they are too complex to work with thus a sampling driven algorithmic framework is very attractive both in practice and in theory we shall describe how to compute sampling based policies that is policies that are computed based only on observed samples of the demands without any access to or assumptions on the true demand distributions moreover we establish bounds on the number of samples required to guarantee that with high probability the expected cost of the sampling based policies is arbitrarily close i e with arbitrarily small relative error compared to the expected cost of the optimal policies which have full access to the demand distributions the bounds that we develop are general easy to compute and do not depend at all on the specific demand distributions

segmentation problems we study a novel genre of optimization problems which we call segmentation problems motivated in part by certain aspects of clustering and data mining for any classical optimization problem the corresponding segmentation problem seeks to partition a set of cost vectors into several segments so that the overall cost is optimized we focus on two natural and interesting but maxsnp complete problems in this class the hypercube segmentation problem and the catalog segmentation problem and present approximation algorithms for them we also present a general greedy scheme which can be specialized to approximate any segmentation problem

optimal and near optimal policies for lost sales inventory models with at most one replenishment order outstanding in this paper we use policy iteration to explore the behaviour of optimal control policies for lost sales inventory models with the constraint that not more than one replenishment order may be outstanding at any time continuous and periodic review fixed and variable lead times replenishment order sizes which are constrained to be an integral multiple of some fixed unit of transfer and service level constraint models are all considered demand is discrete and for continuous review assumed to derive from a compound poisson process it is demonstrated that in general neither the best s s nor the best r q policy is optimal but that the best policy from within those classes will have a cost which is generally close to that of the optimal policy obtained by policy iteration finally near optimal computationally efficient control procedures for finding s s and r q policies are proposed and their performance illustrated

asymptotically optimal inventory control for assemble to order systems with identical lead times optimizing multiproduct assemble to order ato inventory systems is a long standing difficult problem we consider ato systems with identical component lead times and a general bill of materials we use a related two stage stochastic program sp to set a lower bound on the average inventory cost and develop inventory control policies for the dynamic ato system using this sp we apply the first stage sp optimal solution to specify a base stock replenishment policy and the second stage sp recourse linear program to make allocation decisions we prove that our policies are asymptotically optimal on the diffusion scale so the percentage gap between the average cost from its lower bound diminishes to zero as the lead time grows subject classifications assemble to order inventory management stochastic linear program stochastic control asymptotic optimality diffusion scale area of review stochastic models history received may 2013 revisions received july 2014 january 2015 accepted february 2015

joint demand fulfillment probability in a multi item inventory system with independent order up to policies we consider a multi item inventory system with dependent item demands represented by a multivariate normal distribution and filled under a first come first served rule each item is managed independently through a periodic review order up to policy while all items have the same review cycle we obtain the joint demand fulfillment probability within a pre specified time window we also study the problem of maximizing the joint demand fulfillment probability and discuss a heuristic approach in which equal safety factors equal fractiles are specified for all items finally we present numerical results and an application with actual data

a characterisation of powerfully embedded normal subgroups of ap group abstract letpbe a prime number andgbe ap group 6 have introduced the notion of a powerfully embedded subgroup ofg our main result gives a characterisation of those powerfully embedded subgroups which are contained in the frattini subgroup ofg whenpis odd whenp 2 the subgroups characterised have a weaker property than powerful embedding this result also yields information on the structure of d maximal 2 groups

smoothing point processes as a means to increase throughput unlike the leaky bucket scheme which regulates the input rate the filters we study in this paper reduce the variability of interarrival times subject to a maximum delay constraint for each customer these filters are called smoothing filters for point processes by considering the output processes of various queueing systems we show that an infinite server queue acts as a smoothing filter for a doubly stochastic poisson process and a single server queue with deterministic service times acts as a smoothing filter for a stationary and ergodic point process based on the second point we provide a smoothing algorithm that satisfies a maximum delay constraint the algorithm is shown to be robust and consistent we then consider two examples where these filters can be applied to increase throughput one is the single server loss system with exponential service times the other is the leaky bucket scheme with token buffer size 1 numerical examples and simulations are also given for comparing the tradeoff between delay and throughput our approach is based on the variability ordering and a cut criterion for the majorization ordering the criterion appears to be new and of independent interest

fully polynomial approximation schemes for single item capacitated economic lot sizing problems np hard cases of the single item capacitated lot sizing problem have been the topic of extensive research and continue to receive considerable attention however surprisingly few theoretical results have been published on approximation methods for these problems to the best of our knowledge until now no polynomial approximation method is known that produces solutions with a relative deviation from optimality that is bounded by a constant in this paper we show that such methods do exist by presenting an even stronger result the existence of fully polynomial approximation schemes the approximation scheme is first developed for a quite general model which has concave backlogging and production cost functions and arbitrary monotone holding cost functions subsequently we discuss important special cases of the model and extensions of the approximation scheme to even more general models

asymptotics in the random assignment problem we show that in the usual probabilistic model for the random assignment problem the optimal cost tends to a limit constant in probability and in expectation the method involves construction of an infinite limit structure in terms of which the limit constant is defined but we cannot improve on the known numerical bounds for the limit

supermodularity and affine policies in dynamic robust optimization this paper considers a particular class of dynamic robust optimization problems where a large number of decisions must be made in the first stage which consequently fix the constraints and cost structure underlying a one dimensional linear dynamical system we seek to bridge two classical paradigms for solving such problems namely 1 dynamic programming dp and 2 policies parameterized in model uncertainties also known as decision rules obtained by solving tractable convex optimization problems we show that if the uncertainty sets are integer sublattices of the unit hypercube the dp value functions are convex and supermodular in the uncertain parameters and a certain technical condition is satisfied then decision rules that are affine in the uncertain parameters are optimal we also derive conditions under which such rules can be obtained by optimizing simple i e linear objective functions over the uncertainty sets our results suggest new modeling paradigms for dynamic robust optimiz

approximation algorithms for stochastic inventory control models we consider two classical stochastic inventory control models the periodic review stochastic inventory control problem and the stochastic lot sizing problem the goal is to coordinate a sequence of orders of a single commodity aiming to supply stochastic demands over a discrete finite horizon with minimum expected overall ordering holding and backlogging costs in this paper we address the important problem of finding computationally efficient and provably good inventory control policies for these models in the presence of correlated nonstationary time dependent and evolving stochastic demands this problem arises in many domains and has many practical applications in supply chain management our approach is based on a new marginal cost accounting scheme for stochastic inventory control models combined with novel cost balancing techniques specifically in each period we balance the expected cost of overordering i e costs incurred by excess inventory against the expected cost of underordering i e costs incurred by not satisfying demand on time this leads to what we believe to be the first computationally efficient policies with constant worst case performance guarantees for a general class of important stochastic inventory control models that is there exists a constant c such that for any instance of the problem the expected cost of the policy is at most c times the expected cost of an optimal policy in particular we provide a worst case guarantee of two for the periodic review stochastic inventory control problem and a worst case guarantee of three for the stochastic lot sizing problem our results are valid for all of the currently known approaches in the literature to model correlation and nonstationarity of demands over time

on pro p groups with potent filtrations abstract in this note we prove that if g is pf group of finite exponent then the exponent of the second homology group h 2 g m divides the exponent of g for every profinite trivial z g module m we introduce the notion of the exponential rank of a pro p group and find a bound for the exponential rank of a pf group

discrete time controlled markov processes with average cost criterion a survey this work is a survey of the average cost control problem for discrete time markov processes the authors have attempted to put together a comprehensive account of the considerable research on this problem over the past three decades the exposition ranges from finite to borel state and action spaces and includes a variety of methodologies to find and characterize optimal policies the authors have included a brief historical perspective of the research efforts in this area and have compiled a substantial yet not exhaustive bibliography the authors have also identified several important questions that are still open to investigation

approximating the nonlinear newsvendor and single item stochastic lot sizing problems when data is given by an oracle the single item stochastic lot sizing problem is to find an inventory replenishment policy in the presence of discrete stochastic demands under periodic review and finite time horizon a closely related problem is the single period newsvendor model it is well known that the newsvendor problem admits a closed formula for the optimal order quantity whenever the revenue and salvage values are linear increasing functions and the procurement ordering cost is fixed plus linear the optimal policy for the single item lot sizing model is also well known under similar assumptions in this paper we show that the classical single period newsvendor model with fixed plus linear ordering cost cannot be approximated to any degree of accuracy when either the demand distribution or the cost functions are given by an oracle we provide a fully polynomial time approximation scheme for the nonlinear single item stochastic lot sizing problem when demand distribution is given by an oracle procurement costs are provided as nondecreasing oracles holding backlogging disposal costs are linear and lead time is positive similar results exist for the nonlinear newsvendor problem these approximation schemes are designed by extending the technique of k approximation sets and functions

approximating the stochastic knapsack problem the benefit of adaptivity we consider a stochastic variant of the np hard 0 1 knapsack problem in which item values are deterministic and item sizes are independent random variables with known arbitrary distributions items are placed in the knapsack sequentially and the act of placing an item in the knapsack instantiates its size our goal is to compute a solution policy that maximizes the expected value of items successfully placed in the knapsack where the final overflowing item contributes no value we consider both nonadaptive policies that designate a priori a fixed sequence of items to insert and adaptive policies that can make dynamic choices based on the instantiated sizes of items placed in the knapsack thus far an important facet of our work lies in characterizing the benefit of adaptivity for this purpose we advocate the use of a measure called the adaptivity gap the ratio of the expected value obtained by an optimal adaptive policy to that obtained by an optimal nonadaptive policy we bound the adaptivity

schur multipliers and power endomorphisms of groups abstract we obtain new bounds for the exponent of the schur multiplier of a given p group we prove that the exponent of the schur multiplier can be bounded by a function depending only on the exponent of a given group as a consequence we show that the exponent of the schur multiplier of any group of exponent four divides eight and that this bound is best possible the notion of the exponential rank of a p group is introduced we show that powerful p groups have exponential rank either zero or one

profinite groups and the fixed points of coprime automorphisms abstract the main result of the paper is the following theorem let q be a prime and a an elementary abelian group of order q 3 suppose that a acts coprimely on a profinite group g and assume that c g a is locally nilpotent for each a a then the group g is locally nilpotent

the unitary cover of a finite group and the exponent of the schur multiplier abstract for a finite group we introduce a particular central extension the unitary cover having minimal exponent among those satisfying the projective lifting property we obtain new bounds for the exponent of the schur multiplier relating to subnormal series and we discover new families for which the bound is the exponent of the group finally we show that unitary covers are controlled by the zelmanov solution of the restricted burnside problem for 2 generator groups

sequential cavity method for computing free energy and surface pressure we propose a new method for the problems of computing free energy and surface pressure for various statistical mechanics models on a lattice d our method is based on representing the free energy and surface pressure in terms of certain marginal probabilities in a suitably modified sublattice of d then recent deterministic algorithms for computing marginal probabilities are used to obtain numerical estimates of the quantities of interest the method works under the assumption of strong spatial mixing ssp which is a form of a correlation decay

comparison properties of stochastic decision free petri nets stochastic comparison problems arising in the analysis of stochastic decision free petri nets are treated the analysis is based on the evolution equations satisfied by firing times various structural properties are obtained including the association of the firing times and their stochastic and convex monotonicity with respect to the holding time sequences the initial marking and the topology the association and the stochastic monotonicity properties are extended to the counters using an inversion relation it is also proven that the counters and the throughput are stochastically concave in the initial marking provided that the holding times are independent identically distributed and belong to a subclass of the log concave distribution functions that are introduced various bounds for the asymptotic cycle time and the throughput are then derived from these stochastic ordering results comparative results are presented for the marking and response times

finding optimal s s policies is about as simple as evaluating a single policy in this paper a new algorithm for computing optimal s s policies is derived based upon a number of new properties of the infinite horizon cost function c s s as well as a new upper bound for optimal order up to levels s and a new lower bound for optimal reorder levels s the algorithm is simple and easy to understand its computational complexity is only 2 4 times that required to evaluate a specific single s s policy the algorithm applies to both periodic review and continuous review inventory systems

some computations of non abelian tensor products of groups on considere le produit tensoriel generalise g h des groupes g h on presente le calcul de g g

a stochastic programming approach for planning horizons of infinite horizon capacity planning problems planning horizon is a key issue in production planning different from previous approaches based on markov decision processes we study the planning horizon of capacity planning problems within the framework of stochastic programming we first consider an infinite horizon stochastic capacity planning model involving a single resource linear cost structure and discrete distributions for general stochastic cost and demand data non markovian and non stationary we give sufficient conditions for the existence of an optimal solution furthermore we study the monotonicity property of the finite horizon approximation of the original problem we show that the optimal objective value and solution of the finite horizon approximation problem will converge to the optimal objective value and solution of the infinite horizon problem when the time horizon goes to infinity these convergence results together with the integrality of decision variables imply the existence of a planning horizon we also develop a useful formula to calculate an upper bound on the planning horizon then by decomposition we show the existence of a planning horizon for a class of very general stochastic capacity planning problems which have complicated decision structure

stochastic lot sizing problem with inventory bounds and constant order capacities abstract in this paper we study the stochastic version of lot sizing problems with inventory bounds and order capacities customer demands inventory bounds and costs are subject to uncertainty and dependent with each other throughout the finite planning horizon two models in stochastic programming are developed the first one has inventory bound constraints and the second one has both inventory bound and order capacity constraints we explore structural properties of the two models and develop o n 2 and o n 2 t log n dynamic programming algorithms for them respectively our model also generalizes the deterministic lot sizing problem with inventory bounds for some cases when applied to the deterministic versions our algorithms outperform existing deterministic algorithms

optimality gap of constant order policies decays exponentially in the lead time for lost sales models inventory models with lost sales and large lead times have traditionally been considered intractable due to the curse of dimensionality recently goldberg and coauthors laid the foundations for a new approach to solving these models by proving that as the lead time grows large a simple constant order policy is asymptotically optimal however the bounds proven there require the lead time to be very large before the constant order policy becomes effective in contrast to the good numerical performance demonstrated by zipkin even for small lead time values in this work we prove that for the infinite horizon variant of the same lost sales problem the optimality gap of the same constant order policy actually converges exponentially fast to zero with the optimality gap decaying to zero at least as fast as the exponential rate of convergence of the expected waiting time in a related single server queue to its steady state value we also derive simple and explicit bounds for the optimality gap and demons

an efficient algorithm for computing optimal s s policies this paper presents an algorithm to compute an optimal s s policy under standard assumptions stationary data well behaved one period costs discrete demand full backlogging and the average cost criterion the method is iterative starting with an arbitrary given s s policy and converging to an optimal policy in a finite number of iterations any of the available approximations can thus be used as an initial solution each iteration requires only modest computations also a lower bound on the true optimal cost can be computed and used in a termination test empirical testing suggests very fast convergence

notes on l m convex functions and the separation theorems the concepts of l convex function and m convex function have recently been introduced by murota as generalizations of submodular function and base polyhedron respectively and discrete separation theorems are established for l convex concave functions and for m convex concave functions as generalizations of frank s discrete separation theorem for submodular supermodular set functions and edmonds matroid intersection theorem this paper shows the equivalence between murota s l convex functions and favati and tardella s submodular integrally convex functions and also gives alternative proofs of the separation theorems that provide a geometric insight by relating them to the ordinary separation theorem in convex analysis

approximately counting hamilton paths and cycles in dense graphs we describe fully polynomial randomized approximation schemes for the problems of determining the number of hamilton paths and cycles in an n vertex graph with minimum degree frac 1 2 a n for any fixed a 0 we show that the exact counting problems are p complete we also describe fully polynomial randomized approximation schemes for counting paths and cycles of all sizes in such graphs

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

on elements of order p in powerful p groups abstract we investigate the set 1 p of elements of order at most p in a powerful p group p and prove that 1 p p p p as a corollary we obtain a necessary and sufficient condition for 1 p to be of exponent p we give an example to show that for p 2 there is a powerful 2 group such that 1 p is not of exponent 2

the distribution based p value for the outlier sum in differential gene expression analysis outlier sums were proposed by tibshirani hastie 2007 and wu 2007 for detecting outlier genes where only a small subset of disease samples shows unusually high gene expression but they did not develop their distributional properties and formal statistical inference in this study a new outlier sum for detection of outlier genes is proposed its asymptotic distribution theory is developed and the p value based on this outlier sum is formulated its analytic form is derived on the basis of the large sample theory we compare the proposed method with existing outlier sum methods by power comparisons our method is applied to dna microarray data from samples of primary breast tumors examined by huang et al 2003 the results show that the proposed method is more efficient in detecting outlier genes

fixed points of coprime operator groups abstract let m be a positive integer and a an elementary abelian group of order q r with r 2 acting on a finite q group g we show that if for some integer d such that 2 d r 1 the dth derived group of c g a has exponent dividing m for any a a then g d has m q r bounded exponent and if r 1 c g a has exponent dividing m for any a a then r 1 g has m q r bounded exponent

an fptas for a single item capacitated economic lot sizing problem with monotone cost structure we present a fully polynomial time approximation scheme fptas for a capacitated economic lot sizing problem with a monotone cost structure an fptas delivers a solution with a given relative error in time polynomial in the problem size and in 1 such a scheme was developed by van hoesel and wagelmans 8 for a capacitated economic lot sizing problem with monotone concave convex production and backlogging cost functions we omit concavity and convexity restrictions furthermore we take advantage of a straightforward dynamic programming algorithm applied to a rounded problem

a simple fptas for a single item capacitated economic lot sizing problem with a monotone cost structure abstract the single item capacitated economic lot sizing cels problem is a fundamental problem of production and inventory management the first fully polynomial approximation scheme fptas for this problem with concave cost functions was developed by van hoesel and wagelmans c p m van hoesel a p m wagelmans fully polynomial approximation schemes for single item capacitated economic lot sizing problems mathematics of operations research 26 2001 339 357 chubanov et al s chubanov m y kovalyov e pesch an fptas for a single item capacitated economic lot sizing problem mathematical programming series a 106 2006 453 466 later presented a sophisticated fptas for the general case of the cels problem with a monotone cost structure in this paper we present a better fptas for this case the ideas and presentation of our fptas are simple and straightforward its running time is about n 4 e 2 times faster than that of chubanov et al 5 where n is the number of production periods and e is the anticipated relative error of the approximate solution

a classification of finite 2 groups with exactly three involutions we finish here a classification of finite 2 groups with exactly three involutions we clear up this remaining very difficult situation and determine completely the structure of g in terms of two generators and relations

a general framework for designing approximation schemes for combinatorial optimization problems with many objectives combined into one in this paper we present a general framework for designing approximation schemes for combinatorial optimization problems in which the objective function is a combination of more than one function examples of such problems include those in which the objective function is a product or ratio of two linear functions parallel machine scheduling problems with the makespan objective robust versions of weighted multiobjective optimization problems and assortment optimization problems with logit choice models the main idea behind our approximation schemes is the construction of an approximate pareto optimal frontier of the functions that constitute the given objective using this idea we give the first fully polynomial time approximation schemes for the max min resource allocation problem with a fixed number of agents combinatorial optimization problems in which the objective function is the sum of a fixed number of ratios of linear functions or the product of a fixed number of linear functions and assor

polynomial identities in graded group rings restricted lie algebras and p adic analytic groups let g be any finitely generated group and let k be a field of characteristic p 0 it is shown that the graded group ring gr kg satisfies a nontrivial polynomial identity if and only if the pro p completion of g is p adic analytic i e can be given the structure of a lie group over the p adic field q p the proof applies theorems of lazard quillen and passman as well as results on engel lie algebras and on dimension subgroups in positive characteristic

approximation algorithm for the stochastic multiperiod inventory problem via a look ahead optimization approach we consider the single item multiperiod stochastic inventory problem with nonstationary and correlated demands we provide the first proof that a well known myopic policy which we call look ahead optimization la is in fact an approximation algorithm for solving the large dynamic program that arises from this problem we prove that la provides the tightest known approximation bound for this problem the expected cost of la is at most twice the expected holding cost plus the expected backorder cost of an optimal policy we introduce a new cost accounting technique and a new class of invariances on the ongoing performance of la relative to that of an optimal policy we use these invariances to extend the myopic optimality of la to a global bound on its performance we allow convex and nonlinear holding and backorder cost functions integral order quantities and positive lead times under a technical condition on the growth of demand we show in computational experiments that la has excellent average cas

a deterministic fully polynomial time approximation scheme for counting integer knapsack solutions made easy given n elements with nonnegative integer weights w w 1 w n an integer capacity c and positive integer ranges u u 1 u n we consider the counting version of the classic integer knapsack problem find the number of distinct multisets whose weights add up to at most c we give a deterministic algorithm that estimates the number of solutions to within relative error in time polynomial in n log u and 1 where u max i u i more precisely our algorithm runs in o n 3 log 2 u log n log u time this is an improvement of n 2 and 1 up to log terms over the best known deterministic algorithm by gopalan et al 2011 5 our algorithm is relatively simple and its analysis is rather elementary our results are achieved by means of a careful formulation of the problem as a dynamic program using the notion of binding constraints

extremal splittings of point processes the sequence with nth term defined by n 1 p np is an extremal zero one valued sequence of asymptotic mean p in the following sense for example if a fraction p of customers from a point process with iid interarrival times is sent to an exponential server queue according to a prespecified splitting sequence then the long term average queue size is minimized when the above sequence is used the proof involves consideration of the lower convex envelope j which is a function on rm of a function j on zm an explicit representation is given for j in terms of j for j in a broad class of functions which we call multimodular the expected queue size just before an arrival considered as a function of the zero one splitting sequence is shown to belong to this class

approximating the permanent a randomised approximation scheme for the permanent of a 0 1s presented the task of estimating a permanent is reduced to that of almost uniformly generating perfect matchings in a graph the latter is accomplished by simulating a markov chain whose states are the matchings in the graph for a wide class of 0 1 matrices the approximation scheme is fully polynomial i e runs in time polynomial in the size of the matrix and a parameter that controls the accuracy of the output this class includes all dense matrices those that contain sufficiently many 1 s and almost all sparse matrices in some reasonable probabilistic model for 0 1 matrices of given density for the approach sketched above to be computationally efficient the markov chain must be rapidly mixing informally it must converge in a short time to its stationary distribution a major portion of the paper is devoted to demonstrating that the matchings chain is rapidly mixing apparently the first such result for a markov chain with genuinely c

a fully polynomial time approximation scheme for single item stochastic inventory control with discrete demand the single item stochastic inventory control problem is to find an inventory replenishment policy in the presence of independent discrete stochastic demands under periodic review and finite time horizon in this paper we prove that this problem is intractable and design for it a fully polynomial time approximation scheme

computational geometry an introduction from the reviews this book offers a coherent treatment at the graduate textbook level of the field that has come to be known in the last decade or so as computational geometry the book is well organized and lucidly written a timely contribution by two founders of the field it clearly demonstrates that computational geometry in the plane is now a fairly well understood branch of computer science and mathematics it also points the way to the solution of the more challenging problems in dimensions higher than two

provably near optimal balancing policies for multi echelon stochastic inventory control models we develop the first algorithmic approach to compute provably good ordering policies for a multi echelon stochastic inventory system facing correlated nonstationary and evolving demands over a finite horizon specifically we study the serial system our approach is computationally efficient and provides worst case guarantees that is the expected cost of the algorithms is guaranteed to be within a constant factor of the optimal expected cost depending on the assumption the constant varies between two and three our algorithmic approach is based on an innovative scheme to account for costs in a multi echelon multi period environment as well as repeatedly balancing between opposing cost the cost accounting scheme called a cause effect cost accounting scheme is significantly different from traditional cost accounting schemes in that it reallocates costs with the goal of assigning every unit of cost to the decision that caused the cost to be incurred we believe it will have additional applications

autocommutator subgroups of finite groups abstract we present some new results about the absolute center and autocommutator subgroup of a group concepts which were introduced in an earlier paper

optimal control of a high volume assemble to order system we consider an assemble to order system with a high volume of prospective customers arriving per unit time our objective is to maximize expected infinite horizon discounted profit by choosing product prices component production capacities and a dynamic policy for sequencing customer orders for assembly we prove that a myopic discrete review sequencing policy which allocates scarce components among orders for different products to minimize instantaneous physical and financial holding costs is asymptotically optimal furthermore we prove that optimal prices and production capacity nearly balance the supply and demand for components i e it is economically optimal to operate the system in heavy traffic so system performance is characterized by a diffusion approximation the diffusion approximation exhibits state space collapse its dimension equals the number of components rather than the number of components plus the number of products these results complement the existing assemble to order literature which focuses on managing component inventory and assumes fifo sequencing of orders for assembly

extension of a schur theorem to groups with a central factor with a bounded section rank abstract a well known result reported by schur states that the derived subgroup of a group is finite provided its central factor is finite here we show that if the p section rank of the central factor of a locally generalized radical group is bounded then so is the p section rank of its derived subgroup we also give an explicit expression for this bound

fixed dimensional stochastic dynamic programs an approximation scheme and an inventory application we study fixed dimensional stochastic dynamic programs in a discrete setting over a finite horizon under the primary assumption that the cost to go functions are discrete l convex we propose a pseudo polynomial time approximation scheme that solves this problem to within an arbitrary prespecified additive error of e 0 the proposed approximation algorithm is a generalization of the explicit enumeration algorithm and offers us full control in the trade off between accuracy and running time the main technique we develop for obtaining our scheme is approximation of a fixed dimensional l convex function on a bounded rectangular set using only a selected number of points in its domain furthermore we prove that the approximation function preserves l convexity finally to apply the approximate functions in a dynamic program we bound the error propagation of the approximation our approximation scheme is illustrated on a well known problem in inventory theory the single product problem with lost sales and lead times we demonstrate the practical value of our scheme by implementing our approximation scheme and the explicit enumeration algorithm on instances of this inventory problem

on the growth of subalgebras in lie p algebras abstract let l be a finitely generated lie p algebra over a finite field f then the number a n l of p subalgebras of finite codimension n in l is finite we say that l has psg polynomial p subalgebras growth if the growth of a n l is bounded above by some polynomial in f n we show that if l has psg then the lower central series of l stabilises after a finite number of steps on the other hand if l is nilpotent then l has psg we deduce the following group theoretic result let g be a group and let g p denote a pro p completion of g then the associated lie p algebra l p g of g has psg if and only if g p is a p adic analytic lie group

max product for maximum weight matching convergence correctness and lp duality max product belief propagation bp is an iterative message passing algorithm for finding the maximum a posteriori map assignment of a discrete probability distribution specified by a graphical model despite the spectacular success of the algorithm in many application areas such as iterative decoding and combinatorial optimization which involve graphs with many cycles theoretical results about both the correctness and convergence of the algorithm are known in only a few cases see section i for references in this paper we prove the correctness and convergence of max product for finding the maximum weight matching mwm in bipartite graphs even though the underlying graph of the mwm problem has many cycles somewhat surprisingly we show that the max product algorithm converges to the correct mwm as long as the mwm is unique we provide a bound on the number of iterations required and show that for a graph of size n the computational cost of the algorithm scales as o n3 which is the same as the computational cost of the best known algorithms for finding the mwm we also provide an interesting relation between the dynamics of the max product algorithm and the auction algorithm which is a well known distributed algorithm for solving the mwm problem

some baer invariants of free nilpotent groups abstract we present an explicit structure for the baer invariant of a free nth nilpotent group the nth nilpotent product of infinite cyclic groups z n z n n z with respect to the variety v with the set of words v c 1 1 c 2 1 for all c 1 c 2 and 2 c 2 c 1 2 n 2 also an explicit formula for the polynilpotent multiplier of a free nth nilpotent group is given for any class row c 1 c 2 c t where c 1 n

asymptotic optimality of constant order policies for lost sales inventory models with large lead times lost sales inventory models with large lead times which arise in many practical settings are notoriously difficult to optimize due to the curse of dimensionality in this paper we show that when lead times are large a very simple constant order policy first studied by reiman performs nearly optimally the main insight of our work is that when the lead time is very large such a significant amount of randomness is injected into the system between when an order for more inventory is placed and when the order is received that being smart algorithmically provides almost no benefit our main proof technique combines a novel coupling for suprema of random walks with arguments from queueing theory

profit estimation error in the newsvendor model under a parametric demand distribution we consider the newsvendor model in which uncertain demand is assumed to follow a probabilistic distribution with known functional form but unknown parameters these parameters are estimated unbia

an adaptive algorithm for finding the optimal base stock policy in lost sales inventory systems with censored demand we consider a periodic review single location single product inventory system with lost sales and positive replenishment lead times it is well known that the optimal policy does not possess a simple structure motivated by recent results showing that base stock policies perform well in these systems we study the problem of finding the best base stock policy in such a system in contrast to the classical inventory literature we assume that the manager does not know the demand distribution a priori but must make the replenishment decision in each period based only on the past sales censored demand data we develop a nonparametric adaptive algorithm that generates a sequence of order up to levels whose running average of the inventory holding and lost sales penalty cost converges to the cost of the optimal base stock policy and we establish the cubic root convergence rate of the algorithm our analysis is based on recent advances in stochastic online convex optimization and on the uniform ergodicity of markov chains associated with bases stock policies

groups satisfying semigroup laws and nilpotent by burnside varieties abstract we investigate the structure of groups satisfying apositive law that is an identity of the formu v whereuandvare positive words the main question here is whether all such groups are nilpotent by finite exponent we answer this question affirmatively for a large class c of groups including soluble and residually finite groups showing that moreover the nilpotency class and the finite exponent in question are bounded solely in terms of the length of the positive law it follows in particular that if a variety of groups is locally nilpotent by finite then it must in fact be contained in the product of a nilpotent variety by a locally finite variety of finite exponent we deduce various other corollaries for instance that a torsion free residually finite n engel group is nilpotent of class bounded in terms ofn we also consider incidentally a question of bergman as to whether a positive law holding in a generating subsemigroup of a group must in fact be a law in the whole group showing that it has an affirmative answer for soluble groups

adaptive trimmed t statistics for identifying predominantly high expression in a microarray experiment often interesting candidate tumor markers are not only genes that show homogeneously higher expression hhe in tumor samples compared to control samples but also genes with only predominantly higher expression phe i e genes which exhibit higher expression in at least 80 per cent of tumor samples standard parametric test statistics used in the analysis of microarray experiments may fail with phe as a consequence of the mixture of distributions present in the tumor group as alternative we consider trimmed t statistics which compare group mean values after removing outliers in each group the trimming proportion can be chosen adaptively either based on a boxplot outlier detection rule or by optimization over a series of tests with varying trimming proportions the trimmed t statistics can be plugged into the significance analysis of microarrays sam procedure yielding the modified boxplot rule test modbox and the modified optimization test modopt respectively by means of simulation of microarray experiments we show that modopt is superior to contenders in detecting phe while there is only little loss in efficiency under hhe compared to sam analysis of a real microarray experiment revealed that out of nearly 29 000 genes about 417 genes exhibiting phe are detected by modopt but missed by sam

closed form solutions for robust inventory management we propose and analyze robust optimization models of an inventory management problem where cumulative purchase inventory and shortage costs over n periods are minimized for correlated nonidentically distributed demand we assume that the means and covariance matrix of stochastic demand are known the distributions are not needed we derive closed form ordering quantities for both symmetric and asymmetric uncertainty sets under capacitated inventory constraints in both static and dynamic settings the behaviors of our robust strategies differ qualitatively depending on the symmetry of the uncertainty set for instance considering our simplest static problem 1 if the uncertainty set is symmetric then there is positive ordering in all periods whereas 2 if the set is asymmetric then there is a set of periods in the middle of the planning horizon with zero orders we also link the symmetry of the uncertainty set to the symmetry of the demand distribution finally we present encouraging computati

near optimal algorithms for the assortment planning problem under dynamic substitution and stochastic demand assortment planning of substitutable products is a major operational issue that arises in many industries such as retailing airlines and consumer electronics we consider a single period joint assortment and inventory planning problem under dynamic substitution with stochastic demands and provide complexity and algorithmic results as well as insightful structural characterizations of near optimal solutions for important variants of the problem first we show that the assortment planning problem is np hard even for a very simple consumer choice model where each consumer is willing to buy only two products in fact we show that the problem is hard to approximate within a factor better than 1 1 e second we show that for several interesting and practical customer choice models one can devise a polynomial time approximation scheme ptas i e the problem can be solved efficiently to within any level of accuracy to the best of our knowledge this is the first efficient algorithm with provably near

an approximation scheme for stochastic linear programming and its application to stochastic integer programs stochastic optimization problems attempt to model uncertainty in the data by assuming that the input is specified by a probability distribution we consider the well studied paradigm of 2 stage models with recourse first given only distributional information about some of the data one commits on initial actions and then once the actual data is realized according to the distribution further recourse actions can be taken we show that for a broad class of 2 stage linear models with recourse one can for any e 0 in time polynomial in 1 e and the size of the input compute a solution of value within a factor 1pe of the optimum in spite of the fact that exponentially many second stage scenarios may occur in conjunction with a suitable rounding scheme this yields the first approximation algorithms for 2 stage stochastic integer optimization problems where the underlying random data is given by a black box and no restrictions are placed on the costs in the two stages our rounding approach for stochastic integer programs shows that an approximation algorithm for a deterministic analogue yields with a small constant factor loss provably near optimal solutions for the stochastic generalization among the range of applications we consider are stochastic versions of the multicommodity flow set cover vertex cover and facility location problems

controlling the false discovery rate a practical and powerful approach to multiple testing summary the common approach to the multiplicity problem calls for controlling the familywise error rate fwer this approach though has faults and we point out a few a different approach to problems of multiple significance testing is presented it calls for controlling the expected proportion of falsely rejected hypotheses the false discovery rate this error rate is equivalent to the fwer when all hypotheses are true but is smaller otherwise therefore in problems where the control of the false discovery rate rather than that of the fwer is desired there is potential for a gain in power a simple sequential bonferronitype procedure is proved to control the false discovery rate for independent test statistics and a simulation study shows that the gain in power is substantial the use of the new procedure and the appropriateness of the criterion are illustrated with examples

a 2 approximation algorithm for stochastic inventory control models with lost sales in this paper we describe the first computationally efficient policies for stochastic inventory models with lost sales and replenishment lead times that admit worst case performance guarantees in particular we introduce dual balancing policies for lost sales models that are conceptually similar to dual balancing policies recently introduced for a broad class of inventory models in which demand is backlogged rather than lost that is in each period we balance two opposing costs the expected marginal holding costs against the expected marginal lost sales cost specifically we show that the dual balancing policies for the lost sales models provide a worst case performance guarantee of two under relatively general demand structures in particular the guarantee holds for independent not necessarily identically distributed demands and for models with correlated demands such as the ar 1 model and the multiplicative autoregressive demand model the policies and the worst case guarantee extend to models with capacity constraints on the size of the order and stochastic lead times our analysis has several novel elements beyond the balancing ideas for backorder models

correlation decay in random decision networks we consider a decision network on an undirected graph in which each node corresponds to a decision variable and each node and edge of the graph is associated with a reward function whose value depends only on the variables of the corresponding nodes the goal is to construct a decision vector that maximizes the total reward this decision problem encompasses a variety of models including maximum likelihood inference in graphical models markov random fields combinatorial optimization on graphs economic team theory and statistical physics the network is endowed with a probabilistic structure in which rewards are sampled from a distribution our aim is to identify sufficient conditions on the network structure and rewards distributions to guarantee average case polynomiality of the underlying optimization problem additionally we wish to characterize the efficiency of a decentralized solution generated on the basis of local information we construct a new decentralized algorithm called cavity expans

minimizing delays in the gi g 1 queue what service time distribution minimizes delays in a gi g 1 queue with given renewal arrival process and given mean service time it is natural to conjecture that the deterministic service time distribution with unit mass on the mean is optimal for all objective functions of the form efw where f is a nondecreasing function of the steady state delay w however we show that this conjecture is false in fact for hyperexponential interarrival time distributions mixtures of two exponential distributions the service time distribution minimizing the average delay maximizes the proportion of customers delayed

an analysis of belief propagation on the turbo decoding graph with gaussian densities motivated by its success in decoding turbo codes we provide an analysis of the belief propagation algorithm on the turbo decoding graph with gaussian densities in this context we are able to show that under certain conditions the algorithm converges and that somewhat surprisingly though the density generated by belief propagation may differ significantly from the desired posterior density the means of these two densities coincide since computation of posterior distributions is tractable when densities are gaussian use of belief propagation in such a setting may appear unwarranted indeed our primary motivation for studying belief propagation in this context stems from a desire to enhance our understanding of the algorithm s dynamics in a non gaussian setting and to gain insights into its excellent performance in turbo codes nevertheless even when the densities are gaussian belief propagation may sometimes provide a more efficient alternative to traditional inference methods