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

a note on node aggregation and benders decomposition node aggregation methods have been previously studied as a means for approximating large scale transportation problems in this paper we show how the approximations inherent in benders decomposition method can be combined with node aggregation in optimizing large scale capacitated plant location problems

reflection sieve obstructions in finite groups of type dn abstract in 4 we constructed certain homology representations of a finite group g of type an bn or cn and showed that these representations can be used to sift out the reflection compound characters of g in the present note we show that for a group g of type dn each reflection compound character k 2 k n 2 determines a unique obstruction character k which occurs with positive multiplicity in every homology representation containing k

fuzzy versus statistical linear regression abstract statistical linear regression and fuzzy linear regression have been developed from different perspectives and thus there exist several conceptual and methodological differences between the two approaches the characteristics of both methods in terms of basic assumptions parameter estimation and application are described and contrasted their descriptive and predictive capabilities are also compared via a simulation experiment to identify the conditions under which one outperforms the other it turns out that statistical linear regression is superior to fuzzy linear regression in terms of predictive capability whereas their comparative descriptive performance depends on various factors associated with the data set size quality and proper specificity of the model aptness of the model heteroscedasticity autocorrelation nonrandomness of error terms specifically fuzzy linear regression performance becomes relatively better vis a vis statistical linear regression as the size of the data set diminishes and the aptness of the regression model deteriorates fuzzy linear regression may thus be used as a viable alternative to statistical linear regression in estimating regression parameters when the data set is insufficient to support statistical regression analysis and or the aptness of the regression model is poor e g due to vague relationship among variables and poor model specification

primal dual aggregation and disaggregation for stochastic linear programs a multistage stochastic linear program can be approximated by replacing the underlying information structure by a coarser structure given the ordinary lagrangian for the original problem this amounts to restricting both the primal and dual variables to be adapted to the coarser information structure the resulting primal problem has the same form as the original but with aggregated constraints and decisions mutual upper and lower bounds on the optimal values of the original and aggregated problems are obtained via partially aggregated problems formed by restricting only the primal variables or only the dual variables to the coarser structure the introduction of the partially aggregated problems leads to a general disaggregation framework for developing heuristic approaches to solving the original problem

a local convergence proof for the iterative aggregation method the iterative aggregation method for the solution of a system of linear algebraic equations x ax b where a 0 b 0 s 0 and s a s is proved to be locally convergent it is shown that the method can be considered a consistent nonstationary iterative method where the iteration matrix depends on the current iterate and that some norm of the iteration matrix is less than one in the vicinity of the solution

consistency of the disaggregation process in hierarchical planning this paper deals with the planning process of a manufacturing system in which the aggregation of products sharing similar characteristics leads to a hierarchical structure it gives special attention to the consistency of decisions in a two level structure and presents necessary and sufficient conditions for consistency of a disaggregation procedure using a subset of these conditions improves upon the disaggregation procedure proposed by bitran and hax bitran g r a c hax 1981 disaggregation and resource allocation using convex knapsack problems with bounded variables mgmt sci 27 431 441

weights improvement in column aggregation abstract in the present paper we study how the weights used in column aggregation of linear programs may be changed in order to improve the objective value a procedure similar to postoptimal change of the coefficients of a basic variable is developed

iterative aggregation disaggregation procedures for discounted semi markov reward processes the equation v q mv where m is a matrix with nonnegative elements and spectral radius less than one arises in markovian decision processes and input output models in this paper we solve the equation using an iterative aggregation disaggregation procedure that alternates between solving an aggregated problem and disaggregating the variables one block at a time in terms of the aggregate variables of the other blocks the disaggregated variables are then used to guide the choice of weights in the subsequent aggregation computational experiments on randomly generated and inventory problems indicate that this algorithm is significantly faster than successive approximations when the spectral radius of m is near one and is slower in unstructured problems with spectral radii in the neighborhood of 0 8 the algorithm appears promising for large structured problems where it can often reduce computational time and main memory storage requirements and offer greater robustness to initial values

multi phase dynamic constraint aggregation for set partitioning type problems dynamic constraint aggregation is an iterative method that was recently introduced to speed up the linear relaxation solution process of set partitioning type problems this speed up is mostly due to the use at each iteration of an aggregated problem defined by aggregating disjoint subsets of constraints from the set partitioning model this aggregation is updated when needed to ensure the exactness of the overall approach in this paper we propose a new version of this method called the multi phase dynamic constraint aggregation method which essentially adds to the original method a partial pricing strategy that involves multiple phases this strategy helps keeping the size of the aggregated problem as small as possible yielding a faster average computation time per iteration and fewer iterations we also establish theoretical results that provide some insights explaining the success of the proposed method tests on the linear relaxation of simultaneous bus and driver scheduling problems involving up to 2 000 set partitioning constraints show that the partial pricing strategy speeds up the original method by an average factor of 4 5

error bounds for the aggregated convex programming problem applied mathematical programming problems are often approximations of larger more detailed problems one criterion to evaluate an approximating program is the magnitude of the difference between the optimal objective values of the original and the approximating program the approximation we consider is variable aggregation in a convex program bounds are derived on the difference between the two optimal objective values previous results of geoffrion and zipkin are obtained by specializing our results to linear programming also we apply our bounds to a convex transportation problem

application of general semi infinite programming to lapidary cutting problems we consider a volume maximization problem arising in gemstone cutting industry the problem is formulated as a general semi infinite program gsip and solved using an interior point method developed by stein o stein bi level strategies in semi infinite programming kluwer academic publishers boston 2003 it is shown that the convexity assumption needed for the convergence of the algorithm can be satisfied by appropriate modelling clustering techniques are used to reduce the number of container constraints which is necessary to make the subproblems practically tractable an iterative process consisting of gsip optimization and adaptive refinement steps is then employed to obtain an optimal solution which is also feasible for the original problem some numerical results based on real world data are also presented

hierarchical knowledge gradient for sequential sampling we propose a sequential sampling policy for noisy discrete global optimization and ranking and selection in which we aim to efficiently explore a finite set of alternatives before selecting an alternative as best when exploration stops each alternative may be characterized by a multi dimensional vector of categorical and numerical attributes and has independent normal rewards we use a bayesian probability model for the unknown reward of each alternative and follow a fully sequential sampling policy called the knowledge gradient policy this policy myopically optimizes the expected increment in the value of sampling information in each time period we propose a hierarchical aggregation technique that uses the common features shared by alternatives to learn about many alternatives from even a single measurement this approach greatly reduces the measurement effort required but it requires some prior knowledge on the smoothness of the function in the form of an aggregation function and computational issues limit the number of alternatives that can be easily considered to the thousands we prove that our policy is consistent finding a globally optimal alternative when given enough measurements and show through simulations that it performs competitively with or significantly better than other policies

stabilized dynamic constraint aggregation for solving set partitioning problems abstract dynamic constraint aggregation dca and dual variable stabilization dvs are two methods that can reduce the negative impact of degeneracy when solving linear programs the first uses a projection to reduce the primal space whereas the second acts in the dual space in this paper we develop a new method called stabilized dynamic constraint aggregation sdca that combines dca and dvs for solving set partitioning problems it allows to fight degeneracy from both primal and dual perspectives simultaneously to assess the effectiveness of sdca we report computational results obtained for highly degenerate multi depot vehicle scheduling problem instances solved by column generation these results indicate that sdca can reduce the average computational time of the master problem by a factor of up to 7 with respect to the best of the two combined methods furthermore they show that its performance is robust with regard to increasing levels of degeneracy in test problems

an adaptive partition based approach for solving two stage stochastic programs with fixed recourse we study an adaptive partition based approach for solving two stage stochastic programs with fixed recourse a partition based formulation is a relaxation of the original stochastic program and we study a finitely converging algorithm in which the partition is adaptively adjusted until it yields an optimal solution a solution guided refinement strategy is developed to refine the partition by exploiting the relaxation solution obtained from a partition in addition to refinement we show that in the case of stochastic linear programs it is possible to merge some components in a partition without weakening the corresponding relaxation bound thus allowing the partition size to be kept small we also show that for stochastic linear programs with simple recourse there exists a small partition that yields an optimal solution the size of this partition is independent of the number of scenarios used in the model our computational results show that the proposed adaptive partition based approach converges v

state space size considerations for disease progression models markov models of disease progression are widely used to model transitions in patients health state over time usually patients health status may be classified according to a set of ordered health states modelers lump together similar health states into a finite and usually small number of health states that form the basis of a markov chain disease progression model this increases the number of observations used to estimate each parameter in the transition probability matrix however lumping together observably distinct health states also obscures distinctions among them and may reduce the predictive power of the model moreover as we demonstrate precision in estimating the model parameters does not necessarily improve as the number of states in the model declines this paper explores the tradeoff between lumping error introduced by grouping distinct health states and sampling error that arises when there are insufficient patient data to precisely estimate the transition probability matrix copyright 2013 john wiley sons ltd

constraint aggregation principle in convex optimization a general constraint aggregation technique is proposed for convex optimization problems at each iteration a set of convex inequalities and linear equations is replaced by a single surrogate inequality formed as a linear combination of the original constraints after solving the simplified subproblem new aggregation coefficients are calculated and the iteration continues this general aggregation principle is incorporated into a number of specific algorithms convergence of the new methods is proved and speed of convergence analyzed next dual interpretation of the method is provided and application to decomposable problems is discussed finally a numerical illustration is given

surrogate dual multiplier search procedures in integer programming search procedures for optimal lagrange multipliers are highly developed and provide good bounds in branch and bound procedures that have led to the successful application of lagrangean duality in integer programming although the surrogate dual generally provides a better objective bound there has been little development of surrogate multiplier search procedures this paper develops and empirically analyzes several surrogate multiplier search procedures results indicate that the procedures can produce possibly superior bounds in an amount of time comparable to other techniques our discussion also highlights the similarity of the procedures to some well known lagrangean search techniques

on decomposition of modular representations from cohen macaulay geometries abstract chain spaces for a cohen macaulay complex split into subspaces corresponding to cycles for truncations of the complex precisely when the cycles are non degenerate in the natural bilinear form application to the tits building provides a simple geometric approach to the well known splitting of ind g b 1 for a lie type group g in natural characteristic an application to a sporadic geometry is also indicated

minimizing effective resistance of a graph the effective resistance between two nodes of a weighted graph is the electrical resistance seen between the nodes of a resistor network with branch conductances given by the edge weights the effective resistance comes up in many applications and fields in addition to electrical network analysis including for example markov chains and continuous time averaging networks in this paper we study the problem of allocating edge weights on a given graph in order to minimize the total effective resistance i e the sum of the resistances between all pairs of nodes we show that this is a convex optimization problem and can be solved efficiently either numerically or in some cases analytically we show that optimal allocation of the edge weights can reduce the total effective resistance of the graph compared to uniform weights by a factor that grows unboundedly with the size of the graph we show that among all graphs with n nodes the path has the largest value of optimal total effective resistance and the complete graph has the least

duality for representations of a reductive group over a finite field the purpose of this paper is to construct a duality operation for representations of a reductuve group over a finite field its effect very roughly speaking is to interchange irreducible representations of small degree with ones of large degree for example the unit and steinberg representation at level of characters this operation has been also considered by alvis 11 curtis 2 and kawanaka 4 we shall now fix some notation g will denote a connected reductive group defined over a finite field fq iv s its weyl group and f the set of orbits of the frobenius map on s the subsets of parametrize the classes of parabolic subgroup of g which are defined over fq let 9 be the class corresponding to z c s thus 9 is the class of bore1 subgroups and s g we denote by g the group of f rational points of g and by 9 the set of parabolic subgroups in 9 with are defined over fq similarly if p e 3 we denote by p its group of fq rational points and by u the group of f rational points of its unipotent radical let k be an algebraically closed field of characteristic zero all g modules will be over k let e be a g module for each z c we define

accelerating column generation for variable sized bin packing problems abstract in this paper we study different strategies to stabilize and accelerate the column generation method when it is applied specifically to the variable sized bin packing problem or to its cutting stock counterpart the multiple length cutting stock problem many of the algorithms for these problems discussed in the literature rely on column generation processes that are known to converge slowly due to primal degeneracy and the excessive oscillations of the dual variables in the sequel we introduce new dual optimal inequalities and explore the principle of model aggregation as an alternative way of controlling the progress of the dual variables two algorithms based on aggregation are proposed the first one relies on a row aggregated lp while the second one solves iteratively sequences of doubly aggregated models working with these approximations in the various stages of an iterative solution process has proven to be an effective way of achieving faster convergence the computational experiments were conducted on a broad range of instances many of them published in the literature they show a significant reduction of the number of column generation iterations and computing time

the p median problem for cluster analysis a comparative test using the mixture model approach recently mulvey and crowder mulvey j h crowder 1979 cluster analysis an application of lagrangian relaxation management sci 25 329 340 suggested that the p median problem might be useful for cluster analysis problems where the goal is to group objects described by a vector of characteristics in such a way that objects in the same group are somehow more alike than objects in different groups the intent of this paper is to test mulvey and crowder s proposal using the mixture model approach i e by applying a number of algorithms including one for the p median problem to a set of objects randomly sampled from a number of known multivariate populations and comparing the ability of each algorithm to detect the original populations in order to evaluate the results a generalized partition comparison measure and its distribution are developed using this measure results from various algorithms are compared

robust reduction of a class of large scale linear programs a set of new tests for linear programming presolving analysis is described these tests are applicable for linear programs with box constraints and positive or zero coefficients partial applicability to general linear programming lp problems is discussed in a special section the aim is to detect and remove redundant rows and columns the tests are based on the solution of some auxiliary lp problems with one constraint and upper bounds on the variables a comparison with the klein holm numerical test is presented the tests are applied iteratively to the primal and dual lp problems the method is also applicable to lp problems with coefficients belonging to some range of uncertainty providing a robust procedure for scale reduction a detailed numerical example and results of numerical experiments are presented

value function approximation using multiple aggregation for multiattribute resource management we consider the problem of estimating the value of a multiattribute resource where the attributes are categorical or discrete in nature and the number of potential attribute vectors is very large the problem arises in approximate dynamic programming when we need to estimate the value of a multiattribute resource from estimates based on monte carlo simulation these problems have been traditionally solved using aggregation but choosing the right level of aggregation requires resolving the classic tradeoff between aggregation error and sampling error we propose a method that estimates the value of a resource at different levels of aggregation simultaneously and then uses a weighted combination of the estimates using the optimal weights which minimizes the variance of the estimate while accounting for correlations between the estimates is computationally too expensive for practical applications we have found that a simple inverse variance formula adjusted for bias which effectively assumes the estimates are independent produces near optimal estimates we use the setting of two levels of aggregation to explain why this approximation works so well

an iterative aggregation disaggregation algorithm for solving linear equations a general procedure is given for solving large sets of linear equations by first rewriting them in a form suitable for aggregation of both the variables and equations followed by disaggregation a computational algorithm which iteratively aggregates and disaggregates is shown to converge geometrically to the exact solution provided the original problem has a structure suitable for such aggregation the algorithm exhibits fast computation times small main memory requirements and robustness to the starting point a rigorous foundation for aggregation and disaggregation is provided by the equations employed by this algorithm

acceleration by aggregation of successive approximation methods abstract methods of successive approximation for solving linear systems or minimization problems are accelerated by aggregation disaggregation processes these processes which modify the iterates being produced are characterized by a two directional flow of information between the original higher dimensional problem and a lower dimensional aggregated version this technique is characterized by means of galerkin approximations and this in turn permits analysis of the method a deterministic as well as probabilistic analysis is given of a number of specific aggregation disaggregation examples numerical experiments have been performed and these confirm the analysis and demonstrate the acceleration

an iterative aggregation procedure for markov decision processes an iterative aggregation procedure is described for solving large scale finite state finite action markov decision processes mdps at each iteration an aggregate master problem and a sequence of smaller subproblems are solved the weights used to form the aggregate master problem are based on the estimates from the previous iteration each subproblem is a finite state finite action mdp with a reduced state space and unequal row sums global convergence of the algorithm is proven under very weak assumptions the proof relates this technique to other iterative methods that have been suggested for general linear programs

on the imbalance of inventories in multi echelon systems a central depot periodically orders or produces new stock and withdraws inventory in order to allocate it among several demand points each experiencing random demands over a finite planning horizon this system gives rise to a dynamic program with a state space of very large dimension we determine a near optimal allocation policy and construct a tractable approximate dynamic program to help determine good orders and withdrawals this is done by exploring the effects of a measure of the imbalance of inventories at the demand points numerical results are included

an improved disaggregation method for transportation problems a method is given to disaggregate the solution to an aggregated transportation problem the resulting solution to the original problem is feasible all integer and has lower cost that those of solutions produced by earlier methods

a method to calculate steady state distributions of large markov chains by aggregating states this paper develops an efficient iterative algorithm to calculate the steady state distribution of nearly all irreducible discrete time markov chains computational experiences suggest that for large markovian systems more than 130 states the proposed algorithm can be ten times faster than standard gaussian elimination in finding solutions to an accuracy of 0 1 the proposed algorithm is developed in three stages first we develop a very efficient algorithm for determining steady state distributions of a restricted class of markovian systems a second result establishes a relationship between a general irreducible markovian system and a system in the restricted class of markovian systems finally we combine the two results to produce an efficient iterative algorithm to solve markov systems the paper concludes with a discussion of the observed performance of the algorithm

a unipotent support for irreducible representations 1 1 let g f be the group of fq rational points of a connected reductive algebraic group g defined over a finite field f throughout this paper we shall assume that the characteristic of the ground field is large the following is part of problem ii in l9 let p be an irreducible complex character of g f show that there is a unique umpotent class c in g which has the property that cga c cfqi p g 0 and has maximal dimension among classes with this property one of the results of this paper is a solution of this problem we also prove the following refinement if g e gf is such that tr g p 0 then the unipotent part of g lies in c or in a conjugacy class of dimension dim c we call c the unipotent support of p another map from the set of irreducible representations of gf to unipotent classes in g has been defined in l8 13 43 this was the composition of two maps one from irreducible representations of gf to a set of parameters in the langlands dual of g and one from this set of parameters to unipotent classes in g another conjectural definition of this map which did not pass through the langlands dual has been proposed by kawanaka kl k2 k3 in terms of generalized gelfand graev representations in analogy with real lie groups he uses the name wave front set for the unipotent class attached to a representation one of the results of this paper is a proof see section 11 of kawanaka s conjecture that his wave front set is well defined and that it coincides with the one defined in l8 this has been known earlier from the work of kawanaka in certain special cases we also show that the unipotent support of an irreducible representation coincides with the wave front set of the dual representation

lower bounds for the economic lot scheduling problem using aggregation abstract the problem considered in this paper is to choose cyclical production patterns for several products which are produced on a common production facility by aggregating data dynamic programming can be used to obtain lower bounds which in general are much stronger than those obtained when the capacity constraint is not recognized lower bounds are of interest when generating possible solutions and also in connection with branch and bound methods

randomized distributed edge coloring via an extension of the chernoff hoeffding bounds certain types of routing scheduling and resource allocation problems in a distributed setting can be modeled as edge coloring problems we present fast and simple randomized algorithms for edge coloring a graph in the synchronous distributed point to point model of computation our algorithms compute an edge coloring of a graph g with n nodes and maximum degree delta with at most 1 6 delta o log 1 delta n colors with high probability arbitrarily close to 1 for any fixed delta 0 they run in polylogarithmic time the upper bound on the number of colors improves upon the 2 delta 1 coloring achievable by a simple reduction to vertex coloring to analyze the performance of our algorithms we introduce new techniques for proving upper bounds on the tail probabilities of certain random variables the chernoff hoeffding bounds are fundamental tools that are used very frequently in estimating tail probabilities however they assume stochastic independence among certain random variables which may not always hold our results extend the chernoff hoeffding bounds to certain types of random variables which are not stochastically independent we believe that these results are of independent interest and merit further study

zero duality gap in surrogate constraint optimization a concise review of models abstract surrogate constraint relaxation was proposed in the 1960s as an alternative to the lagrangian relaxation for solving difficult optimization problems the duality gap in the surrogate relaxation is always as good as the duality gap in the lagrangian relaxation over the years researchers have proposed procedures to reduce the gap in the surrogate constraint our aim is to review models that close the surrogate duality gap five research streams that provide procedures with zero duality gap are identified and discussed in each research stream we will review major results discuss limitations and suggest possible future research opportunities in addition relationships between models if they exist are also discussed

iterative aggregation and disaggregation algorithm for pseudo polynomial network flow models with side constraints this paper develops a general solution framework based on aggregation techniques to solve np hard problems that can be formulated as a circulation model with specific side constraints the size of the extended mixed integer linear programming formulation is generally pseudo polynomial to efficiently solve exactly these large scale models we propose a new iterative aggregation and disaggregation algorithm at each iteration it projects the original model onto an aggregated one producing an approximate model the process iterates to refine the current aggregated model until the opti mality is proved the computational experiments on two hard optimization problems a variant of the vehicle routing problem and the cutting stock problem show that a generic implementation of the proposed framework allows us to out perform previous known methods

computable error bounds for aggregated markov chains abstract this paper describes a method for computing the steady state probability vector of a nearly completely decomposable markov chain the method is closely related to one proposed by simon and ando and developed by courtois however the method described here does not require the determination of a completely decomposable stochastic approximation to the transition matrix and hence it is applicable to matrices other than stochastic an error analysis of the procedure is given which results in effectively computable error bounds author

an approximation algorithm for max min fair allocation of indivisible goods in this paper we give the first approximation algorithm for the problem of max min fair allocation of indivisible goods an instance of this problem consists of a set of k people and m indivisible goods each person has a known linear utility function over the set of goods which might be different from the utility functions of other people the goal is to distribute the goods among the people and maximize the minimum utility received by them the approximation ratio of our algorithm is omega frac 1 sqrt k log 3 k as a crucial part of our algorithm we design and analyze an iterative method for rounding a fractional matching on a tree which might be of independent interest we also provide better bounds when we are allowed to exclude a small fraction of the people from the problem

an o log n log log n approximation algorithm for the asymmetric traveling salesman problem we present a randomized o log n log log n approximation algorithm for the asymmetric traveling salesman problem atsp this provides the first asymptotic improvement over the long standing log n approximation bound stemming from the work of frieze et al 1982 frieze am galbiati g maffioki f 1982 on the worst case performance of some algorithms for the asymmetric traveling salesman problem networks 12 1 23 39 the key ingredient of our approach is a new connection between the approximability of the atsp and the notion of so called thin trees to exploit this connection we employ maximum entropy rounding a novel method of randomized rounding of lp relaxations of optimization problems we believe that this method might be of independent interest

aggregation and disaggregation techniques and methodology in optimization a fundamental issue in the use of optimization models is the tradeoff between the level of detail and the ease of using and solving the model aggregation and disaggregation techniques have proven to be valuable tools for manipulating data and determining the appropriate policies to employ for this tradeoff furthermore aggregation and disaggregation techniques offer promise for solving large scale optimization models supply a set of promising methodologies for studying the underlying structure of both univariate and multivariate data sets and provide a set of tools for manipulating data for different levels of decision makers in this paper we develop a general framework for aggregation and disaggregation methodology survey previous work regarding aggregation and disaggregation techniques for optimization problems illuminate the appropriate role of aggregation and disaggregation methodology for optimization applications and propose future research directions

some combinatorial and algebraic properties of coxeter complexes and tits buildings a des groupes de coxeter et des groupes a bn paire sont associes certaines structures combinatoires les complexes de coxeter et les constructions de tits on etudie ces structures du point de vue des complexes de cohen macaulay et des complexes decorticables

computational proof of the mackey formula for q 2 let g be a connected reductive group defined over a finite field with q elements we prove that the mackey formula for the lusztig induction and restriction holds in g whenever q 2 or g does not have a component of type e

on the integrality ratio for the asymmetric traveling salesman problem we improve the lower bound on the integrality ratio of the held karp bound for asymmetric tsp with triangle inequality from 4 3 to 2

on state aggregation to approximate complex value functions in large scale markov decision processes many small electronic devices such as cell phones and wireless sensors have restrictive memory space computing power and battery the pervasive applications of these devices in industry military and our daily lives require simple policies that are easy to implement and can be executed in a reasonably short time the markov decision process mdp provides a general framework for these policy optimization problems with complexity constraint in many cases we can use a powerful computer to find the optimal or a good policy and the value function first and then approximate by a simple one the approximation usually depends on heuristics or experiences because the relationship between the complexity of a function and the approximation error is not clear in general in this paper we assume the optimal value function is known or a reasonably good estimate is available and consider how to approximate a complex value function due to the broad application of state aggregation in the large scale mdp we focus on piecewise constant approximate value functions and use the number of aggregated states to measure the complexity of a value function we quantify how the complexity of a value function affects the approximation error when the optimal value function is known for sure we develop an algorithm that finds the best simple state aggregation within polynomial time when we have estimates of the optimal value function we apply ordinal optimization to find good simple state aggregations with high probability the algorithms are demonstrated on a node activation policy optimization problem in wireless sensor network we hope this work can shed some insight on how to find simple policies with good performances

disaggregation and resource allocation using convex knapsack problems with bounded variables the allocation of a specific amount of a given resource among competitive alternatives can often be modelled as a knapsack problem this model formulation is extremely efficient because it allows convex cost representation with bounded variables to be solved without great computational efforts practical applications of this problem abound in the fields of operations management finance manpower planning marketing etc in particular knapsack problems emerge in hierarchical planning systems when a first level of decisions need to be further allocated among specific activities which have been previously treated in an aggregate way in this paper we provide a recursive procedure to solve such problems the method differs from classical optimization algorithms of convex programming in that it determines at each iteration the optimal value of at least one variable applications and computational results are presented

approximation algorithms for asymmetric tsp by decomposing directed regular multigraphs a directed multigraph is said to be d regular if the indegree and outdegree of every vertex is exactly d by hall s theorem one can represent such a multigraph as a combination of at most n2 cycle covers each taken with an appropriate multiplicity we prove that if the d regular multigraph does not contain more than d 2 copies of any 2 cycle then we can find a similar decomposition into n2 pairs of cycle covers where each 2 cycle occurs in at most one component of each pair our proof is constructive and gives a polynomial algorithm to find such a decomposition since our applications only need one such a pair of cycle covers whose weight is at least the average weight of all pairs we also give an alternative simpler algorithm to extract a single such pair this combinatorial theorem then comes handy in rounding a fractional solution of an lp relaxation of the maximum traveling salesman problem tsp problem the first stage of the rounding procedure obtains two cycle covers that do not share a 2 cycle with weight at least twice the weight of the optimal solution then we show how to extract a tour from the 2 cycle covers whose weight is at least 2 3 of the weight of the longest tour this improves upon the previous 5 8 approximation with a simpler algorithm utilizing a reduction from maximum tsp to the shortest superstring problem we obtain a 2 5 approximation algorithm for the latter problem which is again much simpler than the previous one for minimum asymmetric tsp the same technique gives two cycle covers not sharing a 2 cycle with weight at most twice the weight of the optimum assuming triangle inequality we then show how to obtain from this pair of cycle covers a tour whose weight is at most 0 842 log 2 n larger than optimal this improves upon a previous approximation algorithm with approximation guarantee of 0 999 log 2 n other applications of the rounding procedure are approximation algorithms for maximum 3 cycle cover factor 2 3 previously 3 5 and maximum asymmetric tsp with triangle inequality factor 10 13 previously 3 4

technical note computational viability of a constraint aggregation scheme for integer linear programming problems we present the results of a computational evaluation of two constraint aggregation approaches for solving integer linear programs in some applications particularly set partitioning problems the schemes can aggregate significantly different number of constraints we first discuss the implementation of these approaches using both single and multiple precision arithmetic from these results we empirically show that in practical implementation and evaluation of an aggregation scheme the degree of difficulty encountered in solving the resulting equality constrained knapsack problem is crucial we conclude as contrasted with the optimism expressed in some published works that the aggregation approach has limited value for solving general integer linear programs but may be useful in developing a heuristic algorithm for the set partitioning problem

adaptive aggregation methods for infinite horizon dynamic programming a class of iterative aggregation algorithms for solving infinite horizon dynamic programming problems is proposed the idea is to interject aggregation iterations in the course of the usual successive approximation method an important feature that sets this method apart from earlier ones is that the aggregate groups of states change adaptively from one aggregation iteration to the next depending on the progress of the computation this allows acceleration of convergence in difficult problems involving multiple ergodic classes for which methods using fixed groups of aggregate states are ineffective no knowledge of special problem structure is utilized by the algorithms