identifying codes and covering problems the identifying code problem for a given graph involves finding a minimum set of vertices whose neighborhoods uniquely overlap at any given graph vertex initially introduced in 1998 this problem has demonstrated its fundamental nature through a wide variety of applications such as fault diagnosis location detection and environmental monitoring in addition to deep connections to information theory superimposed and covering codes and tilings this work establishes efficient reductions between the identifying code problem and the well known set covering problem resulting in a tight hardness of approximation result and novel provably tight polynomial time approximations the main results are also extended to r robust identifying codes and analogous set 2r 1 multicover problems finally empirical support is provided for the effectiveness of the proposed approximations including good constructions for well known topologies such as infinite two dimensional grids

capacity constraints across nests in assortment optimization under the nested logit model we consider assortment optimization problems when customers choose according to the nested logit model and there is a capacity constraint limiting the total capacity consumption of all products offered in all nests when each product consumes one unit of capacity our capacity constraint limits the cardinality of the offered assortment for the cardinality constrained case we develop an efficient algorithm to compute the optimal assortment when the capacity consumption of each product is arbitrary we give an algorithm to obtain a 4 approximate solution we show that we can compute an upper bound on the optimal expected revenue for an individual problem instance by solving a linear program in our numerical experiments we consider problem instances involving products with arbitrary capacity consumptions comparing the expected revenues from the assortments obtained by our 4 approximation algorithm with the upper bounds on the optimal expected revenues our numerical results indicate that the 4 approxim

aggregation based cutting planes for packing and covering integer programs in this paper we study the strength of chvatal gomory cg cuts and more generally aggregation cuts for packing and covering integer programs ips aggregation cuts are obtained as follows given an ip formulation we first generate a single implied inequality using aggregation of the original constraints then obtain the integer hull of the set defined by this single inequality with variable bounds and finally use the inequalities describing the integer hull as cutting planes our first main result is to show that for packing and covering ips the cg and aggregation closures can be 2 approximated by simply generating the respective closures for each of the original formulation constraints without using any aggregations on the other hand we use computational experiments to show that aggregation cuts can be arbitrarily stronger than cuts from individual constraints for general ips the proof of the above stated results for the case of covering ips with bounds require the development of some new structural results which may be of independent interest finally we examine the strength of cuts based on k different aggregation inequalities simultaneously the so called multi row cuts and show that every packing or covering ip with a large integrality gap also has a large k aggregation closure rank in particular this rank is always at least of the order of the logarithm of the integrality gap

a polynomial time approximation scheme for the multiple knapsack problem the multiple knapsack problem mkp is a natural and well known generalization of the single knapsack problem and is defined as follows we are given a set of n items and m bins knapsacks such that each item i has a profit p i and a size s i and each bin j has a capacity c j the goal is to find a subset of items of maximum profit such that they have a feasible packing in the bins mkp is a special case of the generalized assignment problem gap where the profit and the size of an item can vary based on the specific bin that it is assigned to gap is apx hard and a 2 approximation for it is implicit in the work of shmoys and tardos math program a 62 1993 pp 461 474 and thus far this was also the best known approximation for mkp the main result of this paper is a polynomial time approximation scheme ptas for mkp apart from its inherent theoretical interest as a common generalization of the well studied knapsack and bin packing problems it appears to be the strongest special case of gap that is not apx hard we substantiate this by showing that slight generalizations of mkp are apx hard thus our results help demarcate the boundary at which instances of gap become apx hard an interesting aspect of our approach is a ptas preserving reduction from an arbitrary instance of mkp to an instance with o log n distinct sizes and profits

gmm with many moment conditions this paper provides a first order asymptotic theory for generalized method of moments gmm estimators when the number of moment conditions is allowed to increase with the sample size and the moment conditions may be weak examples in which these asymptotics are relevant include instrumental variable iv estimation with many possibly weak or uninformed instruments and some panel data models that cover moderate time spans and have correspondingly large numbers of instruments under certain regularity conditions the gmm estimators are shown to converge in probability but not necessarily to the true parameter and conditions for consistent gmm estimation are given a general framework for the gmm limit distribution theory is developed based on epiconvergence methods some illustrations are provided including consistent gmm estimation of a panel model with time varying individual effects consistent limited information maximum likelihood estimation as a continuously updated gmm estimator and consistent iv structural estimation using large numbers of weak or irrelevant instruments some simulations are reported

scheduling orders for multiple product types with due date related objectives abstract we consider m machines in parallel with each machine capable of producing one specific product type there are n orders with each one requesting specific quantities of the various different product types order j has a release date rj and a due date dj the different product types for order j can be produced at the same time we consider various due date related objectives such as the minimization of the maximum lateness lmax and the total number of late orders u j we present polynomial time algorithms for the easy cases and heuristics for np hard cases for minimizing u j we also propose an exact algorithm based on constraint propagation and bounding strategy the effectiveness of the algorithms is demonstrated through an empirical study

the variational iteration method for nonlinear oscillators with discontinuities in this paper he s variational iteration method vim is applied to nonlinear oscillators with discontinuities we illustrate that the vim is very effective and convenient and does not require linearization or small perturbation contrary to the conventional methods in vim only one iteration leads to high accuracy of the solutions moreover we show that the obtained approximate solutions are valid for the whole solution domain and the approximations are uniformly valid not only for small parameters but also for very large parameters

bootstrap inference for instrumental variable models with many weak instruments this study s main contribution is to theoretically analyze the application of bootstrap methods to instrumental variable models when the available instruments may be weak and the number of instruments goes to infinity with the sample size we demonstrate that a standard residual based bootstrap procedure cannot consistently estimate the distribution of the limited information maximum likelihood estimator or fuller 1977 estimator under many many weak instrument sequence the primary reason is that the standard procedure fails to capture the instrument strength in the sample adequately in addition we consider the restricted efficient re bootstrap of davidson and mackinnon 2008 2010 2014 that generates bootstrap data under the null restricted and uses an efficient estimator of the coefficient of the reduced form equation efficient we find that the re bootstrap is also invalid however it effectively mimics more key features in the limiting distributions of interest and thus is less distorted in finite samples than the standard bootstrap procedure finally we propose modified bootstrap procedures that provide a valid distributional approximation for the two estimators with many many weak instruments a monte carlo experiment shows that hypothesis testing based on the asymptotic normal approximation can have severe size distortions in finite samples instead our modified bootstrap procedures greatly reduce these distortions

heuristics for the fixed cost median problem we describe in this paper polynomial heuristics for three important hard problems the discrete fixed cost median problem the plant location problem the continuous fixed cost median problem in a euclidean space and the network fixed cost median problem with convex costs the heuristics for all the three problems guarantee error ratios no worse than the logarithm of the number of customer points the derivation of the heuristics is based on the presentation of all types of median problems discussed as a set covering problem

application of he s amplitude frequency formulation to a nonlinear oscillator with discontinuity in this paper he s amplitude frequency formulation is used to obtain a periodic solution of a nonlinear oscillator with discontinuity the accuracy arrives at 0 73 even when the amplitude tends to infinity

the max min approach to a relativistic equation this paper applies the max min approach proposed by ji huan he to the relativistic oscillator comparison with the exact solution shows that the method is very effective and convenient for solving nonlinear equations

probabilistic properties of the dual structure of the multidimensional knapsack problem and fast statistically efficient algorithms properties of several dual characteristics of the multidimensional knapsack problem such as the probability of existence of optimal and optimal feasible lagrange function generalized saddle points magnitude of relative duality gap etc are investigated for different probabilistic models sufficient conditions of good asymptotic behavior of the dual characteristics are given a fast statistically efficient approximate algorithm with linear running time complexity for problems with random coefficients is presented

approximation algorithms for scheduling unrelated parallel machines we consider the following scheduling problem there arem parallel machines andn independent jobs each job is to be assigned to one of the machines the processing of jobj on machinei requires timep ij the objective is to find a schedule that minimizes the makespan our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum we also present a polynomial approximation scheme for the case that the number of machines is fixed both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations in particular we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints in contrast to our main result we prove that no polynomial algorithm can achieve a worst case ratio less than 3 2 unlessp np we finally obtain a complexity classification for all special cases with a fixed number of processing times

minimum distance approach to inference with many instruments abstract i analyze a linear instrumental variables model with a single endogenous regressor and many instruments i use invariance arguments to construct a new minimum distance objective function with respect to a particular weight matrix the minimum distance estimator is equivalent to the random effects estimator of chamberlain and imbens 2004 and the estimator of the coefficient on the endogenous regressor coincides with the limited information maximum likelihood estimator this weight matrix is inefficient unless the errors are normal and i construct a new more efficient estimator based on the optimal weight matrix finally i show that when the minimum distance objective function does not impose a proportionality restriction on the reduced form coefficients the resulting estimator corresponds to a version of the bias corrected two stage least squares estimator i use the objective function to construct confidence intervals that remain valid when the proportionality restriction is violated

lower bounds and algorithms for the minimum cardinality bin covering problem abstract this paper introduces the minimum cardinality bin covering problem where we are given m identical bins with capacity c and n indivisible items with integer weights wj j 1 n the objective is to minimize the number of items packed into the m bins so that the total weight of each bin is at least equal to c we discuss reduction criteria derive several lower bound arguments and propose construction heuristics as well as a powerful subset sum based improvement algorithm that is even optimal when m 2 moreover we present a tailored branch and bound method which is able to solve instances with up to 20 bins and several hundreds of items within a reasonable amount of time in a comprehensive computational study on a wide range of randomly generated instances our algorithmic approach proved to be much more effective than a commercial solver

a threshold of ln n for approximating set cover given a collection f of subsets of s 1 n setcover is the problem of selecting as few as possiblesubsets from f such that their union covers s and maxk cover is the problem of selecting k subsets from f such that their union has maximum cardinality both these problems arenp hard we prove that 1 o 1 ln n is a threshold below which setcover cannot be approximated efficiently unless np has slightlysuperpolynomial time algorithms this closes the gap up to low orderterms between the ratio of approximation achievable by the greedyalogorithm which is 1 o 1 lnn and provious results of lund and yanakakis that showed hardness ofapproximation within a ratio of log 2 n 2s0 72 ln n for max k cover we show an approximationthreshold of 1 1 e up tolow order terms under assumption that p np

origins of the limited information maximum likelihood and two stage least squares estimators theil basmann and sargan are often credited with the development of the two stage least squares tsls estimator of the coefficients of one structural equation in a simultaneous equations model however anderson and rubin had earlier derived the asymptotic distribution of the limited information maximum likelihood liml estimator by finding the asymptotic distribution of what is essentially the tsls estimator

mixed integer programming with convex concave constraints fixed parameter tractability and applications to multicovering and voting abstract a classic result of lenstra math oper res 1983 says that an integer linear program can be solved in fixed parameter tractable fpt time for the parameterization by the number of variables we extend this result by incorporating piecewise linear convex or concave functions to our mixed integer programs this general technique allows us to analyze the parameterized complexity of a number of classic np hard computational problems in particular we prove that weighted set multicover is in fpt when parameterized by the number of elements to cover and that there exists an fpt time approximation scheme for multiset multicover for the same parameter this is our most technical result further we use our general technique to prove that a number of problems from computational social choice e g problems related to bribery and control in elections are in fpt when parameterized by the number of candidates for bribery this resolves a nearly 10 year old family of open problems and for weighted electoral control of approval voting this improves some previously known xp memberships to fpt memberships

alternative approximations to the distributions of instrumental variable estimators the paper considers the ols the iv and two method of moments estimators mm and mmk of the coefficients of a single equation where the explanatory variables are correlated with the disturbance term the mm and mmk estimators are generalizations of the liml and limlk estimators respectively multivariate first order approximations to the distributions are derived under normality using a parameter sequence where the number of instruments increases as the number of observations increases numerical results show these approximations are more accurate compared to large sample approximations even if the number of instruments is small the moments of the multivariate limit distributions of the mm and mmk estimators can be consistently estimated under a variety of parameter sequences including the large sample sequence the new approximate confidence regions perform well in terms of exact levels compared to traditional ones the iv estimator of the coefficient of a single explanatory endogenous variable is interpreted as a shrinkage estimator which is dominated in practical cases by the mm and mmk estimators in terms of nearness to the true value in the sense of pitman

truthful generalized assignments via stable matching in the generalized assignment problem gap a set of jobs seek to be assigned to a set of machines for every job machine pair there are a specific value and an accommodated capacity for the assignment the objective is to find an assignment that maximizes the total sum of values given that the capacity constraint of every machine is satisfied the gap is a classic optimization problem and has been studied extensively from the algorithmic perspective dughmi and ghosh dughmi s ghosh a 2010 truthful assignment without money acm conf electronic commerce acm new york 325 334 proposed the game theoretical framework in which every job is owned by a selfish agent who aims to maximize the value of his own assignment they gave a logarithmic approximation truthful in expectation mechanism and left open the problem whether there exists a truthful mechanism with a constant approximation factor in this paper we give an affirmative answer to this question and provide a constant approximation mechanism

approximation schemes for deal splitting and covering integer programs with multiplicity constraints we consider the problem of splitting an order for r goods r 1 among a set of sellers each having bounded amounts of the goods so as to minimize the total cost of the deal in deal splitting with packages dsp the sellers offer packages containing combinations of the goods in deal splitting with price tables dst the buyer can generate such combinations using price tables our problems which often occur in online reverse auctions generalize covering integer programs with multiplicity constraints cip where we must fill up an r dimensional bin by selecting with a bounded number of repetitions from a set of r dimensional items such that the overall cost is minimized thus both dsp and dst are np hard already for a single good and hard to approximate for arbitrary number of goods in this paper we focus on finding efficient approximations for dsp and dst instances where the number of goods is some fixed constant in particular we develop polynomial time approximation schemes ptas for several subclasses of instances of practical interest our results include a ptas for cip in fixed dimension and a more efficient combinatorial scheme for cip where the multiplicity constraints are omitted our approximation scheme for cip is based on a non trivial application of the fast scheme for the fractional covering problem proposed by fleischer l fleischer a fast approximation scheme for fractional covering problems with variable upper bounds in proc of the 15th acm siam symposium on discrete algorithm 2004 pp 994 1003

coupled and k sided placements generalizing generalized assignment in modern data centers and cloud computing systems jobs often require resources distributed across nodes providing a wide variety of services motivated by this we study the coupled placement problem in which we place jobs into computation and storage nodes with capacity constraints so as to optimize some costs or profits associated with the placement the coupled placement problem is a natural generalization of the widely studied generalized assignment problem which concerns the placement of jobs into single nodes providing one kind of service we also study a further generalization the k sided placement problem in which we place jobs into k tuples of nodes each node in a tuple offering one of k services for both the coupled and k sided placement problems we consider minimization and maximization versions in the minimization versions mincpand min k ksp the goal is to achieve minimum placement cost while incurring a minimum blowup in the capacity of the individual nodes our first main result is an algorithm for min k ksp that achieves optimal cost while increasing capacities by at most a factor of k 1 k 1 also yielding the first constant factor approximation for mincp in the maximization versions maxcp and max k ksp the goal is to maximize the total weight of the jobs that are placed under hard capacity constraints max k ksp can be expressed as a k column sparse integer program and can be approximated to within a factor of o k factor using randomized rounding of a linear program relaxation we consider alternative combinatorial algorithms that are much more efficient in practice our second main result is a local search based combinatorial algorithm that yields a 15 approximation and o k 2 o k2 approximation for maxcp and max k ksp respectively finally we consider an online version of max k ksp and present algorithms that achieve logarithmic competitive ratio under certain necessary technical assumptions

application of he chengtian s interpolation to bethe equation in this work he chengtian s interpolation which has millennial history is applied to the bethe equation the result shows that the present technology is very convenient and effective

thresholded covering algorithms for robust and max min optimization in a two stage robust covering problem one of several possible scenarios will appear tomorrow and require to be covered but costs are higher tomorrow than today what should you anticipatorily buy today so that the worst case cost summed over both days is minimized we consider the k k robust model where the possible scenarios tomorrow are given by all demand subsets of size k k in this paper we give the following simple and intuitive template for k k robust covering problems having built some anticipatory solution if there exists a single demand whose augmentation cost is larger than some threshold augment the anticipatory solution to cover this demand as well and repeat we show that this template gives good approximation algorithms for k k robust versions of many standard covering problems set cover steiner tree steiner forest minimum cut and multicut our k k robust approximation ratios nearly match the best bounds known for their deterministic counterparts the main technical contribution lies in proving certain net type properties for these covering problems which are based on dual rounding and primal dual ideas these properties might be of some independent interest as a by product of our techniques we also get algorithms for max min problems of the form given a covering problem instance which k k of the elements are costliest to cover for the problems mentioned above we show that their k k max min versions have performance guarantees similar to those for the k k robust problems

consistent estimation of linear panel data models with measurement error abstract measurement error causes a bias towards zero when estimating a panel data linear regression model the panel data context offers various opportunities to derive instrumental variables allowing for consistent estimation we consider three sources of moment conditions i restrictions on the covariance matrix of the errors in the equations ii nonzero third moments of the regressors and iii heteroskedasticity and nonlinearity in the relation between the error ridden regressor and another error free regressor in simulations these approaches appear to work well

periodic solutions for some strongly nonlinear oscillations by he s variational iteration method in this paper we implement a new analytical technique he s variational iteration method for some strongly nonlinear oscillations a correction functional is constructed by using a general lagrange multiplier which can be identified via the variational theory the obtained approximate solutions and periods are compared with exact or numerical results to verify the effectiveness and accuracy of the method

on the hardness of approximating minimization problems we prove results indicating that it is hard to compute efficiently good approximate solutions to the graph coloring set covering and other related minimization problems specifically there is an e 0 such that graph coloring cannot be approximated with ratio n e unless p np set covering cannot be approximated with ratio c log n for any c 1 4 unless np is contained in dtime n poly log n similar results follow for related problems such as clique cover fractional chromatic number dominating set and others

testing overidentifying restrictions with many instruments and heteroskedasticity this paper gives a test of overidentifying restrictions that is robust to many instruments and heteroskedasticity it is based on a jackknife version of the overidentifying test statistic correct asymptotic critical values are derived for this statistic when the number of instruments grows large at a rate up to the sample size it is also shown that the test is valid when the number of instruments is fixed and there is homoskedasticity this test improves on recently proposed tests by allowing for heteroskedasticity and by avoiding assumptions on the instrument projection matrix this paper finds in monte carlo studies that the test is more accurate and less sensitive to the number of instruments than the hausman sargan or gmm tests of overidentifying restrictions

generalized method of moments with many weak moment conditions using many moment conditions can improve efficiency but makes the usual generalized method of moments gmm inferences inaccurate two step gmm is biased generalized empirical likelihood gel has smaller bias but the usual standard errors are too small in instrumental variable settings in this paper we give a new variance estimator for gel that addresses this problem it is consistent under the usual asymptotics and under many weak moment asymptotics is larger than usual and is consistent we also show that the kleibergen 2005 lagrange multiplier and conditional likelihood ratio statistics are valid under many weak moments in addition we introduce a jackknife gmm estimator but find that gel is asymptotically more efficient under many weak moments in monte carlo examples we find that t statistics based on the new variance estimator have nearly correct size in a wide range of cases

parameterized approximation scheme for the multiple knapsack problem the multiple knapsack problem mkp is a well known generalization of the classical knapsack problem we are given a set a of n items and set b of m bins knapsacks such that each item a in a has a size size a and a profit value profit a and each bin b in b has a capacity c b the goal is to find a subset u subset a of maximum total profit such that u can be packed into b without exceeding the capacities the decision version of mkp is strongly np complete since it is a generalization of the classical knapsack and bin packing problem furthermore mkp does not admit a fully time polynomial time approximation scheme fptas even if the number m of bins is two kellerer gave a polynomial time approximation scheme ptas for mkp with identical capacities and chekuri and khanna presented a ptas for mkp with general capacities with running time n o log 1 epsilon epsilon 8 in this paper we propose an efficient ptas eptas with parameterized running time 2 o log 1 epsilon epsilon 5 cdot poly n o m for mkp this also solves an open question by chekuri and khanna

approximation of set multi cover via hypergraph matching abstract for a given b n n and a hypergraph h v e with maximum degree the b set multicover problem which we are concerned with in this paper may be stated as follows find a minimum cardinality subset c e such that no vertex v i v is contained in less than b i hyperedges of c peleg schechtman and wool 1997 conjectured that for any fixed and b min i b i the problem cannot be approximated with a ratio strictly smaller than b 1 unless p np in this paper we show that the conjecture of peleg et al is not true on l uniform hypergraphs by presenting a polynomial time approximation algorithm based on a matching covering duality for hypergraphs due to ray chaudhuri 1960 which we convert into an approximative form the given algorithm yields a ratio smaller than 1 b l 1 b moreover we prove that the lower bound conjectured by peleg et al holds for regular hypergraphs under the unique games conjecture

rebuild of king fang 40 bc musical scales by he s inequality he chengtian s inequality is used to re build the king fang s musical scales 41 24 and 53 31 in 40 bc the technology which might be little known to the west has been used for millennia by ancient chinese and can be powerfully applied to nonlinear problems

fast approximation algorithms for fractional packing and covering problems this paper presents fast algorithms that find approximate solutions for a general class of problems which we call fractional packing and covering problems the only previously known algorithms for solving these problems are based on general linear programming techniques the techniques developed in this paper greatly outperform the general methods in many applications and are extensions of a method previously applied to find approximate solutions to multicommodity flow problems our algorithm is a lagrangian relaxation technique an important aspect of our results is that we obtain a theoretical analysis of the running time of a lagrangian relaxation based algorithm we give several applications of our algorithms the new approach yields several orders of magnitude of improvement over the best previously known running times for algorithms for the scheduling of unrelated parallel machines in both the preemptive and the nonpreemptive models for the job shop problem for the held and karp bound for the tr

a fast algorithm for maximizing a non monotone dr submodular integer lattice function abstract in this paper we consider the problem of maximizing a non monotone and non negative dr submodular function on a bounded integer lattice b x 1 x n z n 0 x k b k 1 k n without any constraint where b b 1 b n z n we design an algorithm for the problem and measure its performance by its approximation ratio and the number of value oracle queries it needs where the latter one is the dominating term in the running time of an algorithm it has been showed that for the problem considered any algorithm achieving an approximation ratio greater than 1 2 requires an exponential number of value oracle queries in the literature there are two algorithms that reach 1 2 approximation guarantee the first algorithm needs o n b oracle queries the second one reduces its number of oracle queries to o n max 1 log b but it needs large storage in this paper we present a randomized approximation algorithm that has an approximation guarantee of 1 2 calls o n max 1 log b oracle queries and does not need large storage improving the results of the literature

on the approximability of dodgson and young elections the voting rules proposed by dodgson and young are both designed to find an alternative closest to being a condorcet winner according to two different notions of proximity the score of a given alternative is known to be hard to compute under either rule in this paper we put forward two algorithms for approximating the dodgson score a combinatorial greedy algorithm and an lp based algorithm both of which yield an approximation ratio of h m 1 where m is the number of alternatives and h m 1 is the m 1 st harmonic number we also prove that our algorithms are optimal within a factor of 2 unless problems in np have quasi polynomial time algorithms despite the intuitive appeal of the greedy algorithm we argue that the lp based algorithm has an advantage from a social choice point of view further we demonstrate that computing any reasonable approximation of the ranking produced by dodgson s rule is np hard this result provides a complexity theoretic explanation of sharp discrepancies that have been observed in the social choice theory literature when comparing dodgson elections with simpler voting rules finally we show that the problem of calculating the young score is np hard to approximate by any factor this leads to an inapproximability result for the young ranking

approximate solutions for nonlinear oscillation of a mass attached to a stretched elastic wire in this paper the approximate solutions of the mathematical model of a mass attached to a stretched elastic wire are presented at the beginning of the study the equation of motion is derived in a detailed way he s max min approach he s frequency amplitude method and the parameter expansion method are implemented to solve the established model the numerical results are further compared with the approximate analytical solutions for both a small and large amplitude of oscillations and a very good agreement is observed the relative errors are computed to illustrate the strength of agreement between the numerical and approximate analytical results

exact algorithms for procurement problems under a total quantity discount structure in this paper we study the procurement problem faced by a buyer who needs to purchase a variety of goods from suppliers applying a so called total quantity discount policy this policy implies that every supplier announces a number of volume intervals and that the volume interval in which the total amount ordered lies determines the discount moreover the discounted prices apply to all goods bought from the supplier not only to those goods exceeding the volume threshold we refer to this cost minimization problem as the total quantity discount tqd problem we give a mathematical formulation for this problem and argue that not only it is np hard but also that there exists no polynomial time approximation algorithm with a constant ratio unless p np apart from the basic form of the tqd problem we describe four variants in a first variant the market share that one or more suppliers can obtain is constrained another variant allows the buyer to procure more goods than strictly needed in order to reach a lower total cost we also consider a setting where the buyer needs to pay a disposal cost for the extra goods bought in a third variant the number of winning suppliers is limited both in general and per product finally we investigate a multi period variant where the buyer not only needs to decide what goods to buy from what supplier but also when to do this while considering the inventory costs we show that the tqd problem and its variants can be solved by solving a series of min cost flow problems finally we investigate the performance of three exact algorithms min cost flow based branch and bound linear programming based branch and bound and branch and cut on randomly generated instances involving 50 suppliers and 100 goods it turns out that even the large instances of the basic problem are solved to optimality within a limited amount of time however we find that different algorithms perform best in terms of computation time for different variants

approximating sparse covering integer programs online a covering integer program cip is a mathematical program of the form min c x ax 1 0 x u x n where all entries in a c u are nonnegative in the online setting the constraints i e the rows of the constraint matrix a arrive over time and the algorithm can only increase the coordinates of x to maintain feasibility as an intermediate step we consider solving the covering linear program clp online where the integrality constraints are dropped our main results are a an o log k competitive online algorithm for solving the clp and b an o log k log l competitive randomized online algorithm for solving the cip here k n and l m respectively denote the maximum number of nonzero entries in any row and column of the constraint matrix a our algorithm is based on the online primal dual paradigm where a novel ingredient is to allow dual variables to increase and decrease throughout the course of the algorithm it is known that this result is the best possible for polynomial t

approximation schemes for knapsack problems with shelf divisions given a knapsack of size k non negative values d and and a set s of items each item e s with size se and value ve we define a shelf as a subset of items packed inside a bin with total items size at most two subsequent shelves must be separated by a shelf divisor of size d the size of a shelf is the total size of its items plus the size of the shelf divisor the shelf knapsack problem sk is to find a subset s s partitioned into shelves with total shelves size at most k and maximum value the class constrained shelf knapsack ccsk is a generalization of the problem sk where each item in s has a class and each shelf in the solution must have only items of the same class we present approximation schemes for the sk and the ccsk problems to our knowledge these are the first approximation results where shelves of non null size are used in knapsack problems

worst case analysis of greedy heuristics for integer programming with nonnegative data we give a worst case analysis for two greedy heuristics for the integer programming problem minimize cx ax b 0 x u x integer where the entries in a b and c are all nonnegative the first heuristic is for the case where the entries in a and b are integral the second only assumes the rows are scaled so that the smallest nonzero entry is at least 1 in both cases we compare the ratio of the value of the greedy solution to that of the integer optimal the error bound grows logarithmically in the maximum column sum of a for both heuristics

the homotopy perturbation method applied for solving complex valued differential equations with strong cubic nonlinearity abstract in this paper the homotopy perturbation method is adopted for solving a complex valued second order strongly nonlinear differential equation homotopy with an imbedding parameter p 0 1 is constructed the perturbation procedure with parameter p transforms the strongly nonlinear differential equation into a system of linear complex valued differential equations whose solutions give the approximate solution of the initial differential equation to illustrate the effectiveness and convenience of the suggested procedure a duffing equation with strong cubic nonlinearity is considered the periodic solution in the first approximation is obtained the solution is compared with the exact one and shows good agreement

greedy facility location algorithms analyzed using dual fitting with factor revealing lp in this article we will formalize the method of dual fitting and the idea of factor revealing lp this combination is used to design and analyze two greedy algorithms for the metric uncapacitated facility location problem their approximation factors are 1 861 and 1 61 with running times of o m log m and o n3 respectively where n is the total number of vertices and m is the number of edges in the underlying complete bipartite graph between cities and facilities the algorithms are used to improve recent results for several variants of the problem

on positive influence dominating sets in social networks in this paper we investigate the positive influence dominating set pids which has applications in social networks we prove that pids is apx hard and propose a greedy algorithm with an approximation ratio of h d where h is the harmonic function and d is the maximum vertex degree of the graph representing a social network

the generalized assignment problem with minimum quantities abstract we consider a variant of the generalized assignment problem gap where the amount of space used in each bin is restricted to be either zero if the bin is not opened or above a given lower bound a minimum quantity we provide several complexity results for different versions of the problem and give polynomial time exact algorithms and approximation algorithms for restricted cases for the most general version of the problem we show that it does not admit a polynomial time approximation algorithm unless p np even for the case of a single bin this motivates to study dual approximation algorithms that compute solutions violating the bin capacities and minimum quantities by a constant factor when the number of bins is fixed and the minimum quantity of each bin is at least a factor 1 larger than the largest size of an item in the bin we show how to obtain a polynomial time dual approximation algorithm that computes a solution violating the minimum quantities and bin capacities by at most a factor 1 1 and 1 1 respectively and whose profit is at least as large as the profit of the best solution that satisfies the minimum quantities and bin capacities strictly in particular for 2 we obtain a polynomial time 1 2 approximation algorithm

jackknife instrumental variable estimation with heteroskedasticity we present a new jackknife estimator for instrumental variable inference with unknown heteroskedasticity it weighs observations such that many instruments consistency is guaranteed while the signal component in the data is maintained we show that this results in a smaller signal component in the many instruments asymptotic variance when compared to estimators that neglect a part of the signal to achieve consistency both many strong instruments and many weak instruments asymptotic distributions are derived using high level assumptions that allow for instruments with identifying power that varies between explanatory variables standard errors are formulated compactly we review briefly known estimators and show in particular that our symmetric jackknife estimator performs well when compared to the hlim and hful estimators of hausman et al in monte carlo experiments

the demand matching problem we examine formulations for the well known b matching problem in the presence of integer demands on the edges a subset m of edges is feasible if for each node v the total demand of edges in m incident to v is at most bv we examine the system of star inequalities for this problem this system yields an exact linear description for b matchings in bipartite graphs for the demand version we show that the integrality gap for this system is at least 2 5 and at most 2 764 for general graphs the gap lies between 3 and 3 264 we also describe a 3 approximation algorithm 2 5 for bipartite graphs for the cardinality version of the problem a fully polynomial approximation scheme is also presented for the problem on a tree thus generalising a well known result for the knapsack problem recently the notion of demand matching has arisen in the design of clos network switches

a study of nonlinear oscillators with u1 3 force by he s variational iteration method this paper applies he s variational iteration method to determine the periodic solutions of oscillators in a u1 3 force with the procedure the excellent approximate frequencies and the corresponding periodic solutions can easily be obtained

a fast asymptotic approximation scheme for bin packing with rejection bin packing with rejection is the following problem given a list of items with associated sizes and rejection costs find a packing into unit bins of a subset of the list such that the number of bins used plus the sum of rejection costs of unpacked items is minimized we show that bin packing with rejection can be reduced to n multiple knapsack problems and based on techniques for the multiple knapsack problem we give a fast asymptotic polynomial time approximation scheme reject pack with time complexity o n o e 2 this improves a recent approximation scheme given by epstein which has time complexity o n o e 4 e 1 we also show that reject pack can be extended to variable sized bin packing with rejection and give an asymptotic polynomial time approximation scheme

upper and lower bounding procedures for the multiple knapsack assignment problem abstract we formulate the multiple knapsack assignment problem mkap as an extension of the multiple knapsack problem mkp as well as of the assignment problem except for small instances mkap is hard to solve to optimality we present a heuristic algorithm to solve this problem approximately but very quickly we first discuss three approaches to evaluate its upper bound and prove that these methods compute an identical upper bound in this process reference capacities are derived which enables us to decompose the problem into mutually independent mkps these mkps are solved euristically and in total give an approximate solution to mkap through numerical experiments we evaluate the performance of our algorithm although the algorithm is weak for small instances we find it prospective for large instances indeed for instances with more than a few thousand items we usually obtain solutions with relative errors less than 0 1 within one cpu second

approximation algorithms for the m dimensional 0 1 knapsack problem worst case and probabilistic analyses abstract we describe a polynomial approximation scheme for an m constraint 0 1 integer programming problem m fixed based on the use of the dual simplex algorithm for linear programming we also analyse the asymptotic properties of a particular random model

solving multiple knapsack problems by cutting planes in this paper we consider the multiple knapsack problem which is defined as follows given a set n of items with weights f i i in n a set m of knapsacks with capacities f k k in m and a profit function c ik i in n k in m find an assignment of a subset of the set of items to the set of knapsacks that yields minimum cost we consider the multiple knapsack problem from a polyhedral point of view the inequalities that we describe here serve as the theoretical basis for a cutting plane algorithm we present some of the implementation details of this algorithm including a discussion and evaluation of different separation and primal heuristics our algorithm is applied to practical problem instances arising in the design of main frame computers in the layout of electronic circuits and in sugar cane alcohol production

the encompassing principle and its application to testing nonnested hypotheses the encompassing principle is used to develop a testing framework which unifies the literature on non nested testing allowing analysis of the relationship between alternative tests and in particular enabling asymptotic and finite sample equivalences and identities to be established easily when they exist as well as embracing nested hypothesis testing the concept of the implicit null hypothesis of a test is introduced to show that the effective acceptance region for some tests extends beyond the acceptance region corresponding to the null of interest and so such tests can be inconsistent against fixed alternatives closely related to the nominal null and alternative the analysis is illustrated by an application to two non nested linear regression models and it is shown that the conventional f test as well as all the one degree of freedom non nested tests has an encompassing interpretation and that the f test is a complete parametric encompassing test

an efficient approximation scheme for variable sized bin packing in the classical bin packing problem one is required to pack a given list of items into the smallest possible number of unit sized bins because this problem is np complete researchers have tried to find efficient approximation algorithms that solve this problem in a reasonable amount of time if we let a i be the number of bins used by algorithm a to pack a list of items i and textit opt i be the minimum number of bins necessary to pack i one defines the asymptotic performance ratio of a to be a i textit opt i as textit opt i tends to infinity de la vega and lueker presented an approximation scheme which for any in 0 yields an approximation algorithm with performance ratio 1 varepsilon this scheme has time complexity polynomial in n the number of items but exponential in 1 varepsilon karmarkar and karp extended this scheme to one that has time complexity polynomial in n and 1 varepsilon in the variable sized bin packing problem one

he s frequency amplitude formulation for the duffing harmonic oscillator he s frequency amplitude formulation is used to solve the duffing harmonic oscillator problem the solution procedure is simple and the result obtained is valid for the whole solution domain with high accuracy

primal dual rnc approximation algorithms for set cover and covering integer programs we build on the classical greedy sequential set cover algorithm in the spirit of the primal dual schema to obtain simple parallel approximation algorithms for the set cover problem and its generalizations our algorithms use randomization and our randomized voting lemmas may be of independent interest fast parallel approximation algorithms were known before for set cover though not for the generalizations considered in this paper

set multi covering via inclusion exclusion set multi covering is a generalization of the set covering problem where each element may need to be covered more than once and thus some subset in the given family of subsets may be picked several times for minimizing the number of sets to satisfy the coverage requirement in this paper we propose a family of exact algorithms for the set multi covering problem based on the inclusion exclusion principle the presented esmc exact set multi covering algorithm takes o 2t n time and o t 1 n space where t is the maximum value in the coverage requirement set the o f n notation omits a polylog f n factor we also propose the other three exact algorithms through different tradeoffs of the time and space complexities to the best of our knowledge this present paper is the first one to give exact algorithms for the set multi covering problem with nontrivial time and space complexities this paper can also be regarded as a generalization of the exact algorithm for the set covering problem given in a bjorklund t husfeldt m koivisto set partitioning via inclusion exclusion siam journal on computing in focs 2006 in press special issue

he s frequency amplitude formulation for nonlinear oscillators with an irrational force in this paper he s frequency amplitude formulation is applied to determine the periodic solution for a nonlinear oscillator system with an irrational force comparison with the exact solution shows that the result obtained is of high accuracy

testing with many weak instruments abstract this paper establishes the asymptotic distributions of the likelihood ratio lr anderson rubin ar and lagrange multiplier lm test statistics under many weak iv asymptotics these asymptotics are relevant when the number of ivs is large and the coefficients on the ivs are relatively small the asymptotic results hold under the null and under suitable alternatives hence power comparisons can be made provided k 3 n 0 as n where n is the sample size and k is the number of instruments these tests have correct asymptotic size this holds no matter how weak the instruments are hence the tests are robust to the strength of the instruments the asymptotic power results show that the conditional lr test is more powerful asymptotically than the ar and lm tests under many weak iv asymptotics

variational iteration method some recent results and new interpretations this paper is an elementary introduction to the concepts of variational iteration method first the main concepts in variational iteration method such as general lagrange multiplier restricted variation correction functional are explained heuristically subsequently the solution procedure is systematically addressed in particular for nonlinear oscillators particular attention is paid throughout the paper to give an intuitive grasp for the method the main motivation is to put things together in a convenient form for later reference and systematic use

he chengtian s inequality and its applications he chengtian 369 447 ad is a famous ancient chinese mathematician and astronomer he proposes an iteration technique for searching for high accurate approximate values for one moon 29399752 days and one year 365150608 days respectively this paper re studies the he chengtian s inequality which might be new to the west

a polynomial approximation scheme for scheduling on uniform processors using the dual approximation approach in this paper we present a polynomial approximation scheme for the minimum makespan problem on uniform parallel processors more specifically the problem is to find a schedule for a set of independent jobs on a collection of machines of different speeds so that the last job to finish is completed as quickly as possible we give a family of polynomial time algorithms a such that a delivers a solution that is within a relative error of e of the optimum the technique employed is the dual approximation approach where infeasible but superoptimal solutions for a related dual problem are converted to the desired feasible but possibly suboptimal solution

random effects estimators with many instrumental variables in this paper we propose a new estimator for a model with one endogenous regressor and many instrumental variables our motivation comes from the recent literature on the poor properties of standard instrumental variables estimators when the instrumental variables are weakly correlated with the endogenous regressor our proposed estimator puts a random coefficients structure on the relation between the endogenous regressor and the instruments the variance of the random coefficients is modelled as an unknown parameter in addition to proposing a new estimator our analysis yields new insights into the properties of the standard two stage least squares tsls and limited information maximum likelihood liml estimators in the case with many weak instruments we show that in some interesting cases tsls and liml can be approximated by maximizing the random effects likelihood subject to particular constraints we show that statistics based on comparisons of the unconstrained estimates of these parameters to the implicit tsls and liml restrictions can be used to identify settings when standard large sample approximations to the distributions of tsls and liml are likely to perform poorly we also show that with many weak instruments liml confidence intervals are likely to have under coverage even though its finite sample distribution is approximately centered at the true value of the parameter in an application with real data and simulations around this data set the proposed estimator performs markedly better than tsls and liml both in terms of coverage rate and in terms of risk

improved approximation guarantees for packing and covering integer programs several important np hard combinatorial optimization problems can be posed as packing covering integer programs the randomized rounding technique of raghavan and thompson is a powerful tool with which to approximate them well we present one elementary unifying property of all these integer linear programs and use the fkg correlation inequality to derive an improved analysis of randomized rounding on them this yields a pessimistic estimator thus presenting deterministic polynomial time algorithms for them with approximation guarantees that are significantly better than those known

dynamic programming based algorithms for set multicover and multiset multicover problems given a universe n containing n elements and a collection of multisets or sets over n the multiset multicover msmc problem or the set multicover smc problem is to cover all elements at least a number of times as specified in their coverage requirements with the minimum number of multisets or sets in this paper we give various exact algorithms for these two problems with or without constraints on the number of times a multiset or set may be chosen first we show that the msmc without multiplicity constraints problem can be solved in o b 1 n f time and polynomial space where b is the maximum coverage requirement and f denotes the total number of given multisets over n the o notation suppresses a factor polynomial in n to our knowledge this is the first known exact algorithm for the msmc without multiplicity constraints problem second by combining dynamic programming and the inclusion exclusion principle we can exactly solve the smc without multiplicity constraints problem in o b 2 n time compared with two recent results in q s hua y wang d yu f c m lau set multi covering via inclusion exclusion theoretical computer science 410 38 40 2009 3882 3892 and j nederlof inclusion exclusion for hard problems master thesis utrecht university the netherlands 2008 respectively ours is the fastest exact algorithm for the smc without multiplicity constraints problem finally by directly using dynamic programming we give the first known exact algorithm for the msmc or the smc with multiplicity constraints problem in o b 1 n f time and o b 1 n space this algorithm can also be easily adapted as a constructive algorithm for the msmc without multiplicity constraints problem

an approximation algorithm for the generalized assignment problem the generalized assignment problem can be viewed as the following problem of scheduling parallel machines with costs each job is to be processed by exactly one machine processing jobj on machinei requires timep ij and incurs a cost ofc ij each machinei is available fort i time units and the objective is to minimize the total cost incurred our main result is as follows there is a polynomial time algorithm that given a valuec either proves that no feasible schedule of costc exists or else finds a schedule of cost at mostc where each machinei is used for at most 2t i time units we also extend this result to a variant of the problem where instead of a fixed processing timep ij there is a range of possible processing times for each machine job pair and the cost linearly increases as the processing time decreases we show that these results imply a polynomial time 2 approximation algorithm to minimize a weighted sum of the cost and the makespan i e the maximum job completion time we also consider the objective of minimizing the mean job completion time we show that there is a polynomial time algorithm that given valuesm andt either proves that no schedule of mean job completion timem and makespant exists or else finds a schedule of mean job completion time at mostm and makespan at most 2t

consistent estimation with a large number of weak instruments this paper analyzes the conditions under which consistent estimation can be achieved in instrumental variables iv regression when the available instruments are weak and the number of instruments k n goes to infinity with the sample size we show that consistent estimation depends importantly on the strength of the instruments as measured by r the rate of growth of the so called concentration parameter and also on k n in particular when k n the concentration parameter can grow even if each individual instrument is only weakly correlated with the endogenous explanatory variables and consistency of certain estimators can be established under weaker conditions than have previously been assumed in the literature hence the use of many weak instruments may actually improve the performance of certain point estimators more specifically we find that the limited information maximum likelihood liml estimator and the bias corrected two stage least squares b2sls estimator are consistent when k n r n 0 while the two stage least squares 2sls estimator is consistent only if k n r n 0 as n these consistency results suggest that liml and b2sls are more robust to instrument weakness than 2sls

local computation lower and upper bounds the question of what can be computed and how efficiently is at the core of computer science not surprisingly in distributed systems and networking research an equally fundamental question is what can be computed in a distributed fashion more precisely if nodes of a network must base their decision on information in their local neighborhood only how well can they compute or approximate a global optimization problem in this paper we give the first polylogarithmic lower bound on such local computation for optimization problems including minimum vertex cover minimum connected dominating set maximum matching maximal independent set and maximal matching in addition we present a new distributed algorithm for solving general covering and packing linear programs for some problems this algorithm is tight with the lower bounds whereas for others it is a distributed approximation scheme together our lower and upper bounds establish the local computability and approximability of a large class of problems characterizing how much local information is required to solve these tasks

comparing apples and oranges query trade off in submodular maximization fast algorithms for submodular maximization problems have a vast potential use in applicative settings such as machine learning social networks and economics though fast algorithms were known for some special cases only recently such algorithms were considered in the general case of maximizing a monotone submodular function subject to a matroid independence constraint the known fast algorithm matches the best possible approximation guarantee while trying to reduce the number of value oracle queries the algorithm performs our main result is a new algorithm for this general case that establishes a surprising trade off between two seemingly unrelated quantities the number of value oracle queries and the number of matroid independence queries performed by the algorithm specifically one can decrease the former by increasing the latter and vice versa while maintaining the best possible approximation guarantee such a trade off is very useful since various applications might incur significantly diff

online primal dual algorithms for covering and packing we study a wide range of online covering and packing optimization problems in an online covering problem a linear cost function is known in advance but the linear constraints that define the feasible solution space are given one by one in rounds in an online packing problem the profit function as well as the packing constraints are not known in advance in each round additional information i e a new variable about the profit function and the constraints is revealed an online algorithm needs to maintain a feasible solution in each round in addition the solutions generated over the different rounds need to satisfy a monotonicity property we provide general deterministic primal dual algorithms for online fractional covering and packing problems we also provide deterministic algorithms for several integral online covering and packing problems our algorithms are designed via a novel online primal dual technique and are evaluated via competitive analysis

fixed parameter tractability and completeness ii on completeness for w 1 abstract for many fixed parameter problems that are trivially solvable in polynomial time such as k dominating set essentially no better algorithm is presently known than the one which tries all possible solutions other problems such as feedback vertex set exhibit fixed parameter tractability for each fixed k the problem is solvable in time bounded by a polynomial of degree c where c is a constant independent of k in a previous paper the w hierarchy of parameterized problems was defined and complete problems were identified for the classes w t for t 2 our main result shows that independent set is complete for w 1

a bicriteria algorithm for the minimum submodular cost partial set multi cover problem abstract this paper presents a bicriteria approximation algorithm for the minimum submodular cost partial set multi cover problem scpsmc the goal of which is to find a minimum cost sub collection of sets to fully cover q percentage of total profit of all elements where the cost on sub collections is a submodular function and an element e with covering requirement r e is fully covered if it belongs to at least r e picked sets assuming that the maximum covering requirement r max max e e r e is a constant and the cost function is nonnegative and submodular we give a deterministic b q e 1 e bicriteria algorithm for scpsmc the output of which fully covers at least 1 e q percentage of the total profit and the performance ratio is b q e where b max e f e r e and f e is the number of sets containing element e

approximating disjoint path problems using packing integer programs in a packing integer program we are given a matrix a and column vectors b c with nonnegative entries we seek a vector x of nonnegative integers which maximizes c t x subject to ax leq b the edge and vertex disjoint path problems together with their unsplittable flow generalization are np hard problems with a multitude of applications in areas such as routing scheduling and bin packing these two categories of problems are known to be conceptually related but this connection has largely been ignored in terms of approximation algorithms we explore the topic of approximating disjoint path problems using polynomial size packing integer programs motivated by the disjoint paths applications we introduce the study of a class of packing integer programs called column restricted we develop improved approximation algorithms for column restricted programs a result that we believe is of independent interest additional approximation algorithms for disjoint paths are presented that are simple to implement and achieve good performance when the input has a special structure

on the asymptotic optimality of the liml estimator with possibly many instruments we consider the estimation of the coefficients of a linear structural equation in a simultaneous equation system when there are many instrumental variables we derive some asymptotic properties of the limited information maximum likelihood liml estimator when the number of instruments is large some of these results are new as well as old and we relate them to results in some recent studies we have found that the variance of the limiting distribution of the liml estimator and its modifications often attain the asymptotic lower bound when the number of instruments is large and the disturbance terms are not necessarily normally distributed that is for the micro econometric models of some cases recently called many instruments and many weak instruments