caratheodory helly and radon numbers for sublattice and related convexities the caratheodory helly and radon numbers are three main invariants in convexity theory these invariants have been determined exactly or approximately for a number of different convexity structures we consider convexity structures defined by the sublattices and by the convex sublattices of finite dimensional euclidian integer and boolean spaces such sublattices arise in submodular optimization lattice programming and in monotone comparative statics of optimization and fixed point problems we also consider integral l natural convexities induced by dual network flow constraint systems we determine the exact caratheodory helly and radon numbers of most of these convexities and very close upper and lower bounds for the other caratheodory numbers our results imply for example that if a set can be obtained with unions and intersections from a given family of subsets of a finite set then it can be obtained with unions and intersections from a small subfamily we also show that finding the cara

maximizing a monotone submodular function subject to a matroid constraint let f 2 x rightarrow cal r be a monotone submodular set function and let x cal i be a matroid we consider the problem rm max s in cal i f s it is known that the greedy algorithm yields a 1 2 approximation m l fisher g l nemhauser and l a wolsey math programming stud no 8 1978 pp 73 87 for this problem for certain special cases e g rm max s leq k f s the greedy algorithm yields a 1 1 e approximation it is known that this is optimal both in the value oracle model where the only access to f is through a black box returning f s for a given set s g l nemhauser and l a wolsey math oper res 3 1978 pp 177 188 and for explicitly posed instances assuming p neq np u feige j acm 45 1998 pp 634 652 in this paper we provide a randomized 1 1 e approximation for any monotone submodular function and an arbitrary matroid the algorithm works in the value oracle model our main tools are a variant of the pipage rounding technique of ageev and sviridenko j combin optim 8 2004 pp 307 328 and a continuous greedy process that may be of independent interest as a special case our algorithm implies an optimal approximation for the submodular welfare problem in the value oracle model j vondrak proceedings of the 38 th acm symposium on theory of computing 2008 pp 67 74 as a second application we show that the generalized assignment problem gap is also a special case although the reduction requires x to be exponential in the original problem size we are able to achieve a 1 1 e o 1 approximation for gap simplifying previously known algorithms additionally the reduction enables us to obtain approximation algorithms for variants of gap with more general constraints

running errands in time approximation algorithms for stochastic orienteering in the stochastic orienteering problem we are given a finite metric space where each node contains a job with some deterministic reward and a random processing time the processing time distributions are known and independent across nodes however the actual processing time of a job is not known until it is completely processed the objective is to compute a nonanticipatory policy to visit nodes and run the corresponding jobs so as to maximize the total expected reward subject to the total distance traveled plus the total processing time being at most a given budget of b this problem combines aspects of the stochastic knapsack problem with uncertain item sizes as well as the deterministic orienteering problem in this paper we consider both nonadaptive and adaptive policies for stochastic orienteering we present a constant factor approximation algorithm for the nonadaptive version and an olog log b approximation algorithm for the adaptive version we extend both these results to directed metrics and a more general sequence orienteering problem finally we address the stochastic orienteering problem when the node rewards are also random and possibly correlated with the processing time and obtain an olog n log b approximation algorithm here n is the number of nodes in the metric all our results for adaptive policies also bound the corresponding adaptivity gaps

hausdorff moment problem and fractional moments a simplified procedure abstract the purpose of this paper is the recovering of a probability density function with support 0 1 from the knowledge of its sequence of moments i e the classical hausdorff moment problem to avoid the well known ill conditioning firstly the moment curve is calculated from the assigned sequence of moments next the unknown density is approximated by maximum entropy maxent technique selecting some proper points on the moment curve exploiting convergence in entropy a simplified quick procedure is suggested to recover the approximate density an application to laplace transform inversion is illustrated

the sample average approximation method for stochastic discrete optimization in this paper we study a monte carlo simulation based approach to stochastic discrete optimization problems the basic idea of such methods is that a random sample is generated and the expected value function is approximated by the corresponding sample average function the obtained sample average optimization problem is solved and the procedure is repeated several times until a stopping criterion is satisfied we discuss convergence rates stopping rules and computational complexity of this procedure and present a numerical example for the stochastic knapsack problem

submodular stochastic probing on matroids in a stochastic probing problem we are given a universe e and a probability that each element e in e is active we determine if an element is active by probing it and whenever a probed element is active we must permanently include it in our solution moreover throughout the process we need to obey inner constraints on the set of elements taken into the solution and outer constraints on the set of all probed elements all previous algorithmic results in this framework have considered only the problem of maximizing a linear function of the active elements here we consider submodular objectives we provide new constant factor approximations for maximizing a monotone submodular function subject to multiple matroid constraints on both the elements that may be taken and the elements that may be probed we also obtain an improved approximation for linear objective functions and show how our approach may be generalized to handle k matchoid constraints

sequence folding lattice tiling and multidimensional coding folding a sequence b into a multidimensional box is a well known method which is used as a multidimensional coding technique the operation of folding is generalized in a way that the sequence b can be folded into various shapes and not just a box the novel definition of folding is based on a lattice tiling for the given shape s and a direction in the d dimensional integer grid necessary and sufficient conditions that a lattice tiling for s combined with a direction define a folding of a sequence into s are derived the immediate and most impressive applications are some new lower bounds on the number of dots in two dimensional synchronization patterns asymptotically optimal such patterns were known only for rectangular shapes we show asymptotically optimal such patterns for a large family of hexagons this is also generalized for multidimensional synchronization patterns the best known patterns in terms of dots for circles and other polygons are also given the technique and its application for two dimensional synchronization patterns raises some interesting problems in discrete geometry we will also discuss these problems it is also shown how folding can be used to construct multidimensional error correcting codes finally by using the new definition of folding new types of multidimensional pseudo random arrays with various shapes are generated

adaptivity in the stochastic blackjack knapsack problem we consider a stochastic variant of the np hard 0 1 knapsack problem in which item values are deterministic and item sizes are independent random variables with known arbitrary distributions items are placed in the knapsack sequentially and the act of placing an item in the knapsack instantiates its size in every stage of insertion if the subset of the items inserted thus far is feasible then it has a total value that equals to the sum of values of all items of this subset otherwise if the subset violates the constraint then its value equals to zero the goal is to compute a policy for insertion of the items that maximizes the expected total value of items placed in the knapsack we consider both non adaptive policies that designate a priori a fixed subset of items to insert and adaptive policies that can make dynamic decisions based on the instantiated sizes of items placed in the knapsack thus far our work characterizes the benefit of adaptivity for this purpose we use a measure called the adaptivity gap the supremum over instances of the ratio between the expected value obtained by an optimal adaptive policy and the expected value obtained by an optimal non adaptive policy first we show a tight bound of 32 on the adaptivity gap for the case of inputs consisting of only two items then we present a non adaptive policy with expected value that is at least 2 1 2 2 1 11 66 times the expected value of the optimal adaptive policy thus the adaptivity gap in this model is at most 11 66 additionally this non adaptive policy is computed in polynomial time finally we consider a special case of the model where all sizes are distributed according to bernoulli distribution with different parameters for this special case we improve our result and bound the adaptivity gap by 8

distributionally robust optimization and its tractable approximations in this paper we focus on a linear optimization problem with uncertainties having expectations in the objective and in the set of constraints we present a modular framework to obtain an approximate solution to the problem that is distributionally robust and more flexible than the standard technique of using linear rules our framework begins by first affinely extending the set of primitive uncertainties to generate new linear decision rules of larger dimensions and is therefore more flexible next we develop new piecewise linear decision rules that allow a more flexible reformulation of the original problem the reformulated problem will generally contain terms with expectations on the positive parts of the recourse variables finally we convert the uncertain linear program into a deterministic convex program by constructing distributionally robust bounds on these expectations these bounds are constructed by first using different pieces of information on the distribution of the underlying uncertainties to develop separate bounds and next integrating them into a combined bound that is better than each of the individual bounds

the irrevocable multiarmed bandit problem this paper considers the multiarmed bandit problem with multiple simultaneous arm pulls and the additional restriction that we do not allow recourse to arms that were pulled at some point in the past but then discarded this additional restriction is highly desirable from an operational perspective and we refer to this problem as the irrevocable multiarmed bandit problem we observe that natural modifications to well known heuristics for multiarmed bandit problems that satisfy this irrevocability constraint have unsatisfactory performance and thus motivated introduce a new heuristic the packing heuristic we establish through numerical experiments that the packing heuristic offers excellent performance even relative to heuristics that are not constrained to be irrevocable we also provide a theoretical analysis that studies the price of irrevocability i e the performance loss incurred in imposing the constraint we propose on the multiarmed bandit model we show that this performance loss is uniformly bounded for a general class of multiarmed bandit problems and indicate its dependence on various problem parameters finally we obtain a computationally fast algorithm to implement the packing heuristic the algorithm renders the packing heuristic computationally cheaper than methods that rely on the computation of gittins indices

probabilistic combinatorial optimization moments semidefinite programming and asymptotic bounds we address the problem of evaluating the expected optimal objective value of a 0 1 optimization problem under uncertainty in the objective coefficients the probabilistic model we consider prescribes limited marginal distribution information for the objective coefficients in the form of moments we show that for a fairly general class of marginal information a tight upper lower bound on the expected optimal objective value of a 0 1 maximization minimization problem can be computed in polynomial time if the corresponding deterministic problem is solvable in polynomial time we provide an efficiently solvable semidefinite programming formulation to compute this tight bound we also analyze the asymptotic behavior of a general class of combinatorial problems that includes the linear assignment spanning tree and traveling salesman problems under knowledge of complete marginal distributions with and without independence we calculate the limiting constants exactly

robust solutions of linear programming problems contaminated with uncertain data optimal solutions of linear programming problems may become severely infeasible if the nominal data is slightly perturbed we demonstrate this phenomenon by studying 90 lps from the well known netlib collection we then apply the robust optimization methodology ben tal and nemirovski 1 3 el ghaoui et al 5 6 to produce robust solutions of the above lps which are in a sense immuned against uncertainty surprisingly for the netlib problems these robust solutions nearly lose nothing in optimality

distributionally robust optimization under moment uncertainty with application to data driven problems stochastic programming can effectively describe many decision making problems in uncertain environments unfortunately such programs are often computationally demanding to solve in addition their solution can be misleading when there is ambiguity in the choice of a distribution for the random parameters in this paper we propose a model that describes uncertainty in both the distribution form discrete gaussian exponential etc and moments mean and covariance matrix we demonstrate that for a wide range of cost functions the associated distributionally robust or min max stochastic program can be solved efficiently furthermore by deriving a new confidence region for the mean and the covariance matrix of a random vector we provide probabilistic arguments for using our model in problems that rely heavily on historical data these arguments are confirmed in a practical example of portfolio selection where our framework leads to better performing policies on the true distribution underlying the daily returns of financial assets

allocating bandwidth for bursty connections in this paper we undertake the first study of statistical multiplexing from the perspective of approximation algorithms the basic issue underlying statistical multiplexing is the following in high speed networks individual connections i e communication sessions are very bursty with transmission rates that vary greatly over time as such the problem of packing multiple connections together on a link becomes more subtle than in the case when each connection is assumed to have a fixed demand we consider one of the most commonly studied models in this domain that of two communicating nodes connected by a set of parallel edges where the rate of each connection between them is a random variable we consider three related problems 1 stochastic load balancing 2 stochastic bin packing and 3 stochastic knapsack in the first problem the number of links is given and we want to minimize the expected value of the maximum load in the other two problems the link capacity and an allowed overflow probability p are given and the objective is to assign connections to links so that the probability that the load of a link exceeds the link capacity is at most p in bin packing we need to assign each connection to a link using as few links as possible in the knapsack problem each connection has a value and we have only one link the problem is to accept as many connections as possible for the stochastic load balancing problem we give an o 1 approximation algorithm for arbitrary random variables for the other two problems we have algorithms restricted to on off sources the most common special case studied in the statistical multiplexing literature with a somewhat weaker range of performance guarantees a standard approach that has emerged for dealing with probabilistic resource requirements is the notion of effective bandwidth this is a means of associating a fixed demand with a bursty connection that represents its distribution as closely as possible our approximation algorithms make use of the standard definition of effective bandwidth and also a new one that we introduce the performance guarantees are based on new results showing that a combination of these measures can be used to provide bounds on the optimal solution

approximations for monotone and nonmonotone submodular maximization with knapsack constraints submodular maximization generalizes many fundamental problems in discrete optimization including max cut in directed undirected graphs maximum coverage maximum facility location and marketing over social networks in this paper we consider the problem of maximizing any submodular function subject to d knapsack constraints where d is a fixed constant we establish a strong relation between the discrete problem and its continuous relaxation obtained through extension by expectation of the submodular function formally we show that for any nonnegative submodular function an approximation algorithm for the continuous relaxation implies a randomized e approximation algorithm for the discrete problem we use this relation to obtain an e 1 e approximation for the problem and a nearly optimal 1 e 1 e approximation ratio for the monotone case for any e 0 we further show that the probabilistic domain defined by a continuous solution can be reduced to yield a polynomial size domain given an oracle for the extension by expectation this leads to a deterministic version of our technique

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

stochastic packing integer programs with few queries we consider a stochastic variant of the packing type integer linear programming problem which contains random variables in the objective vector we are allowed to reveal each entry of the objective vector by conducting a query and the task is to find a good solution by conducting a small number of queries we propose a general framework of adaptive and non adaptive algorithms for this problem and provide a unified methodology for analyzing the performance of those algorithms we also demonstrate our framework by applying it to a variety of stochastic combinatorial optimization problems such as matching matroid and stable set problems

on local search for weighted k set packing given a collection of sets of cardinality at most k with weights for each set the maximum weighted packing problem is that of finding a collection of disjoint sets of maximum total weight we study the worst case behavior of the t local search heuristic for this problem proving a tight bound of k 1 1 t as a consequence for any given r 1 k 1 we can compute in polynomial time a solution whose weight is at least r times the optimal

sampling based approximation algorithms for multistage stochastic optimization stochastic optimization problems provide a means to model uncertainty in the input data where the uncertainty is modeled by a probability distribution over the possible realizations of the actual data we consider a broad class of these problems in which the realized input is revealed through a series of stages and hence are called multistage stochastic programming problems multistage stochastic programming and in particular multistage stochastic linear programs with full recourse is a domain that has received a great deal of attention within the operations research community mostly from the perspective of computational results in application settings our main result is to give the first fully polynomial approximation scheme for a broad class of multistage stochastic linear programming problems with any constant number of stages the algorithm analyzed known as the sample average approximation method is quite simple and is the one most commonly used in practice the algorithm accesses the input by

bounds for the size of radar arrays etablissement de bornes ameliorees superieure et inferieure pour la taille des tableaux radars formes de matrices rectangulaires a elements 0 et 1

approximation algorithms for optimal decision trees and adaptive tsp problems we consider the problem of constructing optimal decision trees given a collection of tests that can disambiguate between a set of m possible diseases each test having a cost and the a priori likelihood of any particular disease what is a good adaptive strategy to perform these tests to minimize the expected cost to identify the disease this problem has been studied in several works with o log m approximations known in the special cases when either costs or probabilities are uniform in this paper we settle the approximability of the general problem by giving a tight o log m approximation algorithm we also consider a substantial generalization the adaptive traveling salesman problem given an underlying metric space a random subset s of vertices is drawn from a known distribution but s is initially unknown we get information about whether any vertex is in s only when it is visited what is a good adaptive strategy to visit all vertices in the random subset s while minimizing the expected dista

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

the price of robustness a robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended under this approach we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes a concern with such an approach is that it might be too conservative in this paper we propose an approach that attempts to make this trade off more attractive that is we investigate ways to decrease what we call the price of robustness in particular we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations an attractive aspect of our method is that the new robust formulation is also a linear optimization problem thus we naturally extend our methods to discrete optimization problems in a tractable way we report numerical results for a portfolio optimization problem a knapsack problem and a problem from the net lib library

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

submodular function maximization via the multilinear relaxation and contention resolution schemes we consider the problem of maximizing a nonnegative submodular set function f 2 n rightarrow mathbb r over a ground set n subject to a variety of packing type constraints including multiple matroid constraints knapsack constraints and their intersections in this paper we develop a general framework that allows us to derive a number of new results in particular when f may be a nonmonotone function our algorithms are based on approximately maximizing the multilinear extension f of f over a polytope p that represents the constraints and then effectively rounding the fractional solution although this approach has been used quite successfully it has been limited in some important ways we overcome these limitations as follows first we give constant factor approximation algorithms to maximize f over a downward closed polytope p described by an efficient separation oracle previously this was known only for monotone functions for nonmonotone functions a constant factor was

monotone submodular maximization over a matroid via non oblivious local search we present an optimal combinatorial 1 1 e approximation algorithm for monotone submodular optimization over a matroid constraint compared to the continuous greedy algorithm g calinescu et al ipco springer berlin 2007 pp 182 196 our algorithm is extremely simple and requires no rounding it consists of the greedy algorithm followed by a local search both phases are run not on the actual objective function but on a related auxiliary potential function which is also monotone and submodular in our previous work on maximum coverage y filmus and j ward focs ieee piscataway nj 2012 pp 659 668 the potential function gives more weight to elements covered multiple times we generalize this approach from coverage functions to arbitrary monotone submodular functions when the objective function is a coverage function both definitions of the potential function coincide our approach generalizes to the case where the monotone submodular function has restricted curvature for any curvatu

stieltjes moment problem via fractional moments stieltjes moment problem is considered to recover a probability density function from the knowledge of its infinite sequence of ordinary moments the approximate density is obtained through maximum entropy technique under the constraint of few fractional moments the latter are numerically obtained from the infinite sequence of ordinary moments and are chosen in such a way as to convey the maximum information content carried by the ordinary moments as a consequence a model with few parameters is obtained and intrinsic numerical instability is avoided it is proved that the approximate density is useful for calculating expected values and some other interesting probabilistic quantities

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

two dimensional synchronization patterns for minimum ambiguity a number of closely related combinatorial problems corresponding to specific assumptions about the type of time frequency sequence which may be appropriate in a particular application are formulated in terms of square or rectangular arrays of dots with appropriate constraints on the two dimensional correlation function the current state of knowledge concerning each of these problems is summarized it is hoped that more general constructions may be found leading to larger families of solutions as well as better computational algorithms for finding individual solutions which may lie outside of the general families

stochastic scheduling with release dates and due dates we consider stochastic scheduling problems in which the processing times of jobs are independent exponentially distributed random variables the release dates are random variables with an arbitrary joint distribution and the due dates are random variables with a joint distribution that satisfies certain conditions our development establishes simple policies that minimize such criteria as the expected weighted sum of completion times and the expected weighted number of late jobs these results contrast markedly with the deterministic counterparts of these models for which no polynomial time algorithms are known

submodular maximization meets streaming matchings matroids and more we study the problem of finding a maximum matching in a graph given by an input stream listing its edges in some arbitrary order where the quantity to be maximized is given by a monotone submodular function on subsets of edges this problem which we call maximum submodular function matching msm is a natural generalization of maximum weight matching mwm which is in turn a generalization of maximum cardinality matching we give two incomparable algorithms for this problem with space usage falling in the semi streaming range they store only o n o n edges using o n log n o nlogn working memory that achieve approximation ratios of 7 75 in a single pass and 3 varepsilon 3 in o varepsilon 3 o 3 passes respectively the operations of these algorithms mimic those of zelke s and mcgregor s respective algorithms for mwm the novelty lies in the analysis for the msm setting in fact we identify a general framework for mwm algorithms that allows this kind of adaptation to the broader setting of msm our framework is not specific to matchings rather we identify a general pattern for algorithms that maximize linear weight functions over independent sets and prove that such algorithms can be adapted to maximize a submodular function the notion of independence here is very general in particular appealing to known weight maximization algorithms we obtain results for submodular maximization over hypermatchings in hypergraphs as well as independent sets in the intersection of multiple matroids

stochastic optimization problems with incomplete information on distribution functions the main purpose of this paper is to discuss numerical optimization procedures based on duality theory for stochastic extremal problems in which the distribution function is only partially known we formulate such problems as minimax problems in which the inner problem involves optimization with respect to probability measures the latter problem is solved using generalized linear programming techniques then we state the dual problem to the initial stochastic optimization problem numerical procedures that avoid the difficulties associated with solving the inner problem are proposed

robust convex optimization we study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set u yet the constraints must hold for all possible values of the data from u the ensuing optimization problem is called robust optimization in this paper we lay the foundation of robust convex optimization in the main part of the paper we show that if u is an ellipsoidal uncertainty set then for some of the most important generic convex optimization problems linear programming quadratically constrained programming semidefinite programming and others the corresponding robust convex program is either exactly or approximately a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods

hamburger moment problem and maximum entropy on the existence conditions the existence of maximum entropy solution for the reduced hamburger moment problem is reconsidered existence conditions previously appeared in literature are revisited allowing an easy way to identifying the existence of maximum entropy solution the obtained results suggest that except for special sequences of moments unknown a priori the maximum entropy solution for the non symmetric reduced hamburger moment problem exists for practical purposes the replacing of the support r with a large enough finite interval finds a theoretical warranty the symmetric case may be formulated as follows once assigned the first 2m moments if maxent density does not exist conclusion drawn uniquely from numerical evidence maxent density with the first 2m 2 moments exists in such a case even if the first 2m moments are known we have to settle for a density which carries less information theoretical results are illustrated through some numerical examples

worst case distribution analysis of stochastic programs we show that for even quasi concave objective functions the worst case distribution with respect to a family of unimodal distributions of a stochastic programming problem is a uniform distribution this extends the so called uniformity principle of barmish and lagoa 1997 where the objective function is the indicator function of a convex symmetric set

submodular unsplittable flow on trees we study the unsplittable flow problem mathsf ufp on trees with a submodular objective function the input to this problem is a tree with edge capacities and a collection of tasks each characterized by a source node a sink node and a demand a subset of the tasks is feasible if the tasks can simultaneously send their demands from the source to the sink without violating the edge capacities the goal is to select a feasible subset of the tasks that maximizes a submodular objective function our main result is an o k log n approximation algorithm for submodular ufp on trees where k denotes the pathwidth of the given tree since every tree has pathwidth o log n we obtain an o log 2 n approximation for arbitrary trees this is the first non trivial approximation guarantee for the problem matching the best known approximation ratio for ufp on trees with a linear objective function our main technical contribution is a new geometric relaxation for ufp on trees that builds on the recent work of bonsma et al 2014 focs anagnostopoulos et al amazing 2 epsilon approximation for unsplittable flow on a path siam pp 26 41 2014 for ufp on paths with a linear objective our relaxation is very structured so it can be combined with the contention resolution framework of chekuri et al 2009 stoc our approach is robust and extends to several related problems such as ufp with bag constraints and the storage allocation problem additionally we study the special case of ufp on trees with a linear objective and upward instances where for each task the source node is a descendant of the sink node such instances generalize ufp on paths we build on the work of bansal et al a quasi ptas for unsplittable flow on line graphs acm pp 721 729 2006 for ufp on paths and obtain a qptas for upward instances when the input data is quasi polynomially bounded we complement this result by showing that unlike the path setting upward instances are mathsf apx hard if the input data is arbitrary

stochastic set packing problem abstract in this paper a stochastic version of the set packing problem spp is studied via scenario analysis we consider a one stage recourse approach to deal with the uncertainty in the coefficients it consists of maximizing in the stochastic spp a composite function of the expected value minus the weighted risk of obtaining a scenario whose objective function value is worse than a given threshold the splitting variable representation is decomposed by dualizing the nonanticipativity constraints that link the deterministic spp with a 0 1 knapsack problem for each scenario under consideration as a result a structured larger pure 0 1 model is created we present several procedures for obtaining good feasible solutions as well as a preprocessing approach for fixing variables the lagrange multipliers updating is performed by using the volume algorithm computational experience is reported for a broad variety of instances which shows that the new approach usually outperforms a state of the art optimization engine producing a comparable optimality gap with smaller several orders of magnitude computing time

optimal selection of customers for a last minute offer we analyze a short term revenue optimization problem involving the targeting of customers for a promotion in which a finite number of perishable items are sold on a last minute offer the goal is to select the subset of customers to whom the offer will be made available in order to maximize the expected return each client replies with a certain probability and reports a specific value that might depend on the customer type so that the selected subset has to balance the risk of not selling all items with the risk of assigning an item to a low value customer we show that threshold strategies which select all those clients with values above a certain optimal threshold might fail to achieve the maximal revenue however using a linear programming relaxation we prove that they attain a constant factor of the optimal value specifically the achieved factor is 1 2 when a single item is to be sold and approaches 1 as the number of available items grows to infinity also for the single item case we propose an upper bound based on a sharper linear relaxation that allows us to obtain a threshold strategy achieving at least 2 3 of the optimal revenue moreover although the complexity status of the problem is open we provide a polynomial time approximation scheme for the single item case

a dynamic traveling salesman problem with stochastic arc costs we propose a dynamic traveling salesman problem tsp with stochastic arc costs motivated by applications such as dynamic vehicle routing in which the cost of a decision is known only probabilistically beforehand but is revealed dynamically before the decision is executed we formulate this as a dynamic program dp and compare it to static counterparts to demonstrate the advantage of the dynamic paradigm over an a priori approach we then apply approximate linear programming alp to overcome the dp s curse of dimensionality obtain a semi infinite linear programming lower bound and discuss its tractability we also analyze a rollout version of the price directed policy implied by our alp and derive worst case guarantees for its performance our computational study demonstrates the quality of a heuristically modified rollout policy using a computationally effective a posteriori bound

maximizing stochastic monotone submodular functions we study the problem of maximizing a stochastic monotone submodular function with respect to a matroid constraint because of the presence of diminishing marginal values in real world problems our model can capture the effect of stochasticity in a wide range of applications we show that the adaptivity gap the ratio between the values of optimal adaptive and optimal nonadaptive policies is bounded and is equal to e e 1 we propose a polynomial time nonadaptive policy that achieves this bound we also present an adaptive myopic policy that obtains at least half of the optimal value furthermore when the matroid is uniform the myopic policy achieves the optimal approximation ratio of 1 1 e this paper was accepted by dimitris bertsimas and yinyu ye optimization

on the adaptivity gap of stochastic orienteering the input to the stochastic orienteering problem gupta et al in soda pp 1522 1538 2012 consists of a budget b and metric v d where each vertex v in v v v has a job with a deterministic reward and a random processing time drawn from a known distribution the processing times are independent across vertices the goal is to obtain a non anticipatory policy originating from a given root vertex to run jobs at different vertices that maximizes expected reward subject to the total distance traveled plus processing times being at most b an adaptive policy can choose the next vertex to visit based on observed random instantiations whereas a non adaptive policy is just given by a fixed ordering of vertices the adaptivity gap is the worst case ratio of the optimal adaptive and non adaptive rewards we prove an varomega left log log b 1 2 right loglogb 1 2 lower bound on the adaptivity gap of stochastic orienteering this provides a negative answer to the o 1 adaptivity gap conjectured by gupta et al 2012 and comes close to the o log log b o loglogb upper bound this result holds even on a line metric we also show an o log log b o loglogb upper bound on the adaptivity gap for the correlated stochastic orienteering problem where the reward of each job is random and possibly correlated to its processing time using this we obtain an improved quasi polynomial time min log n log b cdot tilde o log 2 log b min logn logb o log2logb approximation algorithm for correlated stochastic orienteering

new bounds for the sizes of radar arrays improved upper bounds for the size of radar arrays are found by using the window method new general constructions and computer searched examples are presented the radar array problem is generalized to include arrays with sidelobes greater than 1

approximations to stochastic dynamic programs via information relaxation duality in the analysis of complex stochastic dynamic programs we often seek strong theoretical guarantees on the suboptimality of heuristic policies a common technique for obtaining performance bounds i

technical note the adaptive knapsack problem with stochastic rewards given a set of items with associated deterministic weights and random rewards the adaptive stochastic knapsack problem adaptive skp maximizes the probability of reaching a predetermined target reward level when items are inserted sequentially into a capacitated knapsack before the reward of each item is realized this model arises in resource allocation problems that permit or require sequential allocation decisions in a probabilistic setting one particular application is in obsolescence inventory management in this paper the adaptive skp is formulated as a dynamic programming dp problem for discrete random rewards the paper also presents a heuristic that mixes adaptive and static policies to overcome the curse of dimensionality in the dp the proposed heuristic is extended to problems with normally distributed random rewards the heuristic can solve large problems quickly and its solution always outperforms a static policy the numerical study indicates that a near optimal solution can be obtained by using an algorithm with limited look ahead capabilities

randomized algorithms for online knapsack problems in this paper we study online knapsack problems the input is a sequence of items e 1 e 2 e n each of which has a size and a value given the ith item e i we either put e i into the knapsack or reject it in the removable setting when e i is put into the knapsack some items in the knapsack are removed with no cost if the sum of the size of e i and the total size in the current knapsack exceeds the capacity of the knapsack our goal is to maximize the profit i e the sum of the values of items in the last knapsack we present a simple randomized 2 competitive algorithm for the unweighted non removable case and show that it is the best possible where knapsack problem is called unweighted if the value of each item is equal to its size for the removable case we propose a randomized 2 competitive algorithm despite there is no constant competitive deterministic algorithm we also provide a lower bound 1 1 e 1 368 for the competitive ratio for the unweighted removable case we propose a 10 7 competitive algorithm and provide a lower bound 1 25 for the competitive ratio

approximating graph constrained max cut an instance of the graph constrained max cut mathsf gcmc problem consists of i an undirected graph g v e and ii edge weights c v atopwithdelims 2 rightarrow mathbb r on a complete undirected graph the objective is to find a subset s subseteq v of vertices satisfying some graph based constraint in g that maximizes the weight sum u in s v not in s c uv of edges in the cut s v setminus s the types of graph constraints we can handle include independent set vertex cover dominating set and connectivity our main results are for the case when g is a graph with bounded treewidth where we obtain a frac 1 2 approximation algorithm our algorithm uses an lp relaxation based on the sherali adams hierarchy it can handle any graph constraint for which there is a dynamic program of a specific form using known decomposition results these imply essentially the same approximation ratio for mathsf gcmc under constraints such as independent set dominating set and connectivity on a planar graph g

a systematic construction for radar arrays the radar array problem arises from the need to design frequency hopping sequences with small out of phase autocorrelations it assumes the reflected signals have negligible doppler shifts so the correlations are calculated along the time axis only in this correspondence a systematic construction for radar arrays is provided by means of homogeneous uniform difference matrices a systematic construction for properly centered permutation matrices a special kind of homogeneous uniform difference matrices is also provided which partially solves the open problems posed by zhang and tu

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

risk criteria in a stochastic knapsack problem we consider an investor who wants to allocate funds among several projects each project is expected to yield a certain reward and the objective is that a total reward will achieve a certain given amount called the target this problem is relatively easy to solve when rewards are deterministic but may be hard in a more realistic setting when the rewards are stochastic and the investor wants to maximize the probability of attaining the target we show that by combining dynamic programming with a search procedure the stochastic version of the problem can be solved relatively fast when rewards are normally distributed the procedure is also useful for other risk criteria which involve both the mean and the variance of the total reward

symmetric golomb squares inspection of optimum golomb squares found by exhaustive computer search shows some are symmetric with respect to diagonal reflection this suggests searching for such symmetric golomb squares might be an efficient way to find good squares in this correspondence we find the best symmetric n spl times n golomb squares for n spl les 22 some appear to be the best known

optimal inequalities in probability theory a convex optimization approach we propose a semidefinite optimization approach to the problem of deriving tight moment inequalities for p x in s for a set s defined by polynomial inequalities and a random vector x defined on omega subseteq cal r n that has a given collection of up to kth order moments in the univariate case we provide optimal bounds on p x in s when the first k moments of x are given as the solution of a semidefinite optimization problem in k 1 dimensions in the multivariate case if the sets s and omega are given by polynomial inequalities we obtain an improving sequence of bounds by solving semidefinite optimization problems of polynomial size in n for fixed k we characterize the complexity of the problem of deriving tight moment inequalities we show that it is np hard to find tight bounds for k geq 4 and omega cal r n and for k geq 2 and omega cal r n when the data in the problem is rational for k 1 and omega cal r n we show that we can find tight upper bounds by solving n convex optimization problems when the set s is convex and we provide a polynomial time algorithm when s and omega are unions of convex sets over which linear functions can be optimized efficiently for the case k 2 and omega cal r n we present an efficient algorithm for finding tight bounds when s is a union of convex sets over which convex quadratic functions can be optimized efficiently

recovery of a quantile function from moments the problem of recovering a quantile function of a positive random variable via the values of moments or given the values of its laplace transform is studied two new approximations as well as two new estimates of a quantile function given the sample from underlying distribution are proposed the uniform and l 1 upper bounds of proposed estimates are derived the plots illustrate the behavior of the recovered approximants for the moderate and large sample sizes

geometric packing under nonuniform constraints we study the problem of discrete geometric packing here given weighted regions say in the plane and points with capacities one has to pick a maximum weight subset of the regions such that no point is covered more than its capacity we provide a general framework and an algorithm for approximating the optimal solution for packing in hypergraphs arising out of such geometric settings using this framework we get a flotilla of results on this problem and also on its dual where one wants to pick a maximum weight subset of the points when the regions have capacities for example for the case of fat triangles of similar size we show an o 1 approximation and prove that no ptas is possible

golomb rectangles as folded rulers a correspondence between two dimensional golomb rectangles and one dimensional golomb rulers is given results are translated from rulers to rectangles the extension of these results to an arbitrary number of dimensions is given a new bound is developed on the number of ones in a golomb array of any dimension some new optimum golomb squares rectangles and higher dimensional arrays are given

relaxation analysis for the dynamic knapsack problem with stochastic item sizes we consider a version of the knapsack problem in which an item size is random and revealed only when the decision maker attempts to insert it after every successful insertion the decision maker ca

least squares approximation to the distribution of project completion times with gaussian uncertainty this paper is motivated by the following question how to construct good approximation for the distribution of the solution value to linear optimization problem when the random objective coefficients follow a multivariate normal distribution using stein s identity we show that the least squares normal approximation of the random optimal value can be computed by estimating the persistency values of the corresponding optimization problem we further extend our method to construct a least squares quadratic estimator to improve the accuracy of the approximation in particular to capture the skewness of the objective computational studies show that the new approach provides more accurate estimates of the distributions of project completion times compared to existing methods

genetic search for golomb arrays genetic search with a decision level chromosome is used to search efficiently for golomb 1982 arrays a golomb array a is a binary array with three values for its discrete autocorrelation function m the number of ones in a for no offset 1 when there is partial overlap and 0 otherwise in one dimension a is known as a golomb ruler the method yields improvement because the genetic search considers only feasible solutions in the most likely portion of the search space each chromosome bit represents a decision to place a one in an iterative construction the representation includes all possible arrays since the chromosome bits represent binary decisions in the search space genetic operations such as crossover and mutation always result in feasible descendent chromosomes experimental results include comparison to previously known search methods and several new arrays these results indicate that the proposed search is significantly more efficient than prior methods

on the moment recovered approximations of regression and derivative functions with applications in this paper three formulas for recovering the conditional mean and conditional variance based on product moments are proposed the upper bounds for the uniform rate of approximations of regression and derivatives of some moment determinate function are derived two cases where the support of underlying functions is bounded and unbounded from above are studied based on the proposed approximations novel nonparametric estimates of the distribution function and its density in multiplicative censoring model are constructed simulation study justifies the consistency of the estimates

on a class of minimax stochastic programs for a particular class of minimax stochastic programming models we show that the problem can be equivalently reformulated into a standard stochastic programming problem this permits the direct use of standard decomposition and sampling methods developed for stochastic programming we also show that this class of minimax stochastic programs is closely related to a large family of mean risk stochastic programs where risk is measured in terms of deviations from a quantile

a combinatorial strongly polynomial algorithm for minimizing submodular functions this paper presents a combinatorial polynomial time algorithm for minimizing submodular functions answering an open question posed in 1981 by grotschel lovasz and schrijver the algorithm employs a scaling scheme that uses a flow in the complete directed graph on the underlying set with each arc capacity equal to the scaled parameter the resulting algorithm runs in time bounded by a polynomial in the size of the underlying set and the length of the largest absolute function value the paper also presents a strongly polynomial version in which the number of steps is bounded by a polynomial in the size of the underlying set independent of the function values

robust optimization methodology and applications robust optimization ro is a modeling methodology combined with computational tools to process optimization problems in which the data are uncertain and is only known to belong to some uncertainty set the paper surveys the main results of ro as applied to uncertain linear conic quadratic and semidefinite programming for these cases computationally tractable robust counterparts of uncertain problems are explicitly obtained or good approximations of these counterparts are proposed making ro a useful tool for real world applications we discuss some of these applications specifically antenna design truss topology design and stability analysis synthesis in uncertain dynamic systems we also describe a case study of 90 lps from the netlib collection the study reveals that the feasibility properties of the usual solutions of real world lps can be severely affected by small perturbations of the data and that the ro methodology can be successfully used to overcome this phenomenon

a robust optimization perspective on stochastic programming in this paper we introduce an approach for constructing uncertainty sets for robust optimization using new deviation measures for random variables termed the forward and backward deviations these deviation measures capture distributional asymmetry and lead to better approximations of chance constraints using a linear decision rule we also propose a tractable approximation approach for solving a class of multistage chance constrained stochastic linear optimization problems an attractive feature of the framework is that we convert the original model into a second order cone program which is computationally tractable both in theory and in practice we demonstrate the framework through an application of a project management problem with uncertain activity completion time

string submodular functions with curvature constraints consider the problem of choosing a string of actions to optimize an objective function that is string submodular it was shown in previous papers that the greedy strategy consisting of a string of actions that only locally maximizes the step wise gain in the objective function achieves at least a 1 e 1 approximation to the optimal strategy this paper improves this approximation by introducing additional constraints on curvature namely total backward curvature total forward curvature and elemental forward curvature we show that if the objective function has total backward curvature sigma then the greedy strategy achieves at least a 1 sigma 1 e sigma approximation of the optimal strategy if the objective function has total forward curvature epsilon then the greedy strategy achieves at least a 1 epsilon approximation of the optimal strategy moreover we consider a generalization of the diminishing return property by defining the elemental forward curvature we also introduce the notion of string matroid and consider the problem of maximizing the objective function subject to a string matroid constraint we investigate two applications of string submodular functions with curvature constraints 1 choosing a string of actions to maximize the expected fraction of accomplished tasks and 2 designing a string of measurement matrices such that the information gain is maximized

price of correlations in stochastic optimization when decisions are made in the presence of high dimensional stochastic data handling joint distribution of correlated random variables can present a formidable task both in terms of sampling and estimation as well as algorithmic complexity a common heuristic is to estimate only marginal distributions and substitute joint distribution by independent product distribution in this paper we study possible loss incurred on ignoring correlations through a distributionally robust stochastic programming model and we quantify that loss as price of correlations poc using techniques of cost sharing from game theory we identify a wide class of problems for which poc has a small upper bound to our interest this class will include many stochastic convex programs uncapacitated facility location steiner tree and submodular functions suggesting that the intuitive approach of assuming independent distribution acceptably approximates the robust model for these stochastic optimization problems additionally we demonstrate hardness of bounding poc via examples of subadditive and supermodular functions that have large poc we find that our results are also useful for solving many deterministic optimization problems like welfare maximization k dimensional matching and transportation problems under certain conditions

robust mean covariance solutions for stochastic optimization we provide a method for deriving robust solutions to certain stochastic optimization problems based on mean covariance information about the distributions underlying the uncertain vector of returns we prove that for a general class of objective functions the robust solutions amount to solving a certain deterministic parametric quadratic program we first prove a general projection property for multivariate distributions with given means and covariances which reduces our problem to optimizing a univariate mean variance robust objective this allows us to use known univariate results in the multidimensional setting and to add new results in this direction in particular we characterize a general class of objective functions the so called one or two point support functions for which the robust objective is reduced to a deterministic optimization problem in one variable finally we adapt a result from geoffrion 1967a to reduce the main problem to a parametric quadratic program in particular our results are true for increasing concave utilities with convex or concave convex derivatives closed form solutions are obtained for special discontinuous criteria motivated by bonus and commission based incentive schemes for portfolio management we also investigate a multiproduct pricing application which motivates extensions of our results for the case of nonnegative and decision dependent returns

approximate multi matroid intersection via iterative refinement we introduce a new iterative rounding technique to round a point in a matroid polytope subject to further matroid constraints this technique returns an independent set in one matroid with limited violations of the constraints of the other matroids in addition to the classical steps of iterative relaxation approaches we iteratively refine involved matroid constraints this leads to more restrictive constraint systems whose structure can be exploited to prove the existence of constraints that can be dropped hence throughout the iterations we both tighten constraints and later relax them by dropping constraints under certain conditions due to the refinement step we can deal with considerably more general constraint classes than existing iterative relaxation and rounding methods which typically involve a single matroid polytope with additional simple cardinality constraints that do not overlap too much we show that our rounding method combined with an application of a matroid intersection algorithm yields the first 2 approximation for finding a maximum weight common independent set in 3 matroids moreover our 2 approximation is lp based and settles the integrality gap for the natural relaxation of the problem prior to our work no upper bound better than 3 was known for the integrality gap which followed from the greedy algorithm we also discuss various other applications of our techniques including an extension that allows us to handle a mixture of matroid and knapsack constraints

cross monotonic cost sharing methods for connected facility location games we present cost sharing methods for connected facility location games that are cross monotonic and competitive and that recover a constant fraction of the cost of the constructed solution the novelty of this paper is that we use randomized algorithms and that we share the expected cost among the participating users as a consequence our cost sharing methods are simple and achieve attractive approximation ratios we also provide a primal dual cost sharing method for the connected facility location game with opening costs

two dimensional patterns with distinct differences constructions bounds and maximal anticodes a two dimensional 2 d grid with dots is called a configuration with distinct differences if any two lines which connect two dots are distinct either in their length or in their slope these configurations are known to have many applications such as radar sonar physical alignment and time position synchronization rather than restricting dots to lie in a square or rectangle as previously studied we restrict the maximum distance between dots of the configuration the motivation for this is a new application of such configurations to key distribution in wireless sensor networks we consider configurations in the hexagonal grid as well as in the traditional square grid with distances measured both in the euclidean metric and in the manhattan or hexagonal metrics we note that these configurations are confined inside maximal anticodes in the corresponding grid we classify maximal anticodes for each diameter in each grid we present upper bounds on the number of dots in a pattern with distinct differences contained in these maximal anticodes our bounds settle in the negative a question of golomb and taylor on the existence of honeycomb arrays of arbitrarily large size we present constructions and lower bounds on the number of dots in configurations with distinct differences contained in various 2 d shapes such as anticodes by considering periodic configurations with distinct differences in the square grid

a new series of dense graphs of high girth let k 1 be an odd integer t b k 2 4 c and q be a prime power we construct a bipartite q regular edge transitive graph cd k q of order v 2qk t 1 and girth g k 5 if e is the the number of edges of cd k q then e v 1 k t 1 these graphs provide the best known asymptotic lower bound for the greatest number of edges in graphs of order v and girth at least g g 5 g 6 11 12 for g 24 this represents a slight improvement on bounds established by margulis and lubotzky phillips sarnak for 5 g 23 g 6 11 12 it improves on or ties existing bounds

beyond competitive analysis the competitive analysis of online algorithms has been criticized as being too crude and unrealistic we propose refinements of competitive analysis in two directions the first restricts the power of the adversary by allowing only certain input distributions while the other allows for comparisons between information regimes for online decision making we illustrate the first with an application to the paging problem as a byproduct we characterize completely the work functions of this important special case of the k server problem we use the second refinement to explore the power of lookahead in server and task systems

the dynamic and stochastic knapsack problem with deadlines in this paper a dynamic and stochastic model of the well known knapsack problem is developed and analyzed the problem is motivated by a wide variety of real world applications objects of random weight and reward arrive according to a stochastic process in time the weights and rewards associated with the objects are distributed according to a known probability distribution each object can either be accepted to be loaded into the knapsack of known weight capacity or be rejected the objective is to determine the optimal policy for loading the knapsack within a fixed time horizon so as to maximize the expected accumulated reward the optimal decision rules are derived and are shown to exhibit surprising behavior in some cases it is also shown that if the distribution of the weights is concave then the decision rules behave according to intuition

golomb rectangles there are a number of communications problems involving two dimensional arrays with constrained correlation functions golomb s two classes of radar arrays and sonar arrays are examined improved bounds optimal cases and computational procedures are reported

reconstruction of conditional expectations from product moments with applications in this paper it is shown under conditions associated with the moment problem that a sequence of product moments uniquely determines a conditional expectation of two random variables then a numerical procedure is derived to reconstruct a conditional expectation in terms of a sequence of its product moments

improvements and generalizations of stochastic knapsack and markovian bandits approximation algorithms we study the multi armed bandit problem with arms which are markov chains with rewards in the finite horizon setting the celebrated gittins indices do not apply and the exact solution is intractable we provide approximation algorithms for the general model of markov decision processes with nonunit transition times when preemption is allowed we provide a 1 2 e approximation along with an example showing this is tight when preemption isn t allowed we provide a 1 12 approximation which improves to a 4 27 approximation when transition times are unity our model captures the markovian bandits model of gupta et al the stochastic knapsack model of dean et al and the budgeted learning model of guha and munagala our algorithms improve existing results in all three areas in our analysis we encounter and overcome to our knowledge a new obstacle an algorithm that provably exists via analytical arguments but cannot be found in polynomial time

submodular maximization over multiple matroids via generalized exchange properties submodular function maximization is a central problem in combinatorial optimization generalizing many important np hard problems including max cut in digraphs graphs and hypergraphs certain constraint satisfaction problems maximum entropy sampling and maximum facility location problems our main result is that for any k 2 and any e 0 there is a natural local search algorithm that has approximation guarantee of 1 k e for the problem of maximizing a monotone submodular function subject to k matroid constraints this improves upon the 1 k 1 approximation of fisher nemhauser and wolsey obtained in 1978 fisher m g nemhauser l wolsey 1978 an analysis of approximations for maximizing submodular set functions ii math programming stud 8 73 87 also our analysis can be applied to the problem of maximizing a linear objective function and even a general nonmonotone submodular function subject to k matroid constraints we show that in these cases the approximation guarantees of our algorithms are 1 k 1 e and 1 k 1 1 k 1 e respectively our analyses are based on two new exchange properties for matroids one is a generalization of the classical rota exchange property for matroid bases and another is an exchange property for two matroids based on the structure of matroid intersection