centrality of trees for capacitated k k center we consider the capacitated k k center problem in this problem we are given a finite set of locations in a metric space and each location has an associated non negative integer capacity the goal is to choose open k k locations called centers and assign each location to an open center to minimize the maximum over all locations of the distance of the location to its assigned center the number of locations assigned to a center cannot exceed the center s capacity the uncapacitated k k center problem has a simple tight 2 2 approximation from the 80 s in contrast the first constant factor approximation for the capacitated problem was obtained only recently by cygan hajiaghayi and khuller who gave an intricate lp rounding algorithm that achieves an approximation guarantee in the hundreds in this paper we give a simple algorithm with a clean analysis and prove an approximation guarantee of 9 9 it uses the standard lp relaxation and comes close to settling the integrality gap after necessary preprocessing which is narrowed down to either 7 8 7 8 or 9 9 the algorithm proceeds by first reducing to special tree instances and then uses our best possible algorithm to solve such instances our concept of tree instances is versatile and applies to natural variants of the capacitated k k center problem for which we also obtain improved algorithms finally we give evidence to show that more powerful preprocessing could lead to better algorithms by giving an approximation algorithm that beats the integrality gap for instances where all non zero capacities are the same

improved approximation algorithms for the uncapacitated facility location problem we consider the uncapacitated facility location problem in this problem there is a set of locations at which facilities can be built a fixed cost fi is incurred if a facility is opened at location i furthermore there is a set of demand locations to be serviced by the opened facilities if the demand location j is assigned to a facility at location i then there is an associated service cost proportional to the distance between i and j cij the objective is to determine which facilities to open and an assignment of demand points to the opened facilities so as to minimize the total cost we assume that the distance function c is symmetric and satisfies the triangle inequality for this problem we obtain a 1 2 e approximation algorithm where 1 2 e approx 1 736 which is a significant improvement on the previously known approximation guarantees the algorithm works by rounding an optimal fractional solution to a linear programming relaxation our techniques use properties of optimal solutions to the linear program randomized rounding as well as a generalization of the decomposition techniques of shmoys tardos and aardal proceedings of the 29th acm symposium on theory of computing el paso tx 1997 pp 265 274

a new rounding procedure for the assignment problem with applications to dense graph arrangement problems we present a randomized procedure for rounding fractional perfect matchings to integral matchings if the original fractional matching satisfies any linear inequality then with high probability the new matching satisfies that linear inequality in an approximate sense this extends the well known lp rounding procedure of raghavan and thompson which is usually used to round fractional solutions of linear programs we use our rounding procedure to design an additive approximation algorithm to the quadratic assignment problem the approximation error of the algorithm is en 2 and it runs in n o log n e2 time we also describe polynomial time approximation schemes ptass for dense subcases of many well known np hard arrangement problems including minimum linear arrangement minimum cut linear arrangement maximum acyclic subgraph and betweenness

on the configuration lp for maximum budgeted allocation we study the maximum budgeted allocation problem i e the problem of selling a set of m indivisible goods to n players each with a separate budget such that we maximize the collected revenue since the natural assignment lp is known to have an integrality gap of frac 3 4 34 which matches the best known approximation algorithms our main focus is to improve our understanding of the stronger configuration lp relaxation in this direction we prove that the integrality gap of the configuration lp is strictly better than frac 3 4 34 and provide corresponding polynomial time roundings in the following restrictions of the problem i the restricted budgeted allocation problem in which all the players have the same budget and every item has the same value for any player it can be sold to and ii the graph mba problem in which an item can be assigned to at most 2 players finally we improve the best known upper bound on the integrality gap for the general case from frac 5 6 56 to 2 sqrt 2 2 approx 0 828 22 2 0 828 and also prove hardness of approximation results for both cases

multiplicity results for some nonlinear schrodinger equations with potentials multiple semiclassical standing waves for a class of nonlinear schrodinger equations with potentials are found by means of a perturbative variational method

existence of a positive solution to kirchhoff type problems without compactness conditions abstract the existence of a positive solution to a kirchhoff type problem on r n is proved by using variational methods and the new result does not require usual compactness conditions a cut off functional is utilized to obtain the bounded palais smale sequences

covering problems with hard capacities we consider the classical vertex cover and set cover problems with hard capacity constraints this means that a set vertex can cover only a limited number of its elements adjacent edges and the number of available copies of each set vertex is bounded this is a natural generalization of the classical problems which also captures resource limitations in practical scenarios we obtain the following results for the unweighted vertex cover problem with hard capacities we give a 3 approximation algorithm that is based on randomized rounding with alterations we prove that the weighted version is at least as hard as the set cover problem yielding an interesting separation between the approximability of weighted and unweighted versions of a natural graph problem a logarithmic approximation factor for both the set cover and the weighted vertex cover problem with hard capacities follows from the work of wolsey combinatorica 2 1982 pp 385 393 on submodular set cover we provide here a simple and intuitive proof for this bound

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

covering moving points with anchored disks consider a set of mobile clients represented by n points in the plane moving at constant speed along n different straight lines we study the problem of covering all mobile clients using a set of k disks centered at k fixed centers each disk exists only at one instant and while it does covers any client within its coverage radius the task is to select an activation time and a radius for each disk such that every mobile client is covered by at least one disk in particular we study the optimization problem of minimizing the maximum coverage radius first we prove that although the static version of the problem is polynomial the kinetic version is np hard moreover we show that the problem is not approximable by a constant factor unless p np we then present a generic framework to solve it for fixed values of k which in turn allows us to solve more general optimization problems our algorithms are efficient for constant values of k

potential theory on lipschitz domains in riemannian manifolds sobolev besov space results and the poisson problem abstract we continue a program to develop layer potential techniques for pde on lipschitz domains in riemannian manifolds building on l p and hardy space estimates established in previous papers here we establish sobolev and besov space estimates on solutions to the dirichlet and neumann problems for the laplace operator plus a potential on a lipschitz domain in a riemannian manifold with a metric tensor smooth of class c 1 for some 0 we treat the inhomogeneous problem and extend it to the setting of manifolds results obtained for the constant coefficient laplace operator on a lipschitz domain in euclidean space with the dirichlet boundary condition by d jerison and c kenig

a 3 approximation algorithm for the facility location problem with uniform capacities we consider the facility location problem where each facility can serve at most u clients we analyze a local search algorithm for this problem which uses only the operations of add delete and swap and prove that any locally optimum solution is no more than 3 times the global optimum this improves on a result of chudak and williamson who proved an approximation ratio of 3 2 sqrt 2 for the same algorithm we also provide an example which shows that any local search algorithm which uses only these three operations cannot achieve an approximation guarantee better than 3

primal dual approximation algorithm for the two level facility location problem via a dual quasi greedy approach the main contribution of this work is to propose a primal dual combinatorial 3 1 e approximation algorithm for the two level facility location problem 2 lflp by exploring the approximation oracle concept this result improves the previous primal dual 6 approximation algorithm for the multilevel facility location problem and also matches the previous primal dual approximation ratio for the single level facility location problem one of the major merits of primal dual type algorithms is their easy adaption to other variants of the facility location problems as a demonstration our primal dual approximation algorithm can be easily adapted to several variants of the 2 lflp including models with stochastic scenario dynamically arrived demands and linear facility cost

the dirichlet problem in non smooth domains dirichlet problem for an elliptic operator 2s laj jaj with smooth coefficients aij in domains d in rm m 3 with non smooth boundaries the domains we treat are given locally in some cm coordinate system by the graph of a continuous function qs with vq5 e li for some p 1 p oo such domains are called li domains see section 1 for the precise definitions note that l domains are usually called lipschitz domains if d is an li domain and x e d we study the elliptic measure a7 associated with s and d at x thus for f a continuous function on ad the formula u x hf x f doh gives the perron wiener brelot generalized soluid

on the besov regularity of conformal maps and layer potentials on nonsmooth domains abstract we study the besov regularity of conformal mappings for domains with rough boundary based on the well posedness for the dirichlet problem with besov data also sharp invertibility results for the classical layer potential operators on sobolev besov spaces on the boundary of curvilinear polygons are obtained

the wiener hopf technique for impenetrable wedges having arbitrary aperture angle diffraction by impenetrable wedges having arbitrary aperture angle is studied by means of the wiener hopf w h technique a system of functional equations called generalized wiener hopf equations gwhe is obtained only for certain values of the aperture angle are these equations recognizable as standard or classical wiener hopf equations cwhe however in all cases a mapping is found that reduces the gwhe to cwhe it means that the diffraction by an impenetrable wedge always reduces to a standard w h factorization the solution for the diffraction by a wedge with given face impedances the malyuzhinets problem is obtained in closed form by an explicit factorization of the kernel

the interval constrained 3 coloring problem in this paper we settle the open complexity status of interval constrained coloring with a fixed number of colors we prove that the problem is already np complete if the number of different colors is 3 previously it has only been known that it is np complete if the number of colors is part of the input and that the problem is solvable in polynomial time if the number of colors is at most 2 we also show that it is hard to satisfy almost all of the constraints for a feasible instance even in the restricted case where each interval is used at most once this implies apx hardness of maximizing the number of simultaneously satisfiable intervals

some interior and exterior boundary value problems for the helmholtz equation in a quadrant boundary transmission problems for two dimensional helmholtz equations in a quadrant q 1 and its complement q c 1 respectively are considered in a sobolev space setting the first problem of a quadrant with dirichlet condition on one face and transmission condition on the other is solved in closed form for the case where all quadrants are occupied by the same medium unique sovlability can also be shown in the case of two different media up to exceptional cases of wave numbers while fredholm property holds in general in the second problem transmission conditions are prescribed on both faces these problems are reduced to equivalent integral equations in l 2 of wiener hopf hankel type which can be solved explicitly by obtaining canonical generalized factorizations of certain non rational 2 s 2 matrix valued symbols

approximation schemes for covering and packing problems in image processing and vlsi a unified and powerful approach is presented for devising polynomial approximation schemes for many strongly np complete problems such schemes consist of families of approximation algorithms for each desired performance bound on the relative error e o with running time that is polynomial when e is fixed though the polynomiality of these algorithms depends on the degree of approximation e being fixed they cannot be improved owing to a negative result stating that there are no fully polynomial approximation schemes for strongly np complete problems unless np p the unified technique that is introduced here referred to as the shifting strategy is applicable to numerous geometric covering and packing problems the method of using the technique and how it varies with problem parameters are illustrated a similar technique independently devised by b s baker was shown to be applicable for covering and packing problems on planar graphs

positive solutions for a quasilinear elliptic equation of kirchhoff type this paper is concerned with the existence of positive solutions to the class of nonlocal boundary value problems of the type m w u 2dx du f x u email protected u 0 email protected email protected where w is a smooth bounded domain of n m is a positive function and f has subcritical growth

plane wave diffraction by a rational wedge in this paper new expressions for the acoustic field produced when a plane wave source of sound is diffracted by a soft hard or mixed soft hard wedge whose angle can be expressed as a rational multiple of pi are given the solution is expressed in terms of geometrical acoustic source terms and real integrals that represent the diffracted field the expressions are in a form that allows easy calculation of the acoustic field uniformly valid expressions for the far field are also given for all values of the angular variable the general result obtained includes as special cases sommerfeld s solution for diffraction by a half plane reiche s result for the diffraction by a right angled wedge and a new representation for the solution of the problem of diffraction by a mixed soft hard half plane

electromagnetic diffraction by arbitrary angle impedance wedges the problem of the diffraction of a plane electromagnetic wave incident at an oblique angle on a wedge of arbitrary angle with general tensor impedance boundary conditions is solved using a semi an

existence of a least energy nodal solution for a schrodinger kirchhoff equation with potential vanishing at infinity in this paper we study a class of nonlocal schrodinger kirchhoff problems involving only continuous functions using a minimization argument and a quantitative deformation lemma we find a least energy nodal or sign changing solution to this problem moreover when the problem presents symmetry we show that it has infinitely many nontrivial solutions

aggregation error bounds for a class of location models many location models involve distances and demand points in their objective function in urban contexts there can be millions of demand points this leads to demand point aggregation which produces error we identify a general model structure that includes most such location models and present a means of obtaining error bounds for all models with this structure the error bounds suggest how to do the demand point aggregation so as to keep the error small

sign changing solutions for a fractional kirchhoff equation abstract using a minimization argument and a quantitative deformation lemma we establish the existence of least energy sign changing solutions for the following nonlinear kirchhoff problem a b u 2 s u v x u k x f u in r 3 where a b 0 are constants s 0 1 s is the fractional laplacian v k are continuous positive functions allowed for vanishing behavior at infinity and f is a continuous function satisfying suitable growth assumptions moreover when the nonlinearity f is odd we obtain the existence of infinitely many nontrivial weak solutions not necessarily nodals

a contribution to the quarter plane problem in diffraction theory abstract the three dimensional analog of sommerfeld s famous half plane problem for a screen 1 x1 0 x2 0 x3 0 is treated by an operator theoretic approach this paper shows that the dirichlet problem is well posed for any complex wave number k r if we look for a solution u h1 r3 of the helmholtz equation k2 u 0 in e 7mid 1 g h 1 2 1 with u 1 g h 1 2 1 an explicit representation formula for the solution is established by use of general wiener hopf operator theory sobolev space properties and hilbert space methods the corresponding neumann problem can be solved by analogy

remarks on an elliptic equation of kirchhoff type abstract we present a short survey on the nonlocal elliptic boundary value problem m u 2 d x u f x u in u 0 on where is a smooth bounded domain of r n m is a positive function and f has subcritical growth

nodal solutions for singularly perturbed equations with critical exponential growth abstract the existence and concentration behavior of nodal solutions are established for the equation 2 u v z u f u in where is a domain in r 2 not necessarily bounded v is a positive holder continuous function and f c 1 is an odd function having critical exponential growth

a uniqueness result for a nonlocal equation of kirchhoff type and some related open problem abstract we study the uniqueness of solution for the following boundary value problem involving a nonlocal equation of kirchhoff type a b u 2 d x u f u in u 0 here is a bounded open set in r n with smooth boundary a b are positive real numbers and f r r is a continuous function in particular we give an answer to an open problem recently proposed by b ricceri

a ptas for the cardinality constrained covering with unit balls in this paper we address the cardinality constrained covering with unit balls problem given a positive integer l and a set of n points in r d partition them into a minimum number of parts such that each part contains at most l points and it can be covered by a unit ball of the given p metric developing a constant factor approximation algorithm for this problem is an old open problem by proving a structural property in the problem and applying the shifting strategy and dynamic programming we derive the first email protected d approximation n o 1 e d time algorithm for this problem when d is a fixed constant

wave diffraction by wedges having arbitrary aperture angle abstract the problem of plane wave diffraction by a wedge sector having arbitrary aperture angle has a very long and interesting research background in fact we may recognize significant research on this topic for more than one century despite this fact up to now no clear unified approach was implemented to treat such a problem from a rigourous mathematical way and in a consequent appropriate sobolev space setting in the present paper we are considering the corresponding boundary value problems for the helmholtz equation with complex wave number admitting combinations of dirichlet and neumann boundary conditions the main ideas are based on a convenient combination of potential representation formulas associated with weighted mellin pseudo differential operators in appropriate sobolev spaces and a detailed fredholm analysis thus we prove that the problems have unique solutions with continuous dependence on the data which are represented by the single and double layer potentials where the densities are solutions of derived pseudo differential equations on the half line

mellin convolution operators in bessel potential spaces abstract mellin convolution equations acting in bessel potential spaces are considered the study is based upon two results the first one concerns the interaction of mellin convolutions and bessel potential operators bpos in contrast to the fourier convolutions bpos and mellin convolutions do not commute and we derive an explicit formula for the corresponding commutator in the case of mellin convolutions with meromorphic symbols these results are used in the lifting of the mellin convolution operators acting on bessel potential spaces up to operators on lebesgue spaces the operators arising belong to an algebra generated by mellin and fourier convolutions acting on l p spaces fredholm conditions and index formulae for such operators have been obtained earlier by one of the authors and are employed here the results of the present work have numerous applications in boundary value problems for partial differential equations in particular for equations in domains with angular points

existence and multiplicity of non trivial solutions for schrodinger kirchhoff type equations with radial potential abstract in the present paper the following schrodinger kirchhoff type problem p a b r n u 2 d x u v x u q x f u in r n u x 0 as x is studied four existence theorems of nontrivial solutions and a sequence of high energy solutions for problem p are obtained by the mountain pass theorem and symmetric mountain pass theorem

near optimal sensor placements in gaussian processes theory efficient algorithms and empirical studies when monitoring spatial phenomena which can often be modeled as gaussian processes gps choosing sensor locations is a fundamental task there are several common strategies to address this task for example geometry or disk models placing sensors at the points of highest entropy variance in the gp model and a d or e optimal design in this paper we tackle the combinatorial optimization problem of maximizing the mutual information between the chosen locations and the locations which are not selected we prove that the problem of finding the configuration that maximizes mutual information is np complete to address this issue we describe a polynomial time approximation that is within 1 1 e of the optimum by exploiting the submodularity of mutual information we also show how submodularity can be used to obtain online bounds and design branch and bound search procedures we then extend our algorithm to exploit lazy evaluations and local structure in the gp yielding significant speedups we also extend our approach to find placements which are robust against node failures and uncertainties in the model these extensions are again associated with rigorous theoretical approximation guarantees exploiting the submodularity of the objective function we demonstrate the advantages of our approach towards optimizing mutual information in a very extensive empirical study on two real world data sets

existence of solutions for a class of nonlinear schrodinger equations with potential vanishing at infinity abstract in this paper we investigate the existence of positive ground state solution for the following class of elliptic equations u v x u k x f u in r n where n 3 v k are nonnegative continuous functions and f is a continuous function with a quasicritical growth here we prove a hardy type inequality and use it together with variational method to get a ground state solution

lp based approximation algorithms for capacitated facility location in the capacitated facility location problem with hard capacities we are given a set of facilities mathcal f and a set of clients mathcal d in a common metric space each facility i has a facility opening cost f i and capacity u i that specifies the maximum number of clients that may be assigned to this facility we want to open some facilities from the set mathcal f and assign each client to an open facility so that at most u i clients are assigned to any open facility i the cost of assigning client j to facility i is given by the distance c ij and our goal is to minimize the sum of the facility opening costs and the client assignment costs the only known approximation algorithms that deliver solutions within a constant factor of optimal for this np hard problem are based on local search techniques it is an open problem to devise an approximation algorithm for this problem based on a linear programming lower bound or indeed to prove a constant integrality gap for any lp relaxation we make progress on this question by giving a 5 approximation algorithm for the special case in which all of the facility costs are equal by rounding the optimal solution to the standard lp relaxation one notable aspect of our algorithm is that it relies on partitioning the input into a collection of single demand capacitated facility location problems approximately solving them and then combining these solutions in a natural way

mixed boundary value problems of the type of sommerfeld s half plane problem on etudie des modifications du probleme de demi plan de sommerfeld gouverne par deux equations elliptiques propres dans des demi espaces complementaires de r 3 et on pose des conditions aux limites et de transmission sur les deux demi plans qui ensemble forment la frontiere commune de

fast stabbing of boxes in high dimensions abstract we present in this paper a simple yet efficient algorithm for stabbing a set s of n axis parallel boxes in d dimensional space with c s points in output sensitive time o dn log c s and linear space let c s and b s be respectively the minimum number of points required to stab s and the maximum number of pairwise disjoint boxes of s we prove that b s c s c s b s 1 log 2 b s d 1 since finding a minimal set of c s points is np complete as soon as d 1 we obtain a fast precision sensitive heuristic for stabbing s whose quality does not depend on the input size in the case of congruent or constrained isothetic boxes our algorithm reports respectively c s 2 d 1 b s and c s o d b s stabbing points moreover we show that the bounds we get on c s are asymptotically tight and corroborate our results with some experiments we also describe an optimal output sensitive algorithm for finding a minimal size optimal stabbing point set of intervals finally we conclude with insights for further research

on the location and profile of spike layer nodal solutions to nonlinear schrodinger equations abstract the existence and concentration behavior of a nodal solution are established for the equation e 2 u v x u f u in where is a domain in r n not necessarily bounded v is a positive holder continuous function and f c 1 r is an odd function having subcritical and superlinear growth under additional local condition on v it is proved that such a solution has just one positive and one negative peaks which are located around local minima of v

an efficient polynomial time approximation scheme for load balancing on uniformly related machines we consider basic problems of non preemptive scheduling on uniformly related machines for a given schedule defined by a partition of the jobs into m subsets corresponding to the m machines c i c i denotes the completion time of machine i our goal is to find a schedule that minimizes or maximizes sum i 1 m c i p i 1 m c i p for a fixed value of p such that 0 1 p 1 the minimization problem is equivalent to the well known problem of minimizing the ell p l p norm of the vector of the completion times of the machines and for 0 p 1 0 p 1 the maximization problem is of interest our main result is an efficient polynomial time approximation scheme eptas for each one of these problems our schemes use a non standard application of the so called shifting technique we focus on the work total size of jobs assigned to each machine and introduce intervals of work that are forbidden these intervals are defined so that the resulting effect on the goal function is sufficiently small this allows the partition of the problem into sub problems with subsets of machines and jobs whose solutions are combined into the final solution using dynamic programming our results are the first eptas s for this natural class of load balancing problems

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

dependent rounding and its applications to approximation algorithms we develop a new randomized rounding approach for fractional vectors defined on the edge sets of bipartite graphs we show various ways of combining this technique with other ideas leading to improved approximation algorithms for various problems these include low congestion multi path routing richer random graph models for graphs with a given degree sequence improved approximation algorithms for i throughput maximization in broadcast scheduling ii delay minimization in broadcast scheduling as well as iii capacitated vertex cover and fair scheduling of jobs on unrelated parallel machines

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

a multiexchange local search algorithm for the capacitated facility location problem we present a multiexchange local search algorithm for approximating the capacitated facility location problem cflp where a new local improvement operation is introduced that possibly exchanges multiple facilities simultaneously we give a tight analysis for our algorithm and show that the performance guarantee of the algorithm is between 3 2v2 e and 3 2v2 e for any given constant e 0 the previously known best approximation ratio for the cflp is 7 88 as shown by mahdian and pail 2003 universal facility location proc 11th annual eur sympos algorithms esa 409 421 based on the operations proposed by pail et al 2001 facility location with hard capacities proc 42nd ieee sympos foundations comput sci focs 329 338 our upper bound 3 2v2 e also matches the best known ratio obtained by chudak and williamson 1999 improved approximation algorithm for capacitated facility location problems proc 7th conf integer programming combin optim ipco 99 113 for the cflp with uniform capacities in order to obtain the tight bound of our new algorithm we make interesting use of the notion of exchange graph of pail et al and techniques from the area of parallel machine scheduling

the poisson problem with mixed boundary conditions in sobolev and besov spaces in non smooth domains we introduce certain sobolev besov spaces which are particularly well adapted for measuring the smoothness of data and solutions of mixed boundary value problems in lipschitz domains in particular these are used to obtain sharp well posedness results for the poisson problem for the laplacian with mixed boundary conditions on bounded lipschitz domains which satisfy a suitable geometric condition introduced by r brown in 1994 in this context we obtain results which generalize those by d jerison and c kenig 1995 as well as e fabes o mendez and m mitrea 1998 applications to hodge theory and the regularity of green operators are also presented

nonlinear perturbations of a periodic kirchhoff equation in rn abstract this paper is concerned with the existence of positive solutions for the following class of nonlocal problem m r n u 2 d x r n v x u 2 d x u v x u f u u in r n where 5 for n 3 and 1 for n 1 2 is a positive parameter and 0 1 moreover m v and f are continuous functions satisfying some conditions

a plant location guide for the unsure approximation algorithms for min max location problems this paper studies an extension of the k median problem under uncertain demand we are given an n vertex metric space v d and m client sets documentclass aastex usepackage amsbsy usepackage amsfonts usepackage amssymb usepackage bm usepackage mathrsfs usepackage pifont usepackage stmaryrd usepackage textcomp usepackage portland xspace usepackage amsmath amsxtra pagestyle empty declaremathsizes 10 9 7 6 begin document s i subseteq v i 1 m end document the goal is to open a set of k facilities f such that the worst case connection cost over all the client sets is minimized i e documentclass aastex usepackage amsbsy usepackage amsfonts usepackage amssymb usepackage bm usepackage mathrsfs usepackage pifont usepackage stmaryrd usepackage textcomp usepackage portland xspace usepackage amsmath amsxtra pagestyle empty declaremathsizes 10 9 7 6 begin document min f subseteq v f k max i in m left sum j in s i d

a unified approach to scheduling on unrelated parallel machines we develop a single rounding algorithm for scheduling on unrelated parallel machines this algorithm works well with the known linear programming quadratic programming and convex programming relaxations for scheduling to minimize completion time makespan and other well studied objective functions this algorithm leads to the following applications for the general setting of unrelated parallel machines i a bicriteria algorithm for a schedule whose weighted completion time and makespan simultaneously exhibit the current best individual approximations for these criteria ii better than two approximation guarantees for scheduling to minimize the lp norm of the vector of machine loads for all 1

boundary layer methods for lipschitz domains in riemannian manifolds abstract we extend to the variable coefficient case boundary layer techniques that have been successful in the treatment of the laplace equation and certain other constant coefficient elliptic partial differential equations on lipschitz domains in euclidean space we treat the laplace operator on lipschitz domains in a manifold with c 1 metric tensor and study the dirichlet neumann and oblique derivative boundary problems

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