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

using dual approximation algorithms for scheduling problems theoretical and practical results the problem of scheduling a set of n jobs on m identical machines so as to minimize the makespan time is perhaps the most well studied problem in the theory of approximation algorithms for np hard optimization problems in this paper the strongest possible type of result for this problem a polynomial approximation scheme is presented more precisely for each e an algorithm that runs in time o n e 1 e 2 and has relative error at most e is given in addition more practical algorithms for e 1 5 2 k and e 1 6 2 k which have running times o n k log n and o n km 4 log n are presented the techniques of analysis used in proving these results are extremely simple especially in comparison with the baroque weighting techniques used previously the scheme is based on a new approach to constructing approximation algorithms which is called dual approximation algorithms where the aim is to find superoptimal but infeasible solutions and the performance is measured by the degree of infeasibility allowed this notion should find wide applicability in its own right and should be considered for any optimization problem where traditional approximation algorithms have been particularly elusive

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

approximation algorithms for metric facility location problems in this paper we present a 1 52 approximation algorithm for the metric uncapacitated facility location problem and a 2 approximation algorithm for the metric capacitated facility location problem with soft capacities both these algorithms improve the best previously known approximation factor for the corresponding problem and our soft capacitated facility location algorithm achieves the integrality gap of the standard linear programming relaxation of the problem furthermore we will show using a result of thorup that our algorithms can be implemented in quasi linear time

property testing and its connection to learning and approximation in this paper we consider the question of determining whether a function f has property p or is e far from any function with property p a property testing algorithm is given a sample of the value of f on instances drawn according to some distribution in some cases it is also allowed to query f on instances of its choice we study this question for different properties and establish some connections to problems in learning theory and approximation in particular we focus our attention on testing graph properties given access to a graph g in the form of being able to query whether an edge exists or not between a pair of vertices we devise algorithms to test whether the underlying graph has properties such as being bipartite k colorable or having a p clique clique of density p with respect to the vertex set our graph property testing algorithms are probabilistic and make assertions that are correct with high probability while making a number of queries that is independent of the size of the graph moreover the property testing algorithms can be used to efficiently i e in time linear in the number of vertices construct partitions of the graph that correspond to the property being tested if it holds for the input graph

multiple time scale approach to ergodic adiabatic systems another look we reexamine the multiple time scale method as applied to ergodic adiabatic hamiltonian systems solving for the evolution of the phase space density to first order in the slowness parameter we find a term previously overlooked the inclusion of this term resolves a standing discrepancy between the multiple time scale approach to this problem and an approach using a fokker planck equation we apply our solution to the dynamics of a slow system coupled to an ensemble of fast systems following chaotic trajectories

the resurgence properties of the hankel and bessel functions of nearly equal order and argument the aim of this paper is to derive new representations for the hankel functions the bessel functions and their derivatives exploiting the reformulation of the method of steepest descents by berry and howls proc roy soc london ser a 434 657 675 1991 using these representations we obtain a number of properties of the asymptotic expansions of the hankel and bessel functions and their derivatives of nearly equal order and argument including explicit and numerically computable error bounds asymptotics for the late coefficients exponentially improved asymptotic expansions and the smooth transition of the stokes discontinuities

quick k median k center and facility location for sparse graphs we present tilde o m time and space constant factor approximation algorithms for the k median k center and facility location problems with assignment costs being shortest path distances in a weighted undirected graph with n vertices and m edges for all of these location problems tilde o n 2 time and space algorithms were already known but here we are addressing large sparse graphs an application could be placement of content distributing servers on the internet the internet is large and changes so frequently that an tilde o n 2 time solution would likely be outdated long before completion

approximating the weight of the euclidean minimum spanning tree in sublinear time we consider the problem of computing the weight of a euclidean minimum spanning tree for a set of n points in mathbb r d we focus on the setting where the input point set is supported by certain basic and commonly used geometric data structures that can provide efficient access to the input in a structured way we present an algorithm that estimates with high probability the weight of a euclidean minimum spanning tree of a set of points to within 1 eps using only widetilde o sqrt n text poly 1 eps queries for constant d the algorithm assumes that the input is supported by a minimal bounding cube enclosing it by orthogonal range queries and by cone approximate nearest neighbor queries

laplace s method on a computer algebra system with an application to the real valued modified bessel functions we examine a maple implementation of two distinct approaches to laplace s method used to obtain asymptotic expansions of laplace type integrals one algorithm uses power series reversion whereas the other expands all quantities in taylor or puiseux series these algorithms are used to derive asymptotic expansions for the real valued modified bessel functions of pure imaginary order and real argument that mimic the well known corresponding expansions for the unmodified bessel functions

computing the coefficients in laplace s method laplace s method is a preeminent technique in the asymptotic approximation of integrals its utility was enhanced enormously in 1956 when erdelyi found a way to apply watson s lemma and thereby obtain an infinite asymptotic expansion valid in principle for any integral of laplace type erdelyi s formulation requires tedious computation of coefficients c s for each specific application of the method and traditionally this has involved reverting a series recently it was shown that the coefficients c s can be computed via a simple explicit expression that is probably computationally optimal which avoids the reversion approach altogether the formula is made possible by recognizing the central role of faa di bruno s formula alongside watson s lemma in erdelyi s formulation of laplace s classical method laplace s method can now be implemented cleanly and relatively quickly provided one has the luck and the patience to get to the point where implementation becomes automatic the present paper outlines the recent discovery of the role of faa di bruno s formula in laplace s method gives examples of the application of the explicit expression for the coefficients c s and provides grounds for a possible generalization of the result

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

solving the staircase cost facility location problem with decomposition and piecewise linearization abstract facility location problems with staircase costs i e fixed costs that appear at several levels of production are large structured mixed integer programming problems which often are quite hard to solve in this paper solution methods based on convex piecewise linearization and benders decomposition are investigated using convex piecewise linearization only the integer variables that are needed to improve the approximation are introduced and computational tests show that in average only a few are needed benders decomposition can be used to solve the resulting problems and we show how to recalculate benders cuts when new variables are introduced so they still can be used when the approximation is improved

the complexity of approximating the entropy we consider the problem of approximating the entropy of a discrete distribution under several different models of oracle access to the distribution in the evaluation oracle model the algorithm is given access to the explicit array of probabilities specifying the distribution in this model linear time in the size of the domain is both necessary and sufficient for approximating the entropy in the generation oracle model the algorithm has access only to independent samples from the distribution in this case we show that a gamma multiplicative approximation to the entropy can be obtained in o n 1 eta gamma 2 log n time for distributions with entropy omega gamma eta where n is the size of the domain of the distribution and eta is an arbitrarily small positive constant we show that this model does not permit a multiplicative approximation to the entropy in general for the class of distributions to which our upper bound applies we obtain a lower bound of omega n 1 2 gamma 2 we next consider a combined oracle model in which the algorithm has access to both the generation and the evaluation oracles of the distribution in this model significantly greater efficiency can be achieved we present an algorithm for gamma multiplicative approximation to the entropy that runs in o gamma 2 log 2 n h 2 gamma 1 2 time for distributions with entropy omega h for such distributions we also show a lower bound of omega log n h gamma 2 1 gamma 2 finally we consider two special families of distributions those in which the probabilities of the elements decrease monotonically with respect to a known ordering of the domain and those that are uniform over a subset of the domain in each case we give more efficient algorithms for approximating the entropy

on the coefficients that arise from laplace s method laplace s method is one of the best known techniques in the asymptotic approximation of integrals the salient step in the technique s historical development was erdelyi s use of watson s lemma to obtain an infinite asymptotic expansion valid for any laplace type integral published in 1956 erdelyi s expansion contains coefficients cs that must be calculated in each application of laplace s method a tedious process that has traditionally involved the reversion of a series this paper shows that the coefficients cs in fact have a very simple general form in effect we extend erdelyi s theorem our results greatly simplify calculation of the cs in any particular application and clarify the theoretical basis of erdelyi s expansion it turns out that faa di bruno s formula has always played a central role in it we prove or derive the following the correct dimensionless groups erdelyi s expansion is properly expressed in terms of scaled coefficientscs two explicit expressions for cs in terms of combinatorial objects called partial ordinary bell polynomials this form is probably computationally optimal and makes checking for correctness a relatively straightforward process a recursive expression for cs each coefficient cs can be expressed as a polynomial in s where and are quantities in erdelyi s formulation the main insight that emerges is that the traditional approach to laplace s method involving reversion of a series is less efficient and need only be invoked if one is interested in the role of the reversion coefficients in erdelyi s expansion a point which erdelyi himself alluded to we consider as an example an integral that occurs in a variational approach to finding the binding energy of helium dimers we also present a three line computer code to generate the coefficients cs exactly in the general case in a sequel paper to be published in siam review a new representation for the gamma function is obtained and the link with faa di bruno s formula is explained

a diffusion equation for quantum adiabatic systems abstract for ergodic adiabatic quantum systems we study the evolution of energy distribution as the system evolves in time starting from the von neumann equation for the density operator we obtain the quantum analogue of the smoluchowski equation on coarse graining over the energy spectrum this result brings out the precise notion of quantum diffusion

approximate methods for state space models state space models provide an important body of techniques for analyzing time series but their use requires estimating unobserved states the optimal estimate of the state is its conditional expectation given the observation histories and computing this expectation is hard when there are nonlinearities existing filtering methods including sequential monte carlo tend to be either inaccurate or slow in this paper we study a nonlinear filter for nonlinear non gaussian state space models which uses laplace s method an asymptotic series expansion to approximate the state s conditional mean and variance together with a gaussian conditional distribution this laplace gaussian filter lgf gives fast recursive deterministic state estimates with an error which is set by the stochastic characteristics of the model and is we show stable over time we illustrate the estimation ability of the lgf by applying it to the problem of neural decoding and compare it to sequential monte carlo both in simulatio

a sublinear time approximation scheme for bin packing the bin packing problem is defined as follows given a set of n items with sizes 00 we present an algorithm a e that has sampling access to the input instance and outputs a value k such that c o p t k 1 e c o p t 1 where c o p t is the cost of an optimal solution it is clear that uniform sampling by itself will not allow a sublinear time algorithm in this setting a small number of items might constitute most of the total weight and uniform samples will not hit them in this work we use weighted samples where item i is sampled with probability proportional to its weight that is with probability w i iw i in the presence of weighted samples the approximation algorithm runs in o n poly 1 e g 1 e time where g x is an exponential function of x when both weighted sampling and uniform sampling are allowed o n 1 3 poly 1 e g 1 e time suffices in addition to an approximate value to c o p t our algorithm can also output a constant size template of a packing that can later be used to find a near optimal packing in linear time

the geometric phase for chaotic unitary families we consider the geometric phase for a family of quantum classical hamiltonians in which the effect of changing parameters is simply to induce unitary canonical transformations in this case the classical limit of the geometric phase is easily obtained even when the classical motion is chaotic the results agree with those previously obtained for general chaotic families but may be expressed in a simpler form not involving time integrals of correlation functions it is also straightforward to establish some results which are problematic in the general case for example the form of periodic orbit corrections and the closeness of the classical 2 form if the parameters are regarded as dynamical variables evolving slowly so as to maintain adiabaticity they are subject to geometric magnetism but not in contrast to the general case deterministic friction and born oppenheimer forces examples including families of translated and rotated systems are discussed

on the reversion of an asymptotic expansion and the zeros of the airy functions the general theories of the derivation of inverses of functions from their power series and asymptotic expansions are discussed and compared the asymptotic theory is applied to obtain asymptotic expansions of the zeros of the airy functions and their derivatives and also of the associated values of the functions or derivatives a maple code is constructed to generate exactly the coefficients in these expansions the only limits on the number of coefficients are those imposed by the capacity of the computer being used and the execution time that is available the sign patterns of the coefficients suggest open problems pertaining to error bounds for the asymptotic expansions of the zeros and stationary values of the airy functions

local search heuristics for k median and facility location problems we analyze local search heuristics for the metric k median and facility location problems we define the locality gap of a local search procedure for a minimization problem as the maximum ratio of a locally optimum solution obtained using this procedure to the global optimum for k median we show that local search with swaps has a locality gap of 5 furthermore if we permit up to p facilities to be swapped simultaneously then the locality gap is 3 2 p this is the first analysis of a local search for k median that provides a bounded performance guarantee with only k medians this also improves the previous known 4 approximation for this problem for uncapacitated facility location we show that local search which permits adding dropping and swapping a facility has a locality gap of 3 this improves the bound of 5 given by m korupolu c plaxton and r rajaraman analysis of a local search heuristic for facility location problems technical report 98 30 dimacs 1998 we also consider a capacitated facility location problem where each facility has a capacity and we are allowed to open multiple copies of a facility for this problem we introduce a new local search operation which opens one or more copies of a facility and drops zero or more facilities we prove that this local search has a locality gap between 3 and 4