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

stochastic finite element modelling in linear transient heat transfer a stochastic heat transfer variational principle is suggested for transient and steady state heat problems the non statistical formulation is based on the combination of the second order perturbation technique and second moment analysis the principle allows incorporation of system uncertainties into the conventional finite element equations which are solved for the first two probabilistic moments of the nodal random temperature field computational aspects of the problem are discussed numerical algorithms have been worked out and easily adapted to an existing finite element code the formalism presented may be directly employed for a wide class of field problems

modeling uncertainty in steady state diffusion problems via generalized polynomial chaos abstract we present a generalized polynomial chaos algorithm for the solution of stochastic elliptic partial differential equations subject to uncertain inputs in particular we focus on the solution of the poisson equation with random diffusivity forcing and boundary conditions the stochastic input and solution are represented spectrally by employing the orthogonal polynomial functionals from the askey scheme as a generalization of the original polynomial chaos idea of wiener amer j math 60 1938 897 a galerkin projection in random space is applied to derive the equations in the weak form the resulting set of deterministic equations for each random mode is solved iteratively by a block gauss seidel iteration technique both discrete and continuous random distributions are considered and convergence is verified in model problems and against monte carlo simulations

the ordered k median problem surrogate models and approximation algorithms in the last two decades a steady stream of research has been devoted to studying various computational aspects of the ordered k median problem which subsumes traditional facility location problems such as median center p centrum etc through a unified modeling approach given a finite metric space the objective is to locate k facilities in order to minimize the ordered median cost function in its general form this function penalizes the coverage distance of each vertex by a multiplicative weight depending on its ranking or percentile in the ordered list of all coverage distances while antecedent literature has focused on mathematical properties of ordered median functions integer programming methods various heuristics and special cases this problem was not studied thus far through the lens of approximation algorithms in particular even on simple network topologies such as trees or line graphs obtaining non trivial approximation guarantees is an open question the main contribution of this paper is to devise the first provably good approximation algorithms for the ordered k median problem we develop a novel approach that relies primarily on a surrogate model where the ordered median function is replaced by a simplified ranking invariant functional form via efficient enumeration surprisingly while this surrogate model is varomega n varomega 1 hard to approximate on general metrics we obtain an o log n approximation for our original problem by employing local search methods on a smooth variant of the surrogate function in addition an improved guarantee of 2 epsilon is obtained on tree metrics by optimally solving the surrogate model through dynamic programming finally we show that the latter optimality gap is tight up to an o epsilon term

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

a modified stochastic perturbation method for stochastic hyperbolic heat conduction problems abstract the stochastic hyperbolic heat conduction problem is addressed the random field is approximated by using the karhunen loeve expansion truncated after a finite numbers of terms and a modified stochastic perturbation method mspm is proposed for the stochastic hyperbolic heat conduction problems the mspm is based on the stochastic perturbation theory and can provide the mean and covariance of the random temperature including some third and forth order terms compared with the original stochastic perturbation method the mspm is more accurate and is more convenient for the numerical implementation and hence is more applicable for the stochastic hyperbolic heat conduction problem two numerical examples demonstrate that the proposed method can provide very accurate results and can be used for the stochastic hyperbolic heat conduction problems

fillings of unit cotangent bundles we study the topology of exact and stein fillings of the canonical contact structure on the unit cotangent bundle of a closed surface sigma g where g is at least 2 in particular we prove a uniqueness theorem asserting that any stein filling must be s cobordant rel boundary to the disk cotangent bundle of sigma g for exact fillings we show that the rational homology agrees with that of the disk cotangent bundle and that the integral homology takes on finitely many possible values including that of dt sigma g for example if g 1 is square free then any exact filling has the same integral homology and intersection form as dt sigma g

improved combinatorial algorithms for facility location problems we present improved combinatorial approximation algorithms for the uncapacitated facility location problem two central ideas in most of our results are cost scaling and greedy improvement we present a simple greedy local search algorithm which achieves an approximation ratio of 2 414 epsilon in tilde o n 2 epsilon time this also yields a bicriteria approximation tradeoff of 1 gamma 1 2 gamma for facility cost versus service cost which is better than previously known tradeoffs and close to the best possible combining greedy improvement and cost scaling with a recent primal dual algorithm for facility location due to jain and vazirani we get an approximation ratio of 1 853 in tilde o n 3 time this is very close to the approximation guarantee of the best known algorithm which is linear programming lp based further combined with the best known lp based algorithm for facility location we get a very slight improvement in the approximation factor for facility location achieving 1 728 we also consider a variant of the capacitated facility location problem and present improved approximation algorithms for this

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

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

a reduced basis approach for variational problems with stochastic parameters application to heat conduction with variable robin coefficient in this work a reduced basis rb approach is used to solve a large number of boundary value problems bvps parametrized by a stochastic input expressed as a karhunen loeve expansion in order to compute outputs that are smooth functionals of the random solution fields the rb method proposed here for variational problems parametrized by stochastic coefficients bears many similarities to the rb approach developed previously for deterministic systems however the stochastic framework requires the development of new a posteriori estimates for statistical outputs such as the first two moments of integrals of the random solution fields these error bounds in turn permit efficient sampling of the input stochastic parameters and fast reliable computation of the outputs in particular in the many query context

complexity of local search for the p median problem abstract we study the complexity of finding local minima for the p median problem the relationship between swap local optima 0 1 local saddle points and classical karush kuhn tucker conditions is presented it is shown that the local search problems with some neighborhoods are tight pls complete moreover the standard local descent algorithm takes exponential number of iterations in the worst case regardless of the tie breaking and pivoting rules used to illustrate this property we present a family of instances where some local minima may be hard to find computational results with different pivoting rules for random and euclidean test instances are discussed these empirical results show that the standard local descent algorithm is polynomial in average for some pivoting rules

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

on min max r gatherings we consider a min max version of the previously studied r gathering problem with unit demands the problem we consider is a metric facility location problem in which each open facility must serve at least r customers and the maximum of all the facility and connection costs should be minimized rather than their sum this problem is motivated by scenarios in which r customers are required for a facility to be worth opening and the costs represent the time until the facility connection will be available i e we want to have the complete solution ready as soon as possible we present a 3 approximation algorithm for this problem and prove that it cannot be approximated better assuming p np next we consider this problem with the additional natural requirement that each customer will be assigned to a nearest open facility and present a 9 approximation algorithm we further consider previously introduced special cases and variants and obtain improved algorithmic and hardness results

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

one dimensional k center on uncertain data problems on uncertain data have attracted significant attention due to the imprecise nature of many measurement data in this paper we consider the k center problem on one dimensional uncertain data the input is a set p of weighted uncertain points on a real line and each uncertain point is specified by its probability density function pdf which is a piecewise uniform function i e a histogram the goal is to find a set q of k points on the line to minimize the maximum expected distance from the uncertain points of p to their expected closest points in q we present efficient algorithms for this uncertain k center problem and their running times almost match those for the deterministic k center problem

recovery guarantees for exemplar based clustering for a certain class of distributions we prove that the linear programming relaxation of k medoids clustering a variant of k means clustering where means are replaced by exemplars from within the dataset distinguishes points drawn from nonoverlapping balls with high probability once the number of points drawn and the separation distance between any two balls are sufficiently large our results hold in the nontrivial regime where the separation distance is small enough that points drawn from different balls may be closer to each other than points drawn from the same ball in this case clustering by thresholding pairwise distances between points can fail we also exhibit numerical evidence of high probability recovery in a substantially more permissive regime

data stability in clustering a closer look we consider the model introduced by bilu and linial 2010 13 who study problems for which the optimal clustering does not change when distances are perturbed they show that even when a problem is np hard it is sometimes possible to obtain efficient algorithms for instances resilient to certain multiplicative perturbations e g on the order of o n for max cut clustering awasthi et al 2012 6 consider center based objectives and balcan and liang 2012 9 analyze the k median and min sum objectives giving efficient algorithms for instances resilient to certain constant multiplicative perturbations here we are motivated by the question of to what extent these assumptions can be relaxed while allowing for efficient algorithms we show there is little room to improve these results by giving np hardness lower bounds for both the k median and min sum objectives on the other hand we show that constant multiplicative resilience parameters can be so strong as to make the clustering problem trivial leaving only a narrow range of resilience parameters for which clustering is interesting we also consider a model of additive perturbations and give a correspondence between additive and multiplicative notions of stability our results provide a close examination of the consequences of assuming stability in data

better bounds for incremental medians in the incremental version of the well known k medianproblem the objective is to compute an incremental sequence of facility sets f email protected f email protected email protected f n where each f k contains at most k facilities we say that this incremental medians sequence is r competitive if the cost of each f k is at most r times the optimum cost of k facilities the smallest such r is called the competitive ratio of the sequence f k mettu and plaxton ramgopal r mettu c greg plaxton the online median problem in proc 41st symposium on foundations of computer science focs ieee 2000 pp 339 348 ramgopal r mettu c greg plaxton the online median problem siam journal on computing 32 3 2003 816 832 presented a polynomial time algorithm that computes an incremental sequence with competitive ratio 30 they also showed a lower bound of 2 the upper bound on the ratio was improved to 8 in guolong lin chandrashekha nagarajan rajmohan rajamaran david p williamson a general approach for incremental approximation and hierarchical clustering in proc 17th symposium on discrete algorithms soda 2006 pp 1147 1156 and marek chrobak claire kenyon john noga neal young online medians via online bidding in proc 7th latin american theoretical informatics symposium latin in lecture notes in computer science vol 3887 2006 pp 311 322 we improve both bounds in this paper we first show that no incremental sequence can have competitive ratio better than 2 01 and we give a probabilistic construction of a sequence whose competitive ratio is at most 2 42 7 656 we also propose a new approach to the problem that for instances that we refer to as equable achieves an optimal ratio of 2

symplectic manifolds with contact type boundaries an example of a 4 dimensional symplectic manifold with disconnected boundary of contact type is constructed a collection of other results about symplectic manifolds with contact type boundaries are derived using the theory ofj holomorphic spheres in particular the following theorem of eliashberg floer mcduff is proved if a neighbourhood of the boundary of v is symplectomorphic to a neighbourhood ofs2n 1 in standard euclidean space and if vanishes on all 2 spheres inv thenv is diffeomorphic to the ballb2n

polyfolds cobordisms and the strong weinstein conjecture abstract we prove the strong weinstein conjecture for closed contact manifolds that appear as the concave boundary of a symplectic cobordism admitting an essential local foliation by holomorphic spheres

on the arnold conjecture for weakly monotone symplectic manifolds we show the arnold conjecture concerning symplectic fixed points in the case that the symplectic manifold is weakly monotone and all the fixed points are non degenerate in particular the conjecture is true in dimension 2 4 6 if all the fixed points are non degenerate

a nearly linear time approximation scheme for the euclidean k median problem this paper provides a randomized approximation scheme for the k median problem when the input points lie in the d dimensional euclidean space the worst case running time is o 2 o log 1 epsilon varepsilon d 1 n log d 6 n which is nearly linear for any fixed varepsilon and d moreover our method provides the first polynomial time approximation scheme for and uncapacitated facility location instances in d dimensional euclidean space for any fixed d 2 our work extends techniques introduced originally by arora for the euclidean traveling salesman problem tsp to obtain the improvement we develop a structure theorem to describe hierarchical decomposition of solutions the theorem is based on an adaptive decomposition scheme which guesses at every level of the hierarchy the structure of the optimal solution and accordingly modifies the parameters of the decomposition we believe that our methodology is of independent interest and may find applications to further geometric problems

iterative and active graph clustering using trace norm minimization without cluster size constraints this paper investigates graph clustering under the planted partition model in the presence of small clusters traditional results dictate that for an algorithm to provably correctly recover the underlying clusters all clusters must be sufficiently large in particular the cluster sizes need to be n where n is the number of nodes of the graph we show that this is not really a restriction by a refined analysis of a convex optimization based recovery approach we prove that small clusters under certain mild assumptions do not hinder recovery of large ones based on this result we further devise an iterative algorithm to provably recover almost all clusters via a peeling strategy we recover large clusters first leading to a reduced problem and repeat this procedure these results are extended to the partial observation setting in which only a chosen part of the graph is observed the peeling strategy gives rise to an active learning algorithm in which edges adjacent to smaller clusters are queried more often after large clusters are learned and removed we expect that the idea of iterative peeling that is sequentially identifying a subset of the clusters and reducing the problem to a smaller one is useful more broadly beyond the specific implementations based on convex optimization used in this paper

incremental algorithms for facility location and k median in the incremental versions of facility location and k median the demand points arrive one at a time and the algorithm maintains a good solution by either adding each new demand to an existing cluster or placing it in a new singleton cluster the algorithm can also merge some of the existing clusters at any point in time for facility location we consider the case of uniform facility costs where the cost of opening a facility is the same for all points and present the first incremental algorithm which achieves a constant performance ratio using this algorithm as a building block we obtain the first incremental algorithm for k median which achieves a constant performance ratio using o k medians the algorithm is based on a novel merge rule which ensures that the algorithm s configuration monotonically converges to the optimal facility locations according to a certain notion of distance using this property we reduce the general case to the special case when the optimal solution consists of a single facility

floer novikov homology and applications to lagrangian submanifolds we use a non hamiltonian version of lagrangian floer homology to prove that an exact lagrangian submanifold in the cotangent bundle of the 3 torus t 3 must be diffeomorphic to t 3 this improves a previous result of fukaya seidel and smith

approximation algorithms for metric facility location and k median problems using the primal dual schema and lagrangian relaxation we present approximation algorithms for the metric uncapacitated facility location problem and the metric k median problem achieving guarantees of 3 and 6 respectively the distinguishing feature of our algorithms is their low running time o m log m and o m log m l log n respectively where n and m are the total number of vertices and edges in the underlying complete bipartite graph on cities and facilities the main algorithmic ideas are a new extension of the primal dual schema and the use of lagrangian relaxation to derive approximation algorithms

gromov s compactness of pseudo holomorphic curves and symplectic geometry on utilise certaines idees de schoerr et uhlenbeck sur les equations aux derivees partielles pour demontrer la compacite de gromov des courbes pseudoholomorphes

weak and strong fillability of higher dimensional contact manifolds for contact manifolds in dimension three the notions of weak and strong symplectic fillability and tightness are all known to be inequivalent we extend these facts to higher dimensions in particular we define a natural generalization of weak fillings and prove that it is indeed weaker at least in dimension five while also being obstructed by all known manifestations of overtwistedness we also find the first examples of contact manifolds in all dimensions that are not symplectically fillable but also cannot be called overtwisted in any reasonable sense these depend on a higher dimensional analogue of giroux torsion which we define via the existence in all dimensions of exact symplectic manifolds with disconnected contact boundary

clustering under approximation stability a common approach to clustering data is to view data objects as points in a metric space and then to optimize a natural distance based objective such as the k median k means or min sum score for applications such as clustering proteins by function or clustering images by subject the implicit hope in taking this approach is that the optimal solution for the chosen objective will closely match the desired target clustering e g a correct clustering of proteins by function or of images by who is in them however most distance based objectives including those mentioned here are np hard to optimize so this assumption by itself is not sufficient assuming p np to achieve clusterings of low error via polynomial time algorithms in this article we show that we can bypass this barrier if we slightly extend this assumption to ask that for some small constant c not only the optimal solution but also all c approximations to the optimal solution differ from the target on at most some e fraction of points we call this c e approximation stability we show that under this condition it is possible to efficiently obtain low error clusterings even if the property holds only for values c for which the objective is known to be np hard to approximate specifically for any constant c 1 c e approximation stability of k median or k means objectives can be used to efficiently produce a clustering of error o e with respect to the target clustering as can stability of the min sum objective if the target clusters are sufficiently large thus we can perform nearly as well in terms of agreement with the target clustering as if we could approximate these objectives to this np hard value

active clustering of biological sequences given a point set s and an unknown metric d on s we study the problem of efficiently partitioning s into k clusters while querying few distances between the points in our model we assume that we have access to one versus all queries that given a point s s return the distances between s and all other points we show that given a natural assumption about the structure of the instance we can efficiently find an accurate clustering using only o k distance queries our algorithm uses an active selection strategy to choose a small set of points that we call landmarks and considers only the distances between landmarks and other points to produce a clustering we use our procedure to cluster proteins by sequence similarity this setting nicely fits our model because we can use a fast sequence database search program to query a sequence against an entire data set we conduct an empirical study that shows that even though we query a small fraction of the distances between the points we produce clusterings that are close to a desired clustering given by manual classification

on generalizations of network design problems with degree bounds iterative rounding and relaxation have arguably become the method of choice in dealing with unconstrained and constrained network design problems in this paper we extend the scope of the iterative relaxation method in two directions 1 by handling more complex degree constraints in the minimum spanning tree problem namely laminar crossing spanning tree and 2 by incorporating degree bounds in other combinatorial optimization problems such as matroid intersection and lattice polyhedra we give new or improved approximation algorithms hardness results and integrality gaps for these problems our main result is a 1 b o log n approximation algorithm for the minimum crossing spanning tree mcst problem with laminar degree constraints the laminar mcst problem is a natural generalization of the well studied bounded degree mst and is a special case of general crossing spanning tree we give an additive log c m hardness of approximation for general mcst even in the absence of costs c 0 is a fixed constant and m is the number of degree constraints this also leads to a multiplicative log c m hardness of approximation for the robust k median problem anthony et al in math oper res 35 79 101 2010 improving over the previously known factor 2 hardness we then consider the crossing contra polymatroid intersection problem and obtain a 2 2b 1 approximation algorithm where is the maximum element frequency this models for example the degree bounded spanning set intersection in two matroids finally we introduce the crossing latticep olyhedron problem and obtain a 1 b 2 1 approximation algorithm under certain condition this result provides a unified framework and common generalization of various problems studied previously such as degree bounded matroids

clustering to minimize the maximum intercluster distance the problem of clustering a set of points so as to minimize the maximum intercluster distance is studied an o kn approximation algorithm where n is the number of points and k is the number of clusters that guarantees solutions with an objective function value within two times the optimal solution value is presented this approximation algorithm succeeds as long as the set of points satisfies the triangular inequality we also show that our approximation algorithm is best possible with respect to the approximation bound if pz np

contact structures on product five manifolds and fibre sums along circles two constructions of contact manifolds are presented i products of s1 with manifolds admitting a suitable decomposition into two exact symplectic pieces and ii fibre connected sums along isotropic circles baykur has found a decomposition as required for i for all closed oriented 4 manifolds as a corollary we can show that all closed oriented 5 manifolds that are cartesian products of lower dimensional manifolds carry a contact structure for symplectic 4 manifolds we exhibit an alternative construction of such a decomposition this gives us control over the homotopy type of the corresponding contact structure in particular we prove that mathbb cp 2 times s 1 admits a contact structure in every homotopy class of almost contact structures the existence of contact structures is also established for a large class of 5 manifolds with fundamental group mathbb z 2

a spacetime discontinuous galerkin method for hyperbolic heat conduction non fourier conduction models remedy the paradox of infinite signal speed in the traditional parabolic heat equation for applications involving very short length or time scales hyperbolic conduction models better represent the physical thermal transport processes this paper reviews the maxwell cattaneo vernotte modification of the fourier conduction law and describes its implementation within a spacetime discontinuous galerkin sdg finite element method that admits jumps in the primary variables across element boundaries with arbitrary orientation in space and time a causal advancing front meshing procedure enables a patch wise solution procedure with linear complexity in the number of spacetime elements an h adaptive scheme and a special sdg shock capturing operator accurately resolve sharp solution features in both space and time numerical results for one spatial dimension demonstrate the convergence properties of the sdg method as well as the effectiveness of the shock capturing method simulations in two spatial dimensions demonstrate the proposed method s ability to accurately resolve continuous and discontinuous thermal waves in problems where rapid and localized heating of the conducting medium takes place

on solving nonlinear transient heat transfer problems with random parameters abstract a variational methodology is proposed for nonlinear transient heat transfer systems with random parameters defined by their first two probabilistic moments the finite element formalism is based on the combination of the second order perturbation and mean based second moment analysis a hierarchical system of first order partial differential equations is obtained and solved for the first two probabilistic moments of the random temperature field numerical aspects of the problem are discussed and illustrative results are given the approach presented may be employed for a wide class of field problems

the geometry of symplectic energy non squeezing theorem which says that it is impossible to embed a large ball symplectically into a thin cylinder of the form r2 x b2 where b2 is a 2 disc this led to hofer s discovery of symplectic capacities which give a way of measuring the size of subsets in symplectic manifolds recently hofer found a way to measure the size or energy of symplectic diffeomorphisms by looking at the total variation of their generating hamiltonians this gives rise to a bi invariant pseudo norm on the group ham m of compactly supported hamiltonian symplectomorphisms of the manifold m the deep fact is that this pseudo norm is a norm in other words the only symplectomorphism on m with zero energy is the identity map up to now this had been proved only for sufficiently nice symplectic manifolds and by rather complicated analytic arguments in this paper we consider a more geometric version of this energy which was first considered by eliashberg and hofer in connection with their study of the extent to which the interior of a region in a symplectic manifold determines its boundary we prove by a simple geometric argument that both versions of energy give rise to genuine norms on all symplectic manifolds roughly speaking we show that if there were a symplectomorphism of m which had too little energy one could embed a large ball into a thin cylinder m x

an improved 1 1 evolutionary algorithm for k median clustering problem with performance guarantee abstract evolutionary algorithms eas have been widely studied in numerical experiments and successfully applied in practice however most existing eas are designed without a theoretical analysis for their performance guarantee in this paper we define a fitness function to guide the well known 1 1 evolutionary algorithm called 1 1 ea to optimize the np hard k median problem we prove that the 1 1 ea can obtain a performance guarantee of 5 for k median problem in polynomial expected runtime o m n 2 d m a x if all distances between data points and cluster centers are integers where m and n are respectively the cardinalities of the data set and cluster center set and d m a x denotes the largest distance between data points and cluster centers to tackle the general case that the distances between data points and cluster centers are real number we propose an improved 1 1 ea which employs a distance scaling scheme during the evolutionary process our rigorous theoretical analysis shows that the improved 1 1 ea obtains a performance guarantee of 5 e in expected runtime o e 1 m n 2 log m d m a x where e 0 is a constant this study reveals that an appropriate scaling scheme can help eas to obtain a constant performance guarantee in polynomial expected runtime and provides insights into design eas for some np hard problems in addition we conduct experiment to compare the efficiency of the improved 1 1 ea with three clustering algorithms on optimizing three uci datasets

scenario reduction revisited fundamental limits and guarantees the goal of scenario reduction is to approximate a given discrete distribution with another discrete distribution that has fewer atoms we distinguish continuous scenario reduction where the new atoms may be chosen freely and discrete scenario reduction where the new atoms must be chosen from among the existing ones using the wasserstein distance as measure of proximity between distributions we identify those n point distributions on the unit ball that are least susceptible to scenario reduction i e that have maximum wasserstein distance to their closest m point distributions for some prescribed m n we also provide sharp bounds on the added benefit of continuous over discrete scenario reduction finally to our best knowledge we propose the first polynomial time constant factor approximations for both discrete and continuous scenario reduction as well as the first exact exponential time algorithms for continuous scenario reduction

an approximation algorithm for the k median problem with uniform penalties via pseudo solution abstract we present a 1 3 approximation algorithm for the k median problem with uniform penalties extending the recent result by li and svensson for the classical k median problem without penalties one important difference of this work from that of li and svensson is a new definition of sparse instance to exploit the combinatorial structure of our problem

k means under approximation stability one of the most popular algorithms for finding centers for initializing lloyd s heuristic is the k means seeding algorithm the algorithm is a simple sampling procedure that can be described as follows the algorithm picks the first center randomly from among the given points and then for i 2 3 k picks a point to be the ith center with probability proportional to the squared euclidean distance of this point to the nearest center out of the i 1 previously chosen centers the k means seeding algorithm is known to exhibit nice properties it has been noticed that this seeding algorithm tends to perform well when the optimal clusters are separated in some sense intuitively this is because the algorithm gives preference to further away points when picking centers one separation condition that has been studied in the past was due to ostrovsky et al 9 jaiswal and garg 8 showed that if any dataset satisfies the separation condition of 9 then this sampling algorithm gives a constant approximation with probability 1 k on this dataset another separation condition that is strictly weaker than 9 is the approximation stability condition studied by balcan et al 5 in this work we show that the sampling algorithm gives a constant approximation with probability 1 k on any dataset that satisfies the separation condition of 5 and the optimal k clusters are not too small we give a negative result for datasets that have small optimal clusters

symplectic fixed points and holomorphic spheres letp be a symplectic manifold whose symplectic form integrated over the spheres inp is proportional to its first chern class on the loop space ofp we consider the variational theory of the symplectic action function perturbed by a hamiltonian term in particular we associate to each isolated invariant set of its gradient flow an abelian group with a cyclic grading it is shown to have properties similar to the homology of the conley index in locally compact spaces as an application we show that if the fixed point set of an exact diffeomorphism onp is nondegenerate then it satisfies the morse inequalities onp we also discuss fixed point estimates for general exact diffeomorphisms

polyhedral properties of the k median problem on a tree the polyhedral structure of the k median problem on a tree is examined even for very small connected graphs we show that additional constraints are needed to describe the integer polytope a complete description is given of those trees for which an optimal integer lp solution is guaranteed to exist we present a new and simpler demonstration that an lp characterization of the 2 median problem is complete also we provide a simpler proof of the value of a tight worst case bound for the lp relaxation a new class of valid inequalities is identified these inequalities describe a subclass of facets for the k median problem on a general graph also we provide polyhedral descriptions for several types of trees as part of this work we summarize most known results for the k median problem on a tree

facility location with matroid or knapsack constraints in the classical k median problem we are given a metric space and want to open k centers so as to minimize the sum over all the vertices of the distance of each vertex to its nearest open center in this paper we present the first constant factor approximation algorithms for two natural generalizations of this problem that handle matroid or knapsack constraints in the matroid median problem there is an underlying matroid on the vertices and the set of open centers is constrained to be independent in this matroid when the matroid is uniform we recover the k median problem another previously studied special case is the red blue median problem where we have a partition matroid with two parts our algorithm for matroid median is based on rounding a natural linear programming relaxation in two stages and it relies on a connection to matroid intersection in the knapsack median problem centers have weights and the total weight of open centers is constrained to be at most a given capacity when all wei

sharp estimates of the kobayashi metric and gromov hyperbolicity abstract let d 0 be a smooth relatively compact domain in a four dimensional almost complex manifold m j where is a j plurisubharmonic function on a neighborhood of d and strictly j plurisubharmonic on a neighborhood of d we give sharp estimates of the kobayashi metric our approach is based on an asymptotic quantitative description of both the domain d and the almost complex structure j near a boundary point following z m balogh and m bonk z m balogh m bonk gromov hyperbolicity and the kobayashi metric on strictly pseudoconvex domains comment math helv 75 2000 504 533 these sharp estimates provide the gromov hyperbolicity of the domain d

improved algorithms for joint optimization of facility locations and network connections abstract this paper studies a k median steiner forest problem that jointly optimizes the opening of at most k facility locations and their connections to the client locations so that each client is connected by a path to an open facility with the total connection cost minimized the problem has wide applications in the telecommunication and transportation industries but is strongly np hard in the literature only a 2 approximation algorithm is known it being based on a lagrangian relaxation of the problem and using a sophisticated primal dual schema in this study we have developed an improved approximation algorithm using a simple transformation from an optimal solution of a minimum spanning tree problem compared with the existing 2 approximation algorithm our new algorithm not only achieves a better approximation ratio that is easier to be proved but also guarantees to produce solutions of equal or better quality up to 50 percent improvement in some cases in addition for two non trivial special cases where either every location contains a client or all the locations are in a tree shaped network we have developed for the first time in the literature new algorithms that can solve the problem to optimality in polynomial time

the structure of rational and ruled symplectic 4 manifolds this paper investigates the structure of compact symplectic 4 manifolds v w which contain a symplectically embedded copy c of s2 with nonnegative self intersection number such a pair v c w is called minimal if in addition the open manifold v c contains no exceptional curves i e symplectically embedded 2 spheres with self intersection 1 we show that every such pair v c w covers a minimal pair v c c which may be obtained from v by blowing down a finite number of disjoint exceptional curves in v c further the family of manifold pairs v c w under consideration is closed under blowing up and down we next give a complete list of the possible minimal pairs we show that v is symplectomorphic either to p2 with its standard form or to an s2 bundle over a compact surface with a symplectic structure which is uniquely determined by its cohomology class moreover this symplectomorphism may be chosen so that it takes c either to 2 a complex line or quadric in cp2 or in the case when v is a bundle to a fiber or section of the bundle department of mathematics state university of new york at stony brook stony brook new york 11794 3651 e mail address dusa math sunysb edu this content downloaded from 207 46 13 145 on wed 27 apr 2016 05 10 07 utc all use subject to http about jstor org terms

examples of symplectic structures in this paper we construct symplectic forms tilde omega k k geqq 0 on a compact manifold tilde y which have the same homotopy theoretic invariants but which are not diffeomorphic

rationally convex domains and singular lagrangian surfaces in mathbb c 2 we give a complete characterization of those disk bundles over surfaces which embed as rationally convex strictly pseudoconvex domains in mathbb c 2 we recall some classical obstructions and prove some deeper ones related to symplectic and contact topology we explain the close connection to lagrangian surfaces with isolated singularities and develop techniques for constructing such surfaces our proof also gives a complete characterization of lagrangian surfaces with open whitney umbrellas answering a question first posed by givental in 1986

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

on the complexity of some common geometric location problems given n demand points in the plane the p center problem is to find p supply points anywhere in the plane so as to minimize the maximum distance from a demand point to its respective nearest supply point the p median problem is to minimize the sum of distances from demand points to their respective nearest supply points we prove that the p center and the p median problems relative to both the euclidean and the rectilinear metrics are np hard in fact we prove that it is np hard even to approximate the p center problems sufficiently closely the reductions are from 3 satisfiability

symplectic diffeomorphisms and the flux homomorphism let x be a connected smooth manifold without boundary we will write difff ox for the group of all compactly supported diffeomorphisms of x which preserve the symplectic form o2 topologized as usual as the direct limit of the subgroups diff x of all diffeomorphisms with support in the fixed compact set k let diff oox be its identity component the flux is locally a homomorphism from a neighbourhood of the identity in diffo ox to h x r see calabi 6 it gives rise to a global homomorphism s o from the universal cover of difffo0x to hi x f and hence to a homomorphism

the monge ampere equation on almost complex manifolds we study the dirichlet problem for the monge ampere equation on almost complex manifolds we obtain the existence of the unique smooth solution in strictly pseudoconvex domains

problem based optimal scenario generation and reduction in stochastic programming scenarios are indispensable ingredients for the numerical solution of stochastic programs earlier approaches to optimal scenario generation and reduction are based on stability arguments involving distances of probability measures in this paper we review those ideas and suggest to make use of stability estimates based only on problem specific data for linear two stage stochastic programs we show that the problem based approach to optimal scenario generation can be reformulated as best approximation problem for the expected recourse function which in turn can be rewritten as a generalized semi infinite program we show that the latter is convex if either right hand sides or costs are random and can be transformed into a semi infinite program in a number of cases we also consider problem based optimal scenario reduction for two stage models and optimal scenario generation for chance constrained programs finally we discuss problem based scenario generation for the classical newsvendor problem