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

global solution and blow up of the stochastic nonlinear schrodinger system we study a stochastic nonlinear schrodinger system with multiplicative white noise in energy space h1 based on deterministic and stochastic strichartz estimates we prove the local well posedness of a mild solution then we prove the global well posedness in the mass subcritical case and the defocusing case for the mass subcritical case we also investigate the global existence when the l2 norm of the initial value is small enough we also study the blow up phenomenon and give a sharp criterion via a general virial identity

on point covers of c oriented polygons let s be any family of n c oriented polygons of the two dimensional euclidean plane e2 i e bounded intersection of halfplanes whose normal directions of edges belong to a fixed collection of c distinct directions let s denote the packing number of s that is the maximum number of pairwise disjoint objects of s let s be the transversal number of s that is the minimum number of points required so that each object contains at least one of those points we provide linear time algorithms t n c t nc for a fat c oriented polytopes translates or homothets of ed proving that g 2 c o a d g 2 c dd and g 2 c 3d3 2 d respectively

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

approximation algorithms for submodular vertex cover problems with linear submodular penalties using primal dual technique the notion of penalty has been introduced into many combinatorial optimization models in this paper we consider the submodular vertex cover problems with linear and submodular penalties which are two variants of the submodular vertex cover problem where not all the edges are required to be covered by a vertex cover and the uncovered edges are penalized the problem is to determine a vertex subset to cover some edges and penalize the uncovered edges such that the total cost including covering and penalty is minimized to overcome the difficulty of implementing the primal dual framework directly we relax the two dual programs to slightly weaker versions we then present two primal dual approximation algorithms with approximation ratios of 2 and 4 respectively

extracting embedded generalized networks from linear programming problems if a linear program lp possesses a large generalized network gn submatrix this structure can be exploited to decrease solution time the problems of finding maximum sets of gn constraints and finding maximum embedded gn submatrices are shown to be np complete indicating that reliable efficient solution of these problems is difficult therefore efficient heuristic algorithms are developed for identifying such structure and are tested on a selection of twenty three real world problems the best of four algorithms for identifying gn constraint sets finds a set which is maximum in twelve cases and averages 99 1 of maximum on average the gn constraints identified comprise more than 62 3 of the total constraints in these problems the algorithm for identifying embedded gn submatrices finds submatrices whose sizes rows plus columns average 96 8 of an lp upper bound over 91 3 of the total constraint matrix was identified as a gn submatrix in these problems on average

reference points and approximation algorithms in multicriteria discrete optimization abstract mathematical research on multicriteria optimization problems predominantly revolves around the set of pareto optimal solutions in practice on the other hand methods that output a single solution are more widespread reference point methods are a successful example of this approach and are widely used in real world multicriteria optimization a reference point solution is the solution closest to a given reference point in the objective space we study the connection between reference point methods and approximation algorithms for multicriteria optimization problems over discrete sets in particular we establish that in terms of computational complexity computing approximate reference point solutions is polynomially equivalent to approximating the pareto set complementing these results we show for a number of general algorithmic techniques in single criteria optimization how they can be lifted to reference point optimization in particular we lift the link between dynamic programming and fptas as well as certain lp rounding techniques the latter applies e g to set cover and several machine scheduling problems

surrogate constraint normalization for the set covering problem the set covering problem scp is central in a wide variety of practical applications for which finding good feasible solutions quickly often in real time is crucial surrogate constraint normalization is a classical technique used to derive appropriate weights for surrogate constraint relaxations in mathematical programming this framework remains the core of the most effective constructive heuristics for the solution of the scp chiefly represented by the widely used chvatal method this paper introduces a number of normalization rules and demonstrates their superiority to the classical chvatal rule especially when solving large scale and real world instances directions for new advances on the creation of more elaborate normalization rules for surrogate heuristics are also provided

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

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

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

a tale of two methods we describe two widely used methods for the design and analysis of approximation algorithms the primal dual schema and the local ratio technique we focus on the creation of both methods by revisiting two results by bar yehuda and even the linear time primal dual approximation algorithm for set cover and its local ratio interpretation we also follow the evolution of the two methods by discussing more recent studies

instantons in the burgers equation the instanton solution for the forced burgers equation is found this solution describes the exponential tail of the probability distribution function of velocity differences in the region where shock waves are absent that is for large positive velocity differences the results agree with the one found recently by polyakov who used the operator product conjecture if this conjecture is true then our wkb asymptotics of the wyld functional integral should be exact to all orders of perturbation expansion around the instanton solution we also generalized our solution for the arbitrary dimension of the burgers kpz equation as a result we found the asymptotics of the angular dependence of the velocity difference probability distribution function textcopyright 1996 the american physical society

from valid inequalities to heuristics a unified view of primal dual approximation algorithms in covering problems in recent years approximation algorithms based on primal dual methods have been successfully applied to a broad class of discrete optimization problems in this paper we propose a generic primal dual framework to design and analyze approximation algorithms for integer programming problems of the covering type that uses valid inequalities in its design the worst case bound of the proposed algorithm is related to a fundamental relationship called strength between the set of valid inequalities and the set of minimal solutions to the covering problems in this way we can construct an approximation algorithm simply by constructing the required valid inequalities we apply the proposed algorithm to several problems such as covering problems related to totally balanced matrices cyclic scheduling vertex cover general set covering intersections of polymatroids and several network design problems attaining in most cases the best worst case bound known in the literature

on the complexity of learning lexicographic strategies fast and frugal heuristics are well studied models of bounded rationality psychological research has proposed the take the best heuristic as a successful strategy in decision making with limited resources take the best searches for a sufficiently good ordering of cues or features in a task where objects are to be compared lexicographically we investigate the computational complexity of finding optimal cue permutations for lexicographic strategies and prove that the problem is np complete it follows that no efficient that is polynomial time algorithm computes optimal solutions unless p np we further analyze the complexity of approximating optimal cue permutations for lexicographic strategies we show that there is no efficient algorithm that approximates the optimum to within any constant factor unless p np the results have implications for the complexity of learning lexicographic strategies from examples they show that learning them in polynomial time within the model of agnostic probably approximately correct pac learning is impossible unless rp np we further consider greedy approaches for building lexicographic strategies and determine upper and lower bounds for the performance ratio of simple algorithms moreover we present a greedy algorithm that performs provably better than take the best tight bounds on the sample complexity for learning lexicographic strategies are also given in this article

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

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

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

an extension of the lova sz local lemma and its applications to integer programming the lova sz local lemma due to erdods and lova sz infinite and finite sets colloq math soc j bolyai 11 1975 pp 609 627 is a powerful tool in proving the existence of rare events we present an extension of this lemma which works well when the event to be shown to exist is a conjunction of individual events each of which asserts that a random variable does not deviate much from its mean as applications we consider two classes of np hard integer programs minimax and covering integer programs a key technique randomized rounding of linear relaxations was developed by raghavan and thompson combinatorica 7 1987 pp 365 374 to derive good approximation algorithms for such problems we use our extension of the local lemma to prove that randomized rounding produces with nonzero probability much better feasible solutions than known before if the constraint matrices of these integer programs are column sparse e g routing using short paths problems on hypergraphs with small dimension degree this complements certain well known results from discrepancy theory we also generalize the method of pessimistic estimators due to raghavan j comput system sci 37 1988 pp 130 143 to obtain constructive algorithmic versions of our results for covering integer programs

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

on the hardness of optimization in power law graphs our motivation for this work is the remarkable discovery that many large scale real world graphs ranging from internet and world wide web to social and biological networks appear to exhibit a power law distribution the number of nodes y i of a given degree i is proportional to i b where b 0 is a constant that depends on the application domain there is practical evidence that combinatorial optimization in power law graphs is easier than in general graphs prompting the basic theoretical question is combinatorial optimization in power law graphs easy does the answer depend on the power law exponent b our main result is the proof that many classical np hard graph theoretic optimization problems remain np hard on power law graphs for certain values of b in particular we show that some classical problems such as clique and coloring remain np hard for all b 1 moreover we show that all the problems that satisfy the so called optimal substructure property remain np hard for all b 0 this includes classical problems such as minimum vertex cover maximum independent set and minimum dominating set our proofs involve designing efficient algorithms for constructing graphs with prescribed degree sequences that are tractable with respect to various optimization problems

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

circulation statistics in three dimensional turbulent flows we study the large ensuremath lambda limit of the loop dependent characteristic functional z ensuremath lambda mathrm exp i ensuremath lambda ensuremath oint c stackrel ensuremath rightarrow v ensuremath cdot d stackrel ensuremath rightarrow x related to the probability density function pdf of the circulation around a closed contour c the analysis is carried out in the framework of the martin siggia rose field theory formulation of the turbulence problem by means of the saddle point technique axisymmetric instantons labeled by the component ensuremath sigma mathrm zz of the strain field char22 a partially annealed variable in our formalism char22 are obtained for a circular loop in the x ensuremath y plane with radius defined in the inertial range fluctuations of the velocity field around the saddle point solutions are relevant leading to the lorentzian asymptotic behavior z ensuremath lambda ensuremath sim 1 ensuremath lambda 2 the o 1 ensuremath lambda 4 subleading correction and the asymmetry between right and left pdf tails due to parity breaking mechanisms are also investigated

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

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

local majorities coalitions and monopolies in graphs a review this paper provides an overview of recent developments concerning the process of local majority voting in graphs and its basic properties from graph theoretic and algorithmic standpoints

probabilistic analysis of a combined aggregation and math programming heuristic for a general class of vehicle routing and scheduling problems we propose and analyze a heuristic that uses region partitioning and an aggregation scheme for customer attributes load size time windows etc to create a finite number of customer types a math program is solved based on these aggregated customer types to generate a feasible solution to the original problem the problem class we address is quite general and defined by a number of general consistency properties problems in this class include vrps with general distance norms capacitated problems time window vrps pick up and delivery problems combined inventory control and routing problems and arc routing we provide a probabilistic analysis of this heuristic under very general probabilistic assumptions in particular we do not require independence between customer locations and their various attributes the heuristic is a s optimal as the number of customers n tends to infinity further it runs in o n log n time for a fixed relative error and can be designed to be asymptotically optimal w

turbulence without pressure we develop exact field theoretic methods to treat turbulence when the effect of pressure is negligible we find explicit forms of certain probability distributions demonstrate that the breakdown of galilean invariance is responsible for intermittency and establish the operator product expansion we also indicate how the effects of pressure can be turned on perturbatively

robust efficiency measures for linear knapsack problem variants abstract numerous combinatorial optimization applications such as the mobile retailer product mix problem utilize the multidemand multidimensional knapsack problem variant in which there are multiple knapsack constraints requiring a weighted summation of the variables to be less than or equal to a nonnegative value and multiple demand constraints requiring a weighted summation of the variables to be greater than or equal to a nonnegative value the purpose of this paper is to demonstrate that core variables and efficiency measures concepts used in the most efficient solvers to date for binary knapsack problems and some of its variants can be extended to the multidemand multidimensional knapsack problem variant specifically new efficiency measure calculations are provided and their properties are mathematically proven and experimentally demonstrated the contribution of such measures to knapsack problem research is that these measures are applicable to all knapsack problem variants with a single linear objective function and linear constraints of any quantity the applicability of these new measures is demonstrated through the development of three heuristic procedures a fixed core heuristic a genetic algorithm and a kernel search heuristic the results from these tests are compared with the results from a commercial solver and an existing heuristic the findings from these tests demonstrate that the fixed core and kernel search heuristics developed for this paper are the most efficient solvers to date for hard multidemand multidimensional knapsack problems

normalized solutions and asymptotical behavior of minimizer for the coupled hartree equations abstract we prove the existence and nonexistence of l 2 r n normalized solutions of coupled hartree equations which arisen from the studies of the nonlinear optics and multi component bose einstein condensates under certain type trapping potentials by proving some delicate energy estimates we give a precise description on the concentration behavior of minimizer solutions of the system furthermore an optimal blowing up rate for the minimizer solutions of the system is also proved

solving integer programs over monotone inequalities in three variables a framework for half integrality and good approximations abstract we define a class of monotone integer programs with constraints that involve up to three variables each a generic constraint in such integer program is of the form ax by z c where a and b are nonnegative and the variable z appears only in that constraint we devise an algorithm solving such problems in time polynomial in the length of the input and the range of variables u the solution is obtained from a minimum cut on a graph with o nu nodes and o mu arcs where n is the number of variables of the types x and y and m is the number of constraints our algorithm is also valid for nonlinear objective functions nonmonotone integer programs are optimization problems with constraints of the type ax by z c without restriction on the signs of a and b such problems are in general np hard we devise here an algorithm relying on a transformation to the monotone case that delivers half integral superoptimal solutions in polynomial time such solutions provide bounds on the optimum value that can only be superior to bounds provided by linear programming relaxation when the half integral solution can be rounded to an integer feasible solution this is a 2 approximate solution in that the technique is a unified 2 approximation technique for a large class of problems the results apply also for general integer programming problems with worse approximation factors that depend on a quantifier measuring how far the problem is from the class of problems we describe the algorithm described here has a wide array of problem applications an additional important consequence of our results is that nonmonotone problems in the framework are max snp hard and at least as hard to approximate as vertex cover problems that are amenable to the analysis provided here are easily recognized the analysis itself is entirely technical and involves manipulating the constraints and transforming them to a totally unimodular system while losing no more than a factor of 2 in the integrality

computational experience with approximation algorithms for the set covering problem the set covering problem scp is a well known combinatorial optimization problem which is np hard we conducted a comparative study of eight different approximation algorithms for the scp including several greedy variants fractional relaxations randomized algorithms and a neural network algorithm the algorithms were tested on a set of random generated problems with up to 500 rows and 5000 columns and on two sets of problems originating in combinatorial questions with up to 28160 rows and 11264 columns on the random problems and on one set of combinatorial problems the best algorithm among those we tested was the neural network algorithm with greedy variants very close in second and third place on the other set of combinatorial problems the best algorithm was a greedy variant and the neural network performed quite poorly the other algorithms we tested were always inferior to the ones mentioned above

a graph approximation heuristic for the vertex cover problem on planar graphs abstract the vertex cover problem is hard in general and remains so on planar graphs on the other hand it is always solvable on bipartite graphs in this paper we introduce a strategy which creates from an arbitrary planar graph g a minimum size bipartite homeomorph of g say g h we solve the vertex cover problem on g h and convert the solution to a cover on g tests on a set of sample graphs suggest that the resulting heuristic performs quite well

approximation algorithms for the set covering and vertex cover problems we propose a heuristic that delivers in o n 3 steps a solution for the set covering problem the value of which does not exceed the maximum number of sets covering an element times the optimal value

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

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

combinatorics for smaller kernels the differential of a graph let g v e be a graph of order n and let b d be the set of vertices in v d that have a neighbor in the vertex set d the differential of d is defined as d b d d and the differential of a graph g written g is equal to max d d v if g is connected and n 3 g n 5 is known this immediately leads to a linear vertex kernel result in the terminology of parameterized complexity for the problem of deciding whether g k taking k as the parameter we then establish a new combinatorial result which establishes that g n 4 if g is a connected graph of order n 6 and if g contains no induced path of five vertices whose midpoint is a cut vertex and whose endpoints have degree one this technical combinatorial theorem can be used to derive an even smaller linear vertex kernel for general graphs also we show that the related maximization problem allows for a polynomial time factor 14 approximation algorithm

observability transition in real networks we consider the observability model in networks with arbitrary topologies we introduce a system of coupled nonlinear equations valid under the locally treelike ansatz to describe the size of the largest observable cluster as a function of the fraction of directly observable nodes present in the network we perform a systematic analysis on 95 real world graphs and compare our theoretical predictions with numerical simulations of the observability model our method provides almost perfect predictions in the majority of the cases even for networks with very large values of the clustering coefficient potential applications of our theory include the development of efficient and scalable algorithms for real time surveillance of social networks and monitoring of technological networks

efficient approximation algorithms for maximum coverage with group budget constraints abstract given a ground set u with a non negative weight w i for each i u a positive integer k and a collection of sets s which is partitioned into a family of disjoint groups g the goal of the maximum coverage problem with group budget constraints mcg is to select k sets from s such that the total weight of the union of the k sets is maximized and at most one set is selected from each group g g we first present an approximation algorithm with a factor 1 1 e and an exponential time via randomized linear programming rounding technique then we improve the time complexity of the algorithm to o m n 3 5 l k 3 5 q 7 l for s m u n and l being the length of the input by the key idea of modeling the selection of groups as computing a constrained flow in a corresponding auxiliary graph the algorithm is later shown can be extended to solve two generalizations of mcg last but not the least we present another algorithm with a time complexity o m n 3 5 l k 10 5 l and a slightly increased approximation ratio 1 e 1 1 mainly based on the idea of partition where 2 is a parameter tuning which can balance the time complexity and the ratio

a critical review of discrete filled function methods in solving nonlinear discrete optimization problems many real life problems can be modeled as nonlinear discrete optimization problems such problems often have multiple local minima and thus require global optimization methods due to high complexity of these problems heuristic based global optimization techniques are usually required when solving large scale discrete optimization or mixed discrete optimization problems one of the more recent global optimization tools is known as the discrete filled function method nine variations of the discrete filled function method in literature are identified and a review on theoretical properties of each method is given some of the most promising filled functions are tested on various benchmark problems numerical results are given for comparison

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

ground state of n coupled nonlinear schrodinger equations in mathbb r n n leq 3 we establish some general theorems for the existence and nonexistence of ground state solutions of steady state n coupled nonlinear schrodinger equations the sign of coupling constants ij s is crucial for the existence of ground state solutions when all ij s are positive and the matrix is positively definite there exists a ground state solution which is radially symmetric however if all ij s are negative or one of ij s is negative and the matrix is positively definite there is no ground state solution furthermore we find a bound state solution which is non radially symmetric when n 3

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

instantons and intermittency we describe the method for finding the non gaussian tails of the probability distribution function pdf for solutions of a stochastic differential equation such as the convection equation for a passive scalar the random driven navier stokes equation etc the existence of such tails is generally regarded as a manifestation of the intermittency phenomenon our formalism is based on the wkb approximation in the functional integral for the conditional probability of large fluctuation we argue that the main contribution to the functional integral is given by a coupled field force configuration char22 the instanton as an example we examine the correlation functions of the passive scalar u advected by a large scale velocity field ensuremath delta correlated in time we find the instanton determining the tails of the generating functional and show that it is different from the instanton that determines the probability distribution function of high powers of u we discuss the simplest instantons for the navier stokes equation textcopyright 1996 the american physical society

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

generalized submodular cover problems and applications the greedy approach has been successfully applied in the past to produce logarithmic ratio approximations to np hard problems under certain conditions the problems for which these conditions hold are known as submodular cover problems the current paper3extends the applicability of the greedy approach to wider classes of problems the usefulness of our extensions is illustrated by giving new approximate solutions for two different types of problems the first problem is that of finding the spanning tree of minimum weight among those whose diameter is bounded by d a logarithmic ratio approximation algorithm is given for the cases of d 4 and 5 this approximation ratio is also proved to be the best possible unless p np the second type involves some known and new center selection problems for which new logarithmic ratio approximation algorithms are given again it is shown that the ratio must be at least logarithmic unless p np

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

algorithms of discrete optimization and their application to problems with fuzzy coefficients an approach to solving optimization problems with fuzzy coefficients in objective functions and constraints is described it consists in formulating and solving one and the same problem within the framework of mutually related models with constructing equivalent analogs with fuzzy coefficients in objective functions alone it enables one to maximally cut off dominated alternatives from below as well as from above since the approach is applied within the context of fuzzy discrete optimization problems several modified algorithms of discrete optimization are discussed these algorithms are associated with the method of normalized functions are based on a combination of formal and heuristic procedures and allow one to obtain quasi optimal solutions after a small number of steps thus overcoming the computational complexity posed the np completeness of discrete optimization problems the subsequent contraction of the decision uncertainty regions is associated with reduction of the problem to multiobjective decision making in a fuzzy environment with using techniques based on fuzzy preference relations the techniques are also directly applicable to situations in which the decision maker is required to choose alternatives from a set of explicitly available alternatives the results of the paper are of a universal character and can be applied to the design and control of systems and processes of different purposes as well as the enhancement of corresponding cad cam systems and intelligent decision making systems the results of the paper are already being used to solve problems of power engineering

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

rounding algorithms for covering problems in the last 25 years approximation algorithms for discrete optimization problems have been in the center of research in the fields of mathematical programming and computer science recent results from computer science have identified barriers to the degree of approximability of discrete optimization problems unless p np as a result as far as negative results are concerned a unifying picture is emerging on the other hand as far as particular approximation algorithms for different problems are concerned the picture is not very clear different algorithms work for different problems and the insights gained from a successful analysis of a particular problem rarely transfer to another our goal in this paper is to present a framework for the approximation of a class of integer programming problems covering problems through generic heuristics all based on rounding deterministic using primal and dual information or randomized but with nonlinear rounding functions of the optimal solution of a linear programming lp relaxation we apply these generic heuristics to obtain in a systematic way many known as well as new results for the set covering facility location general covering network design and cut covering problems 1998 the mathematical programming society inc published by elsevier science b v

the multidimensional 0 1 knapsack problem an overview abstract the multidimensional 0 1 knapsack problem is one of the most well known integer programming problems and has received wide attention from the operational research community during the last four decades although recent advances have made possible the solution of medium size instances solving this np hard problem remains a very interesting challenge especially when the number of constraints increases this paper surveys the main results published in the literature the focus is on the theoretical properties as well as approximate or exact solutions of this special 0 1 program

clustering through continuous facility location problems we consider the continuous facility location problem conflp given a finite set of clients c r d and a number f r conflp consists in opening a set f r d of facilities each at cost f and connecting each client to an open facility the objective is to minimize the costs of opening facilities and connecting clients we reduce conflp to the standard facility location problem flp by using the so called approximate center sets this reduction preserves the approximation except for an error e and implies 1 488 e and 2 04 e approximations when the connection cost is given by the euclidean distance and the squared euclidean distance respectively moreover we obtain approximate center sets for the case that the connection cost is the th power of the euclidean distance achieving approximations for the corresponding problems for any 1 as a byproduct we also obtain a polynomial time approximation scheme for the k median problem with this cost function for any fixed k

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

blow up behavior of ground states for a nonlinear schrodinger system with attractive and repulsive interactions abstract we consider a nonlinear schrodinger system arising in a two component bose einstein condensate bec with attractive intraspecies interactions and repulsive interspecies interactions in r 2 we get ground states of this system by solving a constrained minimization problem for some kinds of trapping potentials we prove that the minimization problem has a minimizer if and only if the attractive interaction strength a i i 1 2 of each component of the bec system is strictly less than a threshold a furthermore as a 1 a 2 a a the asymptotical behavior for the minimizers of the minimization problem is discussed our results show that each component of the bec system concentrates at a global minimum of the associated trapping potential

the multicovering problem abstract the multicovering problem is min cx subject to ax b x 0 where a is a matrix whose elements are all zero or one and b is a vector of positive integers this problem has many applications in scheduling location and business areas a variety of industrial commercial and military contexts give rise to large problems of this kind frequently at or beyond the limits of current computational tractability our research goal is the development of a special purpose algorithm capable of solving large problems of this kind conceptual and computational inputs to the algorithm arise from three sources from the generalization of computational tools developed for more specific problems in particular the set covering problem from the adaptation of general integer programming and combinatorial optimization methods and from the development within this paper of special techniques designed to take advantage of the structure of the problem studied following a description of the algorithm we give some computational results for randomly generated problems

the greedy procedure for resource allocation problems necessary and sufficient conditions for optimality in many resource allocation problems the objective is to allocate discrete resource units to a set of activities so as to maximize a concave objective function subject to upper bounds on the total amounts allotted to certain groups of activities if the constraints determine a polymatroid and the objective is linear it is well known that the greedy procedure results in an optimal solution in this paper we extend this result to objectives that are weakly concave a property generalizing separable concavity we exhibit large classes of models for which the set of feasible solutions is a polymatroid and for which efficient implementations of the greedy procedure can be given