a greedy randomized adaptive search procedure for maximum independent set an efficient randomized heuristic for a maximum independent set is presented the procedure is tested on randomly generated graphs having from 400 to 3 500 vertices and edge probabilities from 0 2 to 0 9 the heuristic can be implemented trivially in parallel and is tested on an mimd computer with 1 2 4 and 8 processors computational results indicate that the heuristic frequently finds the optimal or expected optimal solution in a fraction of the time required by implementations of simulated annealing tabu search and an exact partial enumeration method

on the optimality of the median cut spectral bisection graph partitioning method recursive spectral bisection rsb is a heuristic technique for finding a minimum cut graph bisection to use this method the second eigenvector of the laplacian of the graph is computed and from it a bisection is obtained the most common method is to use the median of the components of the second eigenvector to induce a bisection we prove here that this median cut method is optimal in the sense that the partition vector induced by it is the closest partition vector in any ls norm for s ge1 to the second eigenvector moreover we prove that the same result also holds for any m partition that is a partition into m and n m vertices when using the mth largest or smallest components of the second eigenvector

generalisation du critere de beurling nyman pour l hypothese de riemann nous generalisons dans cet article le critere de beurling nyman qui concerne la fonction de riemann a une large classe de series de dirichlet nous etablissons donc une correspondance entre la densite d un certain sous espace de fonctions dans et la localisation des zeros d une serie de dirichlet nous utilisons pour obtenir ce resultat la structure de l espace de hardy du demi plan abstract we generalise beurling nyman s criterion already known for the riemann function to a larger class of dirichlet series we reveal a link between the density of some subspace of functions in and the localization of the zeros of a dirichlet series to do so we use the structure of the hardy space of the half plan

a tabu search approach to the constraint satisfaction problem as a general problem solver many combinatorial problems including a variety of combinatorial optimization problems can be naturally formulated as a constraint satisfaction problem csp we develop in this paper a tabu search based algorithm for the csp as a foundation for a general problem solver in addition to the basic components of tabu search we develop a number of elaborations such as an automatic control mechanism for the tabu tenure modification of the penalty function to handle objective functions and enlargement of the neighborhood by allowing swap operations computational results with our algorithm are reported for various problems selected from a wide range of applications i e graph coloring generalized assignment set covering timetabling and nurse scheduling our results appear to be competitive with those of existing algorithms specially developed for the respective problem domains

lifting and separation procedures for the cut polytope the max cut problem and the associated cut polytope on complete graphs have been extensively studied over the last 25 years however in comparison only little research has been conducted for the cut polytope on arbitrary graphs in particular separation algorithms have received only little attention in this study we describe new separation and lifting procedures for the cut polytope on general graphs these procedures exploit algorithmic and structural results known for the cut polytope on complete graphs to generate valid and sometimes facet defining inequalities for the cut polytope on arbitrary graphs in a cutting plane framework we report computational results on a set of well established benchmark problems

two edge connected spanning subgraphs and polyhedra this paper studies the problem of finding a two edge connected spanning subgraph of minimum weight this problem is closely related to the widely studied traveling salesman problem and has applications to the design of reliable communication and transportation networks we discuss the polytope associated with the solutions to this problem we show that when the graph is series parallel the polytope is completely described by the trivial constraints and the so called cut constraints we also give some classes of facet defining inequalities of this polytope when the graph is general

a technique for speeding up the solution of the lagrangian dual we propose techniques for the solution of the lp relaxation and the lagrangean dual in combinatorial optimization and nonlinear programming problems our techniques find the optimal solution value and the optimal dual multipliers of the lp relaxation and the lagrangean dual in polynomial time using as a subroutine either the ellipsoid algorithm or the recent algorithm of vaidya moreover in problems of a certain structure our techniques find not only the optimal solution value but the solution as well our techniques lead to significant improvements in the theoretical running time compared with previously known methods interior point methods ellipsoid algorithm vaidya s algorithm we use our method to the solution of the lp relaxation and the langrangean dual of several classical combinatorial problems like the traveling salesman problem the vehicle routing problem the steiner tree problem thek connected problem multicommodity flows network design problems network flow problems with side constraints facility location problems k polymatroid intersection multiple item capacitated lot sizing problem and stochastic programming in all these problems our techniques significantly improve the theoretical running time and yield the fastest way to solve them

graph partitioning using linear and semidefinite programming graph partition is used in the telecommunication industry to subdivide a transmission network into small clusters we consider both linear and semidefinite relaxations for the equipartition problem and present numerical results on real data from france telecom networks with up 900 nodes and also on randomly generated problems

second order cone relaxations for binary quadratic polynomial programs several types of relaxations for binary quadratic polynomial programs can be obtained using linear second order cone or semidefinite techniques in this paper we propose a general framework to construct conic relaxations for binary quadratic polynomial programs based on polynomial programming using our framework we re derive previous relaxation schemes and provide new ones in particular we present three relaxations for binary quadratic polynomial programs the first two relaxations based on second order cone and semidefinite programming represent a significant improvement over previous practical relaxations for several classes of nonconvex binary quadratic polynomial problems from a practical point of view due to the computational cost semidefinite based relaxations for binary quadratic polynomial problems can be used only to solve small to midsize instances to improve the computational efficiency for solving such problems we propose a third relaxation based purely on second order cone programming computational tests on different classes of nonconvex binary quadratic polynomial problems including quadratic knapsack problems show that the second order cone based relaxation outperforms the semidefinite based relaxations that are proposed in the literature in terms of computational efficiency and it is comparable in terms of bounds

computing the multiplicative group of residue class rings let k be a global field with maximal order 0k and let m0 be an ideal of 0k we present algorithms for the computation of the multiplicative group 0k m0 of the residue class ring 0k m0 and the discrete logarithm therein based on the explicit representation of the group of principal units we show how these algorithms can be combined with other methods in order to obtain more efficient algorithms they are applied to the computation of the ray class group clkm modulo m m0m where m denotes a formal product of real infinite places and also to the computation of conductors of ideal class groups and of discriminants and genera of class fields

solving matching problems with linear programming in this paper we describe an implementation of a cutting plane algorithm for the perfect matching problem which is based on the simplex method the algorithm has the following features it works on very sparse subgraphs ofkn which are determined heuristically global optimality is checked using the reduced cost criterion cutting plane recognition is usually accomplished by heuristics only if these fail the padberg rao procedure is invoked to guarantee finite convergence

an exact combinatorial algorithm for minimum graph bisection we present a novel exact algorithm for the minimum graph bisection problem whose goal is to partition a graph into two equally sized cells while minimizing the number of edges between them our algorithm is based on the branch and bound framework and unlike most previous approaches it is fully combinatorial we introduce novel lower bounds based on packing trees as well as a new decomposition technique that contracts entire regions of the graph while preserving optimality guarantees our algorithm works particularly well on graphs with relatively small minimum bisections solving to optimality several large real world instances with up to millions of vertices for the first time

bc opt a branch and cut code for mixed integer programs a branch and cut mixed integer programming system called bc opt is described incorporating most of the valid inequalities that have been used or suggested for such systems namely lifted 0 1 knapsack inequalities 0 1 gub knapsack and integer knapsack inequalities flowcover and continuous knapsack inequalities path inequalities for fixed charge network flow structure and gomory mixed integer cuts the principal development is a set of interface routines allowing these cut routines to generate cuts for new subsets or aggregations of constraints

on the cut polytope the cut polytopep c g of a graphg v e is the convex hull of the incidence vectors of all edge sets of cuts ofg we show some classes of facet defining inequalities ofp c g we describe three methods with which new facet defining inequalities ofp c g can be constructed from known ones in particular we show that inequalities associated with chordless cycles define facets of this polytope moreover for these inequalities a polynomial algorithm to solve the separation problem is presented we characterize the facet defining inequalities ofp c g ifg is not contractible tok 5 we give a simple characterization of adjacency inp c g and prove that for complete graphs this polytope has diameter one and thatp c g has the hirsch property a relationship betweenp c g and the convex hull of incidence vectors of balancing edge sets of a signed graph is studied

computing stark units for totally real cubic fields a method for computing provably accurate values of partial zeta functions is used to numerically confirm the rank one abelian stark conjecture for some totally real cubic fields of discriminant less than 50000 the results of these computations are used to provide explicit hilbert class fields and some ray class fields for the cubic extensions

graph partitioned spatial priors for functional magnetic resonance images abstract spatial models of functional magnetic resonance imaging fmri data allow one to estimate the spatial smoothness of general linear model glm parameters and eschew pre process smoothing of data entailed by conventional mass univariate analyses recently diffusion based spatial priors harrison l m penny w daunizeau j and friston k j 2008 diffusion based spatial priors for functional magnetic resonance images neuroimage were proposed which provide a way to formulate an adaptive spatial basis where the diffusion kernel of a weighted graph laplacian wgl is used as the prior covariance matrix over glm parameters an advantage of these is that they can be used to relax the assumption of isotropy and stationarity implicit in smoothing data with a fixed gaussian kernel the limitation of diffusion based models is purely computational due to the large number of voxels in a brain volume one solution is to partition a brain volume into slices using a spatial model for each slice this reduces computational burden by approximating the full wgl with a block diagonal form where each block can be analysed separately while fmri data are collected in slices the functional structures exhibiting spatial coherence and continuity are generally three dimensional calling for a more informed partition we address this using the graph laplacian to divide a brain volume into sub graphs whose shape can be arbitrary their shape depends crucially on edge weights of the graph which can be based on the euclidean distance between voxels isotropic or on glm parameters anisotropic encoding functional responses the result is an approximation the full wgl that retains its 3d form and also has potential for parallelism we applied the method to high resolution 1 mm3 fmri data and compared models where a volume was divided into either slices or graph partitions models were optimized using expectation maximization and the approximate log evidence computed to compare these different ways to partition a spatial prior the high resolution fmri data presented here had greatest evidence for the graph partitioned anisotropic model which was best able to preserve fine functional detail

semi supervised learning using greedy max cut graph based semi supervised learning ssl methods play an increasingly important role in practical machine learning systems particularly in agnostic settings when no parametric information or other prior knowledge is available about the data distribution given the constructed graph represented by a weight matrix transductive inference is used to propagate known labels to predict the values of all unlabeled vertices designing a robust label diffusion algorithm for such graphs is a widely studied problem and various methods have recently been suggested many of these can be formalized as regularized function estimation through the minimization of a quadratic cost however most existing label diffusion methods minimize a univariate cost with the classification function as the only variable of interest since the observed labels seed the diffusion process such univariate frameworks are extremely sensitive to the initial label choice and any label noise to alleviate the dependency on the initial observed labels this article proposes a bivariate formulation for graph based ssl where both the binary label information and a continuous classification function are arguments of the optimization this bivariate formulation is shown to be equivalent to a linearly constrained max cut problem finally an efficient solution via greedy gradient max cut ggmc is derived which gradually assigns unlabeled vertices to each class with minimum connectivity once convergence guarantees are established this greedy max cut based ssl is applied on both artificial and standard benchmark data sets where it obtains superior classification accuracy compared to existing state of the art ssl methods moreover ggmc shows robustness with respect to the graph construction method and maintains high accuracy over extensive experiments with various edge linking and weighting schemes

a mesh partitioning algorithm for preserving spatial locality in arbitrary geometries an algorithm for partitioning computational meshes is proposed the morton order space filling curve is modified to achieve improved locality a spatial locality metric is defined to compare results with existing approaches results indicate improved performance of the algorithm in complex geometries a space filling curve sfc is a proximity preserving linear mapping of any multi dimensional space and is widely used as a clustering tool equi sized partitioning of an sfc ignores the loss in clustering quality that occurs due to inaccuracies in the mapping often this results in poor locality within partitions especially for the conceptually simple morton order curves we present a heuristic that improves partition locality in arbitrary geometries by slicing a morton order curve at points where spatial locality is sacrificed in addition we develop algorithms that evenly distribute points to the extent possible while maintaining spatial locality a metric is defined to estimate relative inter partition contact as an indicator of communication in parallel computing architectures domain partitioning tests have been conducted on geometries relevant to turbulent reactive flow simulations the results obtained highlight the performance of our method as an unsupervised and computationally inexpensive domain partitioning tool

how good is recursive bisection the most commonly used p way partitioning method is recursive bisection rb it first divides a graph or a mesh into two equal sized pieces by a good bisection algorithm and then recursively divides the two pieces ideally we would like to use an optimal bisection algorithm because the optimal bisection problem that partitions a graph into two equal sized subgraphs to minimize the number of edges cut is np complete practical rb algorithms use more efficient heuristics in place of an optimal bisection algorithm most such heuristics are designed to find the best possible bisection within allowed time we show that the rb method even when an optimal bisection algorithm is assumed may produce a p way partition that is very far way from the optimal one our negative result is complemented by two positive ones first we show that for some important classes of graphs that occur in practical applications such as well shaped finite element and finite difference meshes rb is within a constant factor of the optimal one almost always second we show that if the balance condition is relaxed so that each block in the p way partition is bounded by 2n p where n is the number of vertices of the graph then a modified rb finds an approximately balanced p way partition whose cost is within an o log p factor of the cost of the optimal p way partition

solving airline crew scheduling problems by branch and cut the crew scheduling problem is one that has been studied almost continually for the past 40 years but all prior approaches have always approximated the problem of finding an optimal schedule for even the smallest of an airline s fleets the problem is especially important today since costs for flying personnel of major u s carriers have grown and now often exceed 1 3 billion a year and are the second largest item next to fuel cost of the total operating cost of major u s carriers thus even small percentage savings amount to substantial dollar amounts we present a branch and cut approach to solving to proven optimality large set partitioning problems arising within the airline industry we first provide some background related to this important application and then describe the approach for solving representative problems in this problem class the branch and cut solver generates cutting planes based on the underlying structure of the polytope defined by the convex hull of the feasible integer points and incorporates these cuts into a tree search algorithm that uses automatic reformulation procedures heuristics and linear programming technology to assist in the solution numerical experiments are reported for a sample of 68 large scale real world crew scheduling problems these problems include both pure set partitioning problems and set partitioning problems with side constraints these base constraints represent contractual labor requirements and have heretofore not been represented explicitly in the construction of crew schedules thus making it impossible to provide any measure of how far the obtained solution was from optimality an interesting result of obtaining less costly schedules is that the crews themselves are happier with the schedules because they spend more of their duty time flying than waiting on the ground

distribution requirements and compactness constraints in school timetabling this paper provides the following contributions first distribution requirements for lessons of different lengths are modelled in a novel way by the use of the so called multiple mode concept with mode identity constraints second we show that several types of constraints may be modelled using the unifying framework of partially renewable resources among these constraints are no class subject room and teacher overlaps class subject room and teacher unavailabilities compactness constraints preassignment constraints lectures to be given simultaneously lunch breaks etc third we present two phase parallel greedy randomized and genetic methods fourth we provide an instance generator for the generation of a representative set of instances fifth the generator along with a statistical model is used for a thorough experimental evaluation of the methods computational results show that the methods solve the instances investigated close to optimality

min max boundary domain decomposition domain decomposition is one of the most effective and popular parallel computing techniques for solving large scale numerical systems in the special case when the amount of computation in a subdomain is proportional to the volume of the subdomain domain decomposition amounts to minimizing the surface area of each subdomain while dividing the volume evenly motivated by this fact we study the following min max boundary multi way partitioning problem given a graph g and an integer k 1 we would like to divide g into k subgraphs g1 gk by removing edges such that i gi g k for all i 1 k and ii the maximum boundary size of any subgraph the set of edges connecting it with other subgraphs is minimized we provide an algorithm that given g a well shaped mesh in d dimensions finds a partition of g into k subgraphs g1 gk such that for all i gi has g k vertices and the number of edges connecting gi with the other subgraphs is o g k 1 1 d our algorithm can find such a partition in o g logk time finally we extend our results to vertex weighted and vertex based graph decomposition our results can be used to simultaneously balance the computational and memory loads on a distributed memory parallel computer without incurring large communication overhead

geometric separators for finite element meshes we propose a class of graphs that would occur naturally in finite element and finite difference problems and we prove a bound on separators for this class of graphs graphs in this class are embedded in d dimensional space in a certain manner for d dimensional graphs our separator bound is o n d 1 d which is the best possible bound we also propose a simple randomized algorithm to find this separator in o n time this separator algorithm can be used to partition the mesh among processors of a parallel computer and can also be used for the nested dissection sparse elimination algorithm

geometric mesh partitioning implementation and experiments we investigate a method of dividing an irregular mesh into equal sized pieces with few interconnecting edges the method s novel feature is that it exploits the geometric coordinates of the mesh vertices it is based on theoretical work of miller teng thurston and vavasis who showed that certain classes of well shaped finite element meshes have good separators the geometric method is quite simple to implement we describe a sc matlab code for it in some detail the method is also quite efficient and effective we compare it with some other methods including spectral bisection

separators for sphere packings and nearest neighbor graphs a collection of n balls in d dimensions forms a k ply system if no point in the space is covered by more than k balls we show that for every k ply system there is a sphere s that intersects at most o k 1 d n 1 1 d balls of and divides the remainder of into two parts those in the interior and those in the exterior of the sphere s respectively so that the larger part contains at most 1 1 d 2 n balls this bound of o k 1 d n 1 1 d is the best possible in both n and k we also present a simple randomized algorithm to find such a sphere in o n time our result implies that every k nearest neighbor graphs of n points in d dimensions has a separator of size o k 1 d n 1 1 d in conjunction with a result of koebe that every triangulated planar graph is isomorphic to the intersection graph of a disk packing our result not only gives a new geometric proof of the planar separator theorem of lipton and tarjan but also generalizes it to higher dimensions the separator algorithm can be used for point location and geometric divide and conquer in a fixed dimensional space

a greedy randomized adaptive search procedure for the two partition problem we present a greedy randomized adaptive search procedure grasp for the network 2 partition problem the heuristic is empirically compared with the kernighan lin k l method on a wide range of instances the grasp approach dominates k l with respect to solution value on a large percentage of the instances tested the ability of grasp to find optimal solutions is assessed by comparing its performance with a general purpose mixed integer programming package

explicit bounds for primes in residue classes let e k be an abelian extension of number fields with e q let and n denote the absolute discriminant and degree of e let denote an element of the galois group of e k we prove the following theorems assuming the extended riemann hypothesis 1 there is a degree 1 prime p of k such that p e k satisfying np 1 o 1 log 2n 2 2 there is a degree 1 prime p of k such that p e k generates the same group as satisfying np 1 o 1 log 2 3 for k q there is a prime p such that p e q satisfying p 1 o 1 log 2 in 1 and 2 we can in fact take p to be unramified in k q a special case of this result is the following 4 if gcd m q 1 the least prime p m mod q satisfies p 1 o 1 q log q 2 it follows from our proof that 1 3 also hold for arbitrary galois extensions provided we replace by its conjugacy class our theorems lead to explicit versions of 1 4 including the following the least prime p m mod q is less than 2 q log q 2

greedy prohibition and reactive heuristics for graph partitioning new heuristic algorithms are proposed for the graph partitioning problem a greedy construction scheme with an appropriate tie breaking rule min max greedy produces initial assignments in a very fast time for some classes of graphs independent repetitions of min max greedy are sufficient to reproduce solutions found by more complex techniques when the method is not competitive the initial assignments are used as starting points for a prohibition based scheme where the prohibition is chosen in a randomized and reactive way with a bias towards more successful choices in the previous part of the run the relationship between prohibition based diversification tabu search and the variable depth kernighan lin algorithm is discussed detailed experimental results are presented on benchmark suites used in the previous literature consisting of graphs derived from parametric models random graphs geometric graphs etc and of real world graphs of large size on the first series of graphs a better performance for equivalent or smaller computing times is obtained while on the large real world instances significantly better results than those of multilevel algorithms are obtained but for a much larger computational effort

combinatorial optimization current successes and directions for the future our ability to solve large important combinatorial optimization problems has improved dramatically in the past decade the availability of reliable software extremely fast and inexpensive hardware and high level languages that make the modeling of complex problems much faster have led to a much greater demand for optimization tools this paper highlights the major breakthroughs and then describes some very exciting future opportunities previously large research projects required major data collection efforts expensive mainframes and substantial analyst manpower now we can solve much larger problems on personal computers much of the necessary data is routinely collected and tools exist to speed up both the modeling and the post optimality analysis with the information technology revolution taking place currently we now have the opportunity to have our tools embedded into supply chain systems that determine production and distribution schedules process design and location allocation decisions these tools can be used industry wide with only minor modifications being done by each user

the equipartition polytope i formulations dimension and basic facets the following basic clustering problem arises in different domains ranging from physics statistics and boolean function minimization given a graphg v e and edge weightsce partition the setv into two sets of 1 2 v and 1 2 v nodes in such a way that the sum of the weights of edges not having both endnodes in the same set is maximized or minimized anequicut is a feasible solution of the above problem and theequicut polytope q g is the convex hull of the incidence vectors of equicuts ing in this paper we give some integer programming formulations of the equicut problem study the dimension of the equicut polytope and describe some basic classes of facet inducing inequalities forq g

zero free regions for dirichlet series in this paper we are interested in explicit zero free discs for some dirichlet series and we also study a general beurling nyman criterion for l functions our results generalize and improve previous results obtained by n nikolski and by a de roton as a concrete application we get for example a beurling nyman type criterion for the siegel zero problem

ranks and representations for spectral graph bisection we investigate the minimum graph bisection size according to spectral decompositions of adjacency matrices in several representations e g laplacian 01 adjacency seidel in addition we introduce a class of adjacency matrix representations by including a multiplicative vertex degree along the diagonal of the seidel adjacency matrix the new representation denoted s c often provided smaller spectral bisections in comparison with the other representations for certain classes of graphs by adjusting the multiplicative parameter c the cut size performance varied predictably between the seidel 01 adjacency and laplacian representations for both random graphs and random geometric graphs with various edge probability parameters furthermore we analyzed the cut sizes over many representations according to each eigenvector rank for large classes of random and real world graphs to give experimental evidence that the quality of spectral bisections depends on the eigenvector rank representation and graph type ultimately we provide evidence that particular ranks and representations are predictably more appropriate for use with certain types of graphs in general the related laplacian and s 2 0 representations produced the smallest cuts for geometric graphs whereas the seidel 01 adjacency and s c with 0 leq c leq1 produced the smallest cuts for random graphs we anticipate that our results will provide motivation for a unifying theorem of spectral bisection across eigenvector ranks adjacency representations and graph types

one half approximation algorithms for the k partition problem the k partition problem seeks to cluster the vertices of an edge weighted graph g v e into k sets of v k vertices each such that the total weight of the edges having both endpoints in the same cluster is maximized bottom up type heuristics based on matchings are presented for this problem these heuristics are shown to yield solutions that are at least one half the weight of the optimal solution for k equal to v 3 and v 4

computing automorphisms of abelian number fields let l q be an abelian number field of degree n most algorithms for computing the lattice of subfields of l require the computation of all the conjugates of this is usually achieved by factoring the minimal polynomial m x of over l in practice the existing algorithms for factoring polynomials over algebraic number fields can handle only problems of moderate size in this paper we describe a fast probabilistic algorithm for computing the conjugates of which is based on p adic techniques given m x and a rational prime p which does not divide the discriminant disc m x of m x the algorithm computes the frobenius automorphism of p in time polynomial in the size of p and in the size of m x by repeatedly applying the algorithm to randomly chosen primes it is possible to compute all the conjugates of

computing class fields via the artin map based on an explicit representation of the artin map for kummer extensions we present a method to compute arbitrary class fields as in the proofs of the existence theorem the problem is first reduced to the case where the field contains sufficiently many roots of unity using kummer theory and an explicit version of the artin reciprocity law we show how to compute class fields in this case we conclude with several examples

solution of large weighted equicut problems given a weighted undirected graph the equicut problem consists of finding a partition of the vertex set into two subsets of equal cardinality such that the sum of the weights of the edges belonging to the cut defined by the partition is minimized the problem is np hard and has several practical applications in recent years a number of algorithms based on metaheuristic techniques have been proposed in this work we first present a survey of the algorithms from the literature then we propose a new tabu search algorithm and compare it with the other heuristics through extensive computational experiments on several classes of graphs with up to 4000 nodes and 320 000 edges the results show that our approach easily determines the optimal solution for small graphs and its average performances are greatly superior to those of the other approximating algorithms

clique web facets for multicut polytopes let g v e be a graph an edge set uv e u si v sj i j where s1 sk is a partition of v is called a multicut with k shores we investigate the polytopes mck n and mck n that are defined as the convex hulls of the incidence vectors of all multicuts with at most k shores and at least k shores respectively of the complete graph kn we introduce a large class of inequalities called clique web inequalities valid for these polytopes and provide a quite complete characterization of those clique web inequalities that define facets of mck n and mck n using general facet manipulation techniques like collapsing and node splitting we construct further new classes of facets for these multicut polytopes we also exhibit a class of clique web facets for which the separation problem can be solved in polynomial time

spectral bounds for unconstrained 1 1 quadratic optimization problems abstract given an unconstrained quadratic optimization problem in the following form qp min x t qx x 1 1 n with q r n n we present different methods for computing bounds on its optimal objective value some of the lower bounds introduced are shown to generally improve over the one given by a classical semidefinite relaxation we report on theoretical results on these new bounds and provide preliminary computational experiments on small instances of the maximum cut problem illustrating their performance

computing ray class groups conductors and discriminants we use the algorithmic computation of exact sequences of abelian groups to compute the complete structure of z k m for an ideal m of a number field k as well as ray class groups of number fields and conductors and discriminants of the corresponding abelian extensions as an application we give several number fields with discriminants less than previously known ones

on computing inverse entries of a sparse matrix in an out of core environment the inverse of an irreducible sparse matrix is structurally full so that it is impractical to think of computing or storing it however there are several applications where a subset of the entries of the inverse is required given a factorization of the sparse matrix held in out of core storage we show how to compute such a subset e ciently by accessing only parts of the factors when there are many inverse entries to compute we need to guarantee that the overall computation scheme has reasonable memory requirements while minimizing the cost of loading the factors this leads to a partitioning problem that we prove is np complete we also show that we cannot get a close approximation to the optimal solution in polynomial time we thus need to develop heuristic algorithms and we propose i a lower bound on the cost of an optimum solution ii an exact algorithm for a particular case iii two other heuristics for a more general case and iv hypergraph partitioning models for the most general setting we illustrate the performance of our algorithms in practice using the mumps software package on a set of real life problems as well as some standard test matrices we show that our techniques can improve the execution time by a factor of 50 key words sparse matrices direct methods for linear systems and matrix inversion multifrontal method graphs and hypergraphs

an efficient memetic algorithm for themax bisection problem the max bisection problem consists in partitioning the vertices of a weighted undirected graph into two equally sized subsets so as to maximize the sum of the weights of crossing edges it is an np hard combinatorial optimization problem that arises in many applications in this paper we present a memetic algorithm for the max bisection problem which integrates a new fast local search procedure a crossover operator and a pool updating strategy these strategies achieve a balance between intensification and diversification extensive experiments were performed on a number of benchmark instances with 800 to 10 000 vertices from the literature the proposed memetic algorithm improved the best known solutions for all benchmark instances tested in this paper the improvement in terms of cut value over the circut by burer et al ranging from 0 02 to 4 15 percent and the average time of our proposed memetic algorithm is much lower than that of circut it shows that the proposed memetic algorithm can find high quality solutions in an acceptable running time

a decomposition method for quadratic zero one programming this paper proposes a decomposition method to compute a lower bound for unconstrained quadratic zero one minimization first we show that any quadratic function can be expressed as a sum of particular quadratic functions whose minima can be computed by a simple branch and bound algorithm then assuming some hypothesis we prove that among all possible decompositions the best one can be found by a lagrangian decomposition method moreover we show that our algorithm gives at least the roof dual bound and should give better results in practice eventually computational results and comparison with pardalos and rodgers algorithm demonstrate the efficiency of our method for medium size problems up to 100 variables

a probe based heuristic for graph partitioning a new heuristic algorithm probe ba which is based on the recently introduced metaheuristic paradigm population reinforced optimization based exploration probe is proposed for solving the graph partitioning problem the exploration part of probe ba is implemented by using the differential greedy algorithm of battiti and bertossi and a modification of the kernighan lin algorithm at the heart of bui and moon s genetic algorithm bfs gba experiments are used to investigate properties of probe and show that probe ba compares favorably with other solution methods based on genetic algorithms randomized reactive tabu search or more specialized multilevel partitioning techniques in addition probe ba finds new best cut values for 10 of the 34 instances in walshaw s graph partitioning archive

applications of cut polyhedra ii abstract in this paper and in its continuation part ii this issue we group within a unified framework many applications of the following polyhedra cut boolean quadric hypermetric and metric polyhedra we treat in particular the following applications l1 and l1 metrics in functional analysis the max cut problem the boole problem and multicommodity flow problems in combinatorial optimization lattice holes in geometry of numbers density matrices of many fermions systems in quantum mechanics we present some other applications in probability theory statistical data analysis and design theory in this first part after introducing the main definitions and operations for cut polyhedra we describe the connections with l1 metrics and with other metric properties and we consider in particular the applications to some classes of metrics arising from graphs normed spaces and lattices

facets for the cut cone i we study facets of the cut conec n i e the cone of dimension 1 2n n 1 generated by the cuts of the complete graph onn vertices actually the study of the facets of the cut cone is equivalent in some sense to the study of the facets of the cut polytope we present several operations on facets and in particular a lifting procedure for constructing facets ofc n 1 from given facets of the lower dimensional conec n after reviewing hypermetric valid inequalities we describe the new class of cycle inequalities and prove the facet property for several subclasses the new class of parachute facets is developed and other known facets and valid inequalities are presented

on the convergence rate of annealing processes for the class of inhomogeneous markov processes arising from simulated annealing it is shown that lim t to infty p x t i exp u i t t exists and is positive for each state i where t t is the temperature and u i is the energy level at i assuming that min u i 0 the method used is to consider the forward equations associated with such processes

a discrete filled function algorithm embedded with continuous approximation for solving max cut problems in this paper a discrete filled function algorithm embedded with continuous approximation is proposed to solve max cut problems a new discrete filled function is defined for max cut problems and properties of the function are studied in the process of finding an approximation to the global solution of a max cut problem a continuation optimization algorithm is employed to find local solutions of a continuous relaxation of the max cut problem and then global searches are performed by minimizing the proposed filled function unlike general filled function methods characteristics of max cut problems are used the parameters in the proposed filled function need not to be adjusted and are exactly the same for all max cut problems that greatly increases the efficiency of the filled function method numerical results and comparisons on some well known max cut test problems show that the proposed algorithm is efficient to get approximate global solutions of max cut problems

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

facets of the bipartite subgraph polytope the bipartite subgraph polytope pbg of a graph g v e is the convex hull of the incidence vectors of all edge sets of bipartite subgraphs of g we show that all complete subgraphs of g of odd order and all so called odd bicycle wheels contained in g induce facets of pbg moreover we describe several methods with which new facet defining inequalities of pbg can be constructed from known ones examples of these methods are contraction of node sets in odd complete subgraphs odd subdivision of edges certain splittings of nodes and subdivision of all edges of a cut using these methods we can construct facet defining inequalities of pbg having coefficients of order v 2

facets of the balanced acyclic induced subgraph polytope a signed graph is a graph whose edges are labelled positive or negative a signed graph is said to be balanced if the set of negative edges form a cut the balanced induced subgraph polytopep g of a graphg is the convex hull of the incidence vectors of all node sets that induce balanced subgraphs ofg in this paper we exhibit various rank facet defining inequalities we describe several methods with which new facet defining inequalities ofp g can be constructed from known ones finding a maximum weighted balanced induced subgraph of a series parallel graph is a polynomial problem we show that for this class of graphsp g may have complicated facet defining inequalities we derive analogous results for the polytope of acyclic induced subgraphs

a branch and cut algorithm for the equicut problem we describe an algorithm for solving the equicut problem on complete graphs the core of the algorithm is a cutting plane procedure that exploits a subset of the linear inequalities defining the convex hull of the incidence vectors of the edge sets that define an equicut the cuts are generated by several separation procedures that will be described in the paper whenever the cutting plane procedure does not terminate with an optimal solution the algorithm uses a branch and cut strategy we also describe the implementation of the algorithm and the interface with the lp solver finally we report on computational results on dense instances with sizes up to 100 nodes

the critical exponents of the two dimensional ising spin glass revisited exact ground state calculations and monte carlo simulations due to an incorrect generation of the bond configuration we obtained incorrect monte carlo results new simulation runs with a corrected and enlarged set of samples has led to the results shown in figures 1 and 2 which replace figures 3 and 4 of the original paper the best data collapse of xl we obtained see figure 1 for and of the binder ratio gl see figure 2 for figure 1 scaling plot of xl for different system sizes and temperatures the value of the correlation length exponent is v 1 5 thus the average value of the correlation length exponent now reads it should be noted that all other results of the paper where independently generated samples were used are not affected by these corrections figure 2 scaling plot of the binder ration gl we obtained the best data collapse for v 1 9 we note that the new estimate is much closer to the results 1 and 2 and is not compatible with the scaling prediction y 1 v

an application of combinatorial optimization to statistical physics and circuit layout design we study the problem of finding ground states of spin glasses with exterior magnetic field and the problem of minimizing the number of vias holes on a printed circuit board or contacts on a chip subject to pin preassignments and layer preferences the former problem comes up in solid state physics and the latter in very large scale integrated vlsi circuit design and in printed circuit board design both problems can be reduced to the max cut problem in graphs based on a partial characterization of the cut polytope we design a cutting plane algorithm and report on computational experience with it our method has been used to solve max cut problems on graphs with up to 1 600 nodes

using network flow techniques to solve an optimization problem from surface physics the solid on solid model provides a commonly used framework for the description of surfaces in recent it has been extended in order to investigate the effect of defects in the bulk on the roughness of the surface the determination of the ground state of this model leads to a combinatorial problem which is reduced to an uncapacitated convex minimum circulation problem we will show that the successive shortest path algorithm solves the problem in polynomial time

modeling and solving the two facility capacitated network loading problem this paper studies a topical and economically significant capacitated network design problem that arises in the telecommunications industry in this problem given point to point communication demand in a network must be met by installing loading capacitated facilities on the arcs loading a facility incurs an arc specific and facility dependent cost this paper develops modeling and solution approaches for loading facilities to satisfy the given demand at minimum cost we consider two approaches for solving the underlying mixed integer program a lagrangian relaxation strategy and a cutting plane approach that uses three classes of valid inequalities that we identify for the problem we show that a linear programming formulation that includes these inequalities always approximates the value of the mixed integer program at least as well as the lagrangian relaxation bound our computational results on a set of prototypical telecommunication data show that including these inequalities considerably improves the gap between the integer programming formulation and its linear programming relaxation from an average of 25 to an average of 8 these results show that strong cutting planes can be an effective modeling and algorithmic tool for solving problems of the size that arise in the telecommunications industry

new approaches for optimizing over the semimetric polytope the semimetric polytope is an important polyhedral structure lying at the heart of several hard combinatorial problems therefore linear optimization over the semimetric polytope is crucial for a number of relevant applications building on some recent polyhedral and algorithmic results about a related polyhedron the rooted semimetric polytope we develop and test several approaches based over lagrangian relaxation and application of non differentiable optimization algorithms for linear optimization over the semimetric polytope we show that some of these approaches can obtain very accurate primal and dual solutions in a small fraction of the time required for the same task by state of the art general purpose linear programming technology in some cases good estimates of the dual optimal solution but not of the primal solution can be obtained even quicker

bounding the probability of success of stochastic methods for global optimization abstract in this paper we establish some bounds for the probability that stimulated annealing produces an optimal or near optimal solution such bounds are given for both asymptotical and finite number of steps in the algorithm and they depend only on the instance of the problem to be treated then we compare its performance with a randomized local search showing that actually simulated annealing behaves worse than such a very simple global optimization technique furthermore since many parallel implementations of simulated annealing exist we also address its behavior in the parallel model of computation even in this case similar bounds hold and we can prove that the most simple parallel version of randomized local search is more likely to find optimal or near optimal solutions than any version of parallel simulated annealing

partitioning planar graphs a common problem in graph theory is that of dividing the vertices of a graph into two sets of prescribed size while cutting a minimum number of edges in this paper this problem is considered as it is restricted to the class of planar graphs let g be a planar graph on n vertices and s in 0 n be given an s partition of g is a partition of the vertex set of g into sets of size s and n s an optimals partition is an s partition that cuts the fewest number of edges the main result of this paper is an algorithm that finds an optimal s partition in time o b 2 n 3 2 4 5b where b is the number of edges cut by an optimal s partition in particular by letting s lfloor n 2 rfloor the immediate corollary that any planar graph with small o log n bisection width may be bisected in polynomial time is obtained furthermore suppose that a planar embedding hat g of g is also given such that the embedding of each biconnected component in hat g is at most m outerplanar such an embedding

project scheduling under partially renewable resource constraints we consider a generalization of the classical resource constrained project scheduling problem we introduce so called partially renewable resources by assuming for each resource a capacity on subsets of periods the concept of partially renewable resources is a fundamental tool in order to make e g timetabling and shift scheduling aspects amenable to project scheduling in addition partially renewable resources serve to model complicated labor regulations furthermore they cover traditional renewable and nonrenewable resource constraints as special cases we consider makespan minimization as objective for the exact solution of the problem we employ a basic enumeration scheme in order to speed up convergence we formulate bounds which take into account future resource consumption of partially renewable resources moreover we generalize the serial scheduling scheme in order to get fast approximation methods a rigorous assessment of the procedures is provided by solving progen instances generated under a full factorial test design besides the well known problem parameters we employ additionally three parameters which control the generation of partially renewable resources

an exact algorithm for graph partitioning an exact algorithm is presented for solving edge weighted graph partitioning problems the algorithm is based on a branch and bound method applied to a continuous quadratic programming formulation of the problem lower bounds are obtained by decomposing the objective function into convex and concave parts and replacing the concave part by an affine underestimate it is shown that the best affine underestimate can be expressed in terms of the center and the radius of the smallest sphere containing the feasible set the concave term is obtained either by a constant diagonal shift associated with the smallest eigenvalue of the objective function hessian or by a diagonal shift obtained by solving a semidefinite programming problem numerical results show that the proposed algorithm is competitive with state of the art graph partitioning codes

laplacian eigenvalues and the maximum cut problem we introduce and study an eigenvalue upper bound g on the maximum cut mc g of a weighted graph the function g has several interesting properties that resemble the behaviour of mc g the following results are presented we show that is subadditive with respect to amalgam and additive with respect to disjoint sum and 1 sum we prove that g is never worse that 1 131 mc g for a planar or more generally a weakly bipartite graph with nonnegative edge weights we give a dual characterization of g and show that g is computable in polynomial time with an arbitrary precision

testing the odd bicycle wheel inequalities for the bipartite subgraph polytope in their paper on facets of the bipartite subgraph polytope barahona grotschel and mahjoub pose the question is there a polynomial time algorithm testing the odd bicycle wheel inequalities this note describes such an algorithm

seizure warning algorithm based on optimization and nonlinear dynamics there is growing evidence that temporal lobe seizures are preceded by a preictal transition characterized by a gradual dynamical change from asymptomatic interictal state to seizure we herein report the first prospective analysis of the online automated algorithm for detecting the preictal transition in ongoing eeg signals such the algorithm constitutes a seizure warning system the algorithm estimates stlmax a measure of the order or disorder of the signal of eeg signals recorded from individual electrode sites the optimization techniques were employed to select critical brain electrode sites that exhibit the preictal transition for the warning of epileptic seizures specifically a quadratically constrained quadratic 0 1 programming problem is formulated to identify critical electrode sites the automated seizure warning algorithm was tested in continuous long term eeg recordings obtained from 5 patients with temporal lobe epilepsy for individual patient we use the first half of seizures to train the parameter settings which is evaluated by roc receiver operating characteristic curve analysis with the best parameter setting the algorithm applied to all cases predicted an average of 91 7 of seizures with an average false prediction rate of 0 196 per hour these results indicate that it may be possible to develop automated seizure warning devices for diagnostic and therapeutic purposes

a grasp heuristic for the mixed chinese postman problem abstract arc routing problems arps consist of finding a traversal on a graph satisfying some conditions related to the links of the graph in the chinese postman problem cpp the aim is to find a minimum cost tour closed walk traversing all the links of the graph at least once both the undirected cpp where all the links are edges that can be traversed in both ways and the directed cpp where all the links are arcs that must be traversed in a specified way are known to be polynomially solvable however if we deal with a mixed graph having edges and arcs the problem turns out to be np hard in this paper we present a heuristic algorithm for this problem the so called mixed cpp mcpp based on greedy randomized adaptive search procedure grasp techniques the algorithm has been tested and compared with other known and recent methods from the literature on a wide collection of randomly generated instances with up to 200 nodes and 600 links producing encouraging computational results as far as we know this is the best heuristic algorithm for the mcpp with respect to solution quality published up to now

a discrete filled function algorithm for approximate global solutions of max cut problems a discrete filled function algorithm is proposed for approximate global solutions of max cut problems a new discrete filled function is defined for max cut problems and the properties of the filled function are studied unlike general filled function methods using the characteristic of max cut problems the parameters in proposed filled function need not be adjusted this greatly increases the efficiency of the filled function method by combining a procedure that randomly generates initial points for minimization of the filled function the proposed algorithm can greatly reduce the calculation cost and be applied to large scale max cut problems numerical results on different sizes and densities test problems indicate that the proposed algorithm is efficient and stable to get approximate global solutions of max cut problems

a cutting plane algorithm for a clustering problem in this paper we consider a clustering problem that arises in qualitative data analysis this problem can be transformed to a combinatorial optimization problem the clique partitioning problem we have studied the latter problem from a polyhedral point of view and determined large classes of facets of the associated polytope these theoretical results are utilized in this paper we describe a cutting plane algorithm that is based on the simplex method and uses exact and heuristic separation routines for some of the classes of facets mentioned before we discuss some details of the implementation of our code and present our computational results we mention applications from e g zoology economics and the political sciences

maximization of submodular functions theory and enumeration algorithms abstract submodular functions are powerful tools to model and solve either to optimality or approximately many operational research problems including problems defined on graphs after reviewing some long standing theoretical results about the structure of local and global maxima of submodular functions cherenin s selection rules and his dichotomy algorithm we revise the above mentioned theory and show that our revision is useful for creating new non binary branching algorithms and finding either approximation solutions with guaranteed accuracy or exact ones

a cutting plane algorithm for the linear ordering problem the linear ordering problem is an np hard combinatorial optimization problem with a large number of applications including triangulation of input output matrices archaeological senation minimizing total weighted completion time in one machine scheduling and aggregation of individual preferences in a former paper we have investigated the facet structure of the 0 1 polytope associated with the linear ordering problem here we report on a new algorithm that is based on these theoretical results the main part of the algorithm is a cutting plane procedure using facet defining inequalities this procedure is combined with various heuristics and branch and bound techniques our computational results compare favorably with the results of existing codes in particular we could triangulate all input output matrices of size up to 60 60 available to us within acceptable time bounds

almost universal codes achieving ergodic mimo capacity within a constant gap this paper addresses the question of achieving capacity with lattice codes in multi antenna block fading channels when the number of fading blocks tends to infinity a design criterion based on the normalized minimum determinant is proposed for division algebra multi block space time codes over fading channels this plays a similar role to the hermite invariant for gaussian channels under maximum likelihood decoding it is shown that this criterion is sufficient to guarantee transmission rates within a constant gap from capacity both for deterministic channels and ergodic fading channels moreover if the number of receive antennas is greater than or equal to the number of transmit antennas the same constant gap is achieved under naive lattice decoding as well in the case of independent identically distributed rayleigh fading the error probability vanishes exponentially fast in contrast to the standard approach in the literature which employs random lattice ensembles the existence results in this paper are derived from the number theory first the gap to capacity is shown to depend on the discriminant of the chosen division algebra then class field theory is applied to build families of algebras with small discriminants the key element in the construction is the choice of a sequence of division algebras whose centers are number fields with small root discriminants

computing the hilbert class field of real quadratic fields using the units appearing in stark s conjectures on the values of l functions at s 0 we give a complete algorithm for computing an explicit generator of the hilbert class field of a real quadratic field

minimizing representations over number fields ii computations in the brauer group abstract finding minimal fields of definition for representations of finite groups is one of the most important unsolved problems of computational representation theory while good methods exist for representations over finite fields it is still an open question in the case of number fields continuing and extending previous work we give a practical method for finding defining fields of minimal degree for absolutely irreducible representations the method is based on techniques from galois cohomology and the use of an explicit form of a weak grunwald wang theorem

zero free regions for dirichlet series ii in this paper we continue some work devoted to explicit zero free discs for a large class of dirichlet series in a previous article such zero free regions were described using some spaces of functions which were defined with some technical conditions here we give two different natural ways in order to remove those technical conditions in particular this allows to right down explicit zero free regions differently and to obtain for them an easier description useful for direct applications