jamming model for the extremal optimization heuristic extremal optimization a recently introduced meta heuristic for hard optimization problems is analysed on a simple model of jamming the model is motivated first by the problem of finding lowest energy configurations for a disordered spin system on a fixed valence graph the numerical results for the spin system exhibit the same phenomenology found in all earlier studies of extremal optimization and our analytical results for the model reproduce many of these features

bipartitioning of random graphs of fixed extensive valence the procedure of fu and anderson ibid vol 19 p 1605 1986 for the application of statistical mechanics to the problem of bipartitioning random graphs is extended to graphs of fixed extensive valence the cost function is shown to be independent of whether the valence is locally or globally constrained

coordination of cluster ensembles via exact methods we present a novel optimization based method for the combination of cluster ensembles for the class of problems with intracluster criteria such as minimum sum of squares clustering mssc we propose a simple and efficient algorithm called examce for this class of problems that is inspired from a set partitioning formulation of the original clustering problem we prove some theoretical properties of the solutions produced by our algorithm and in particular that under general assumptions though the algorithm recombines solution fragments so as to find the solution of a set covering relaxation of the original formulation it is guaranteed to find better solutions than the ones in the ensemble for the mssc problem in particular a prototype implementation of our algorithm found a new better solution than the previously best known for 21 of the test instances of the 40 instance tsplib benchmark data sets used in and and found a worse quality solution than the best known only five times for other published benchmark data sets where the optimal mssc solution is known we match them the algorithm is particularly effective when the number of clusters is large in which case it is able to escape the local minima found by k means type algorithms by recombining the solutions in a set covering context we also establish the stability of the algorithm with extensive computational experiments by showing that multiple runs of examce for the same clustering problem instance produce high quality solutions whose adjusted rand index is consistently above 0 95 finally in experiments utilizing external criteria to compute the validity of clustering examce is capable of producing high quality results that are comparable in quality to those of the best known clustering algorithms

universality of level correlation function of sparse random matrices the statistical properties of sparse random matrices ensembles are investigated by means of a supersymmetric approach with the use of a functional generalization of the hubbard stratonovich transformation when used to calculate the density of states the method is shown to be absolutely equivalent to the replica trick the model turns out to bear a close resemblance to the anderson model on the bethe lattice it possesses a delocalization transition that occurs with an increase in the mean connectivity parameter in the delocalized phase the level level correlation function proves to have a universal dyson form with the full density of states replaced by the contribution from the infinite cluster

taxicab correspondence analysis taxicab correspondence analysis is based on the taxicab singular value decomposition of a contingency table and it shares some similar properties with correspondence analysis it is more robust than the ordinary correspondence analysis because it gives uniform weights to all the points the visual map constructed by taxicab correspondence analysis has a larger sweep and clearer perspective than the map obtained by correspondence analysis two examples are provided

new and efficient dca based algorithms for minimum sum of squares clustering the purpose of this paper is to develop new efficient approaches based on dc difference of convex functions programming and dca dc algorithm to perform clustering via minimum sum of squares euclidean distance we consider the two most widely used models for the so called minimum sum of squares clustering mssc in short that are a bilevel programming problem and a mixed integer program firstly the mixed integer formulation of mssc is carefully studied and is reformulated as a continuous optimization problem via a new result on exact penalty technique in dc programming dca is then investigated to the resulting problem secondly we introduce a gaussian kernel version of the bilevel programming formulation of mssc named gkmssc the gkmssc problem is formulated as a dc program for which a simple and efficient dca scheme is developed a regularization technique is investigated for exploiting the nice effect of dc decomposition and a simple procedure for finding good starting points of dca is developed the proposed dca schemes are original and very inexpensive because they amount to computing at each iteration the projection of points onto a simplex and or onto a ball and or onto a box which are all determined in the explicit form numerical results on real word datasets show the efficiency the scalability of dca and its great superiority with respect to k means and kernel k means standard methods for clustering highlightsthe purpose of this paper is to develop new efficient approaches based on dc programming and dca for clustering we consider the two models for the minimum sum of squares clustering mssc a mixed integer program and a bilevel programming formulation the mixed integer program is reformulated as a dc program via an exact penalty technique for which dca is investigated a gaussian kernel version of the bilevel programming formulation is introduced and a simple and efficient dca scheme is developed the two main features of dca say the effect of dc decomposition and of starting points are carefully studied

fast modified global k means algorithm for incremental cluster construction the k means algorithm and its variations are known to be fast clustering algorithms however they are sensitive to the choice of starting points and are inefficient for solving clustering problems in large datasets recently incremental approaches have been developed to resolve difficulties with the choice of starting points the global k means and the modified global k means algorithms are based on such an approach they iteratively add one cluster center at a time numerical experiments show that these algorithms considerably improve the k means algorithm however they require storing the whole affinity matrix or computing this matrix at each iteration this makes both algorithms time consuming and memory demanding for clustering even moderately large datasets in this paper a new version of the modified global k means algorithm is proposed we introduce an auxiliary cluster function to generate a set of starting points lying in different parts of the dataset we exploit information gathered in previous iterations of the incremental algorithm to eliminate the need of computing or storing the whole affinity matrix and thereby to reduce computational effort and memory usage results of numerical experiments on six standard datasets demonstrate that the new algorithm is more efficient than the global and the modified global k means algorithms

variable neighborhood search principles and applications abstract systematic change of neighborhood within a possibly randomized local search algorithm yields a simple and effective metaheuristic for combinatorial and global optimization called variable neighborhood search vns we present a basic scheme for this purpose which can easily be implemented using any local search algorithm as a subroutine its effectiveness is illustrated by solving several classical combinatorial or global optimization problems moreover several extensions are proposed for solving large problem instances using vns within the successive approximation method yields a two level vns called variable neighborhood decomposition search vnds modifying the basic scheme to explore easily valleys far from the incumbent solution yields an efficient skewed vns svns heuristic finally we show how to stabilize column generation algorithms with help of vns and discuss various ways to use vns in graph theory i e to suggest disprove or give hints on how to prove conjectures an area where metaheuristics do not appear to have been applied before

application of statistical mechanics to np complete problems in combinatorial optimisation recently developed techniques of the statistical mechanics of random systems are applied to the graph partitioning problem the averaged cost function is calculated and agrees well with numerical results the problem bears close resemblance to that of spin glasses the authors find a spin glass transition in the system and the low temperature phase space has an ultrametric structure this sheds light on the nature of hard computation problems

the distribution function of the effective field of the ising spin glass on the bethe lattice for the coordination number z 4 5 6 abstract the integral equation for the effective fields of the spin glass is solved analytically for the j ising model on the bether lattice with coordination number z 4 5 and 6 and symmetric distribution functions are obtained the paramagnetic distribution spin glass distribution with three delta functions and spin glass distributions with a continuous distribution are obtained the effect of the truncation of the algebraic equation for solving the integral equation is discussed

a new nonsmooth optimization algorithm for minimum sum of squares clustering problems the minimum sum of squares clustering problem is formulated as a problem of nonsmooth nonconvex optimization and an algorithm for solving the former problem based on nonsmooth optimization techniques is developed the issue of applying this algorithm to large data sets is discussed results of numerical experiments have been presented which demonstrate the effectiveness of the proposed algorithm

random geometric graphs we analyze graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present the critical connectivity is found numerically by examining the size of the largest cluster we derive an analytical expression for the cluster coefficient which shows that the graphs are distinctly different from standard random graphs even for infinite dimensionality insights relevant for graph bipartitioning are included

average and reliability error exponents in low density parity check codes we present a theoretical method for a direct evaluation of the average and reliability error exponents in low density parity check error correcting codes using methods of statistical physics results for the binary symmetric channel are presented for codes of both finite and infinite connectivity

new diagonal bundle method for clustering problems in large data sets abstract clustering is one of the most important tasks in data mining recent developments in computer hardware allow us to store in random access memory ram and repeatedly read data sets with hundreds of thousands and even millions of data points this makes it possible to use conventional clustering algorithms in such data sets however these algorithms may need prohibitively large computational time and fail to produce accurate solutions therefore it is important to develop clustering algorithms which are accurate and can provide real time clustering in large data sets this paper introduces one of them using nonsmooth optimization formulation of the clustering problem the objective function is represented as a difference of two convex dc functions then a new diagonal bundle algorithm that explicitly uses this structure is designed and combined with an incremental approach to solve this problem the method is evaluated using real world data sets with both large number of attributes and large number of data points the proposed method is compared with two other clustering algorithms using numerical results

relaxation in graph coloring and satisfiability problems using t 0 monte carlo simulation we study the relaxation of graph coloring k col and satisfiability k sat two hard problems that have recently been shown to possess a phase transition in solvability as a parameter is varied a change from exponentially fast to power law relaxation and a transition to freezing behavior are found these changes take place for smaller values of the parameter than the solvability transition results for the coloring problem for colorable and clustered graphs and for the fraction of persistent spins for satisfiability are also presented

roof duality complementation and persistency in quadratic 0 1 optimization the paper is concerned with the primal problem of maximizing a given quadratic pseudo boolean function four equivalent problems are discussed the primal the complementation the discrete rhys lp and the weighted stability problem of a sam graph each of them has a relaxation the roof dual the quadratic complementation the continuous rhys lp and the fractional weighted stability problem of a sam graph the main result is that the four gaps associated with the four relaxations are equal furthermore a solution to any of these problems leads at once to solutions of the other three equivalent ones the four relaxations can be solved in polynomial time by transforming them to a bipartite maximum flow problem the optimal solutions of the roof dual define best linear majorantsp x off having the following persistency property if theith coefficient inp is positive negative thenx i 1 0 in every optimum of the primal problem several characterizations are given for the case where these persistency results cannot be used to fix any variable of the primal on the other hand a class of gap free functions properly including the supermodular ones is exhibited

statistical mechanics of the data compression theorem we analyse the performance of a linear code used for data compression of a slepian wolf type in our framework two correlated data are separately compressed into codewords employing gallager type codes and cast into a communication network through two independent input terminals at the output terminal the received codewords are jointly decoded by a practical algorithm based on the thouless anderson palmer approach our analysis shows that the achievable rate region presented in the data compression theorem is described as first order phase transitions among several phases the typical performance of the practical decoder is also well evaluated by the replica method

a mean field spin glass with short range interactions we formulate and study a spin glass model on the bethe lattice appropriate boundary fields replace the traditional self consistent methods they give our model well defined thermodynamic properties we establish that there is a spin glass transition temperature above which the single site magnetizations vanish and below which the edwards anderson order parameter is strictly positive in a neighborhood below the transition temperature we use bifurcation theory to establish the existence of a nontrivial distribution of single site magnetizations two properties of this distribution are studied the leading perturbative correction to the gaussian scaling form at the transition and the nonperturbative behavior of the tails

optimization of output load in a junction model an optimization problem of minimizing the output load of a junction model with multiple input and output channels is solved by the replica method simulation results show that for the case of extensive connectivity the replica solution obtained by assuming a vanishing solution space is satisfactory however for the case of intensive connectivity the degeneracy of the solution space causes the simulation results to deviate from the replica solution of vanishing volume

graph partitioning and dilute spin glasses the minimum cost solution the authors calculate the ground state cost function energy for graph bipartitioning and spin glasses on networks with fixed finite valences allowing a continuous distribution for the effective field compared with previous calculations using a field distribution in integral multiples of the coupling strength j the ground state energy is higher by less than one per cent although the distribution itself is altered drastically

column generation algorithms for exact modularity maximization in networks finding modules or clusters in networks currently attracts much attention in several domains the most studied criterion for doing so due to newman and girvan phys rev e 69 026113 2004 is modularity maximization many heuristics have been proposed for maximizing modularity and yield rapidly near optimal solution or sometimes optimal ones but without a guarantee of optimality there are few exact algorithms prominent among which is a paper by xu et al eur phys j b 60 231 2007 modularity maximization can also be expressed as a clique partitioning problem and the row generation algorithm of grotschel and wakabayashi math program 45 59 1989 applied we propose to extend both of these algorithms using the powerful column generation methods for linear and non linear integer programming performance of the four resulting algorithms is compared on problems from the literature instances with up to 512 entities are solved exactly moreover the computing time of previously solved problems are reduced substantially

j means a new local search heuristic for minimum sum of squares clustering abstract a new local search heuristic called j m eans is proposed for solving the minimum sum of squares clustering problem the neighborhood of the current solution is defined by all possible centroid to entity relocations followed by corresponding changes of assignments moves are made in such neighborhoods until a local optimum is reached the new heuristic is compared with two other well known local search heuristics k and h m eans as well as with h m eans an improved version of the latter in which degeneracy is removed moreover another heuristic which fits into the variable neighborhood search metaheuristic framework and uses j m eans in its local search step is proposed too results on standard test problems from the literature are reported it appears that j m eans outperforms the other local search methods quite substantially when many entities and clusters are considered

the hyperbolic smoothing clustering method the minimum sum of squares clustering problem is considered the mathematical modeling of this problem leads to a min sum min formulation which in addition to its intrinsic bi level nature has the significant characteristic of being strongly nondifferentiable to overcome these difficulties the resolution method proposed adopts a smoothing strategy using a special c differentiable class function the final solution is obtained by solving a sequence of low dimension differentiable unconstrained optimization subproblems which gradually approach the original problem the use of this technique called hyperbolic smoothing allows the main difficulties presented by the original problem to be overcome a simplified algorithm containing only the essentials of the method is presented for the purpose of illustrating both the reliability and the efficiency of the method a set of computational experiments was performed making use of traditional test problems described in the literature

potts models on feynman diagrams we investigate numerically and analytically potts models on thin random graphs generic feynman diagrams using the idea that such models may be expressed as the limit of a matrix model the thin random graphs in this limit are locally tree like in distinction to the fat random graphs that appear in the planar feynman diagram limit more familiar from discretized models of two dimensional gravity the interest of the thin graphs is that they give mean field theory behaviour for spin models living on them without infinite range interactions or the boundary problems of genuine tree like structures such as the bethe lattice q state potts models display a first order transition in the mean field for q 2 so the thin graph potts models provide a useful test case for exploring discontinuous transitions in mean field theories in which many quantities can be calculated explicitly in the saddle point approximation such discontinuous transitions also appear in multiple ising models on thin graphs and may have implications for the use of the replica trick in spin glass models on random graphs

positivity of the two dimensional brown ravenhall operator we determine the critical coupling of the two dimensional brown ravenhall operator with coulomb potential boundedness from below has essentially been proven by bouzouina however that work contains a trivial error leading to an incorrect constant that is exactly half of the actual critical constant furthermore we show that the operator is in fact positive our proof of that is for the most part analogous to tix s proof of the corresponding result for the three dimensional operator

decomposition and nondifferentiable optimization with the projective algorithm this paper deals with an application of a variant of karmarkar s projective algorithm for linear programming to the solution of a generic nondifferentiable minimization problem this problem is closely related to the dantzig wolfe decomposition technique used in large scale convex programming the proposed method is based on a column generation technique defining a sequence of primal linear programming maximization problems associated with each problem one defines a weighted potential function which is minimized using a variant of the projective algorithm when a point close to the minimum of the potential function is reached a corresponding point in the dual space is constructed which is close to the analytic center of a polytope containing the solution set of the nondifferentiable optimization problem an admissible cut of the polytope corresponding to a new supporting hyperplane of the epigraph of the function to minimize is then generated at this approximate analytic center in the primal space this new cut translates into a new column for the associated linear programming problem the algorithm has performed well on a set of convex nondifferentiable programming problems

modified global k means algorithm for minimum sum of squares clustering problems k means algorithm and its variations are known to be fast clustering algorithms however they are sensitive to the choice of starting points and inefficient for solving clustering problems in large data sets recently a new version of the k means algorithm the global k means algorithm has been developed it is an incremental algorithm that dynamically adds one cluster center at a time and uses each data point as a candidate for the k th cluster center results of numerical experiments show that the global k means algorithm considerably outperforms the k means algorithms in this paper a new version of the global k means algorithm is proposed a starting point for the k th cluster center in this algorithm is computed by minimizing an auxiliary cluster function results of numerical experiments on 14 data sets demonstrate the superiority of the new algorithm however it requires more computational time than the global k means algorithm

phase transitions in semidefinite relaxations statistical inference problems arising within signal processing data mining and machine learning naturally give rise to hard combinatorial optimization problems these problems become intractable when the dimensionality of the data is large as is often the case for modern datasets a popular idea is to construct convex relaxations of these combinatorial problems which can be solved efficiently for large scale datasets semidefinite programming sdp relaxations are among the most powerful methods in this family and are surprisingly well suited for a broad range of problems where data take the form of matrices or graphs it has been observed several times that when the statistical noise is small enough sdp relaxations correctly detect the underlying combinatorial structures in this paper we develop asymptotic predictions for several detection thresholds as well as for the estimation error above these thresholds we study some classical sdp relaxations for statistical problems motivated by graph synchronization and community detection in networks we map these optimization problems to statistical mechanics models with vector spins and use nonrigorous techniques from statistical mechanics to characterize the corresponding phase transitions our results clarify the effectiveness of sdp relaxations in solving high dimensional statistical problems

typical behavior of relays in communication channels the typical behavior of the relay without delay channel under low density parity check coding and its multiple unit generalization termed the relay array is studied using methods of statistical mechanics a demodulate and forward strategy is analytically solved using the replica symmetric ansatz which is exact in the system studied at nishimori s temperature in particular the typical level of improvement in communication performance by relaying messages is shown in the case of a small and a large number of relay units 2007 the american physical society

stability of the two dimensional brown ravenhall operator we prove that the two dimensional brown ravenhall operator is bounded from below when the coupling constant is below a specified critical value a property also referred to as stability as a consequence the operator is then self adjoint the proof is based on the strategy followed by evans et al and lieb and yau with some relevant changes characteristic of the dimension our analysis also yields a sharp kato inequality

nonsmooth dc programming approach to the minimum sum of squares clustering problems abstract this paper introduces an algorithm for solving the minimum sum of squares clustering problems using their difference of convex representations a non smooth non convex optimization formulation of the clustering problem is used to design the algorithm characterizations of critical points stationary points in the sense of generalized gradients and inf stationary points of the clustering problem are given the proposed algorithm is tested and compared with other clustering algorithms using large real world data sets

a repetitive branch and bound procedure for minimum within cluster sums of squares partitioning minimization of the within cluster sums of squares wcss is one of the most important optimization criteria in cluster analysis although cluster analysis modules in commercial software packages typically use heuristic methods for this criterion optimal approaches can be computationally feasible for problems of modest size this paper presents a new branch and bound algorithm for minimizing wcss algorithmic enhancements include an effective reordering of objects and a repetitive solution approach that precludes the need for splitting the data set while maintaining strong bounds throughout the solution process the new algorithm provided optimal solutions for problems with up to 240 objects and eight well separated clusters poorly separated problems with no inherent cluster structure were optimally solved for up to 60 objects and six clusters the repetitive branch and bound algorithm was also successfully applied to three empirical data sets from the classification literature

improved branching disjunctions for branch and bound an analytic center approach abstract in classical branch and bound algorithms the branching disjunction is often based on a single variable which is a special case of the more general approach that involves multiple variables in this paper we present a new approach to generate good general branching disjunctions based on the shape of the polyhedron and interior point concepts the approach is based on approximating the feasible polyhedron using dikin s inscribed ellipsoid calculated using the analytic center from interior point methods we use the fact that the width of the ellipsoid in a given direction has a closed form expression to formulate a quadratic problem whose optimal solution is a thin direction of the ellipsoid while solving a quadratic problem at each node of the branch and bound tree is impractical we use an efficient neighborhood search heuristic for its solution we report computational results on hard mixed integer problems from the literature showing that the proposed approach leads to smaller branch and bound trees and a reduction in the computational time as compared with classical branching and strong branching as the computation of the analytic center is a bottleneck we finally test the approach within a general interior point based benders decomposition where the analytic center is readily available and show clear dominance of the approach over classical branching

statistical mechanics of low density parity check codes we review recent theoretical progress on the statistical mechanics of error correcting codes focusing on low density parity check ldpc codes in general and on gallager and mackay neal codes in particular by exploiting the relation between ldpc codes and ising spin systems with multispin interactions one can carry out a statistical mechanics based analysis that determines the practical and theoretical limitations of various code constructions corresponding to dynamical and thermodynamical transitions respectively as well as the behaviour of error exponents averaged over the corresponding code ensemble as a function of channel noise we also contrast the results obtained using methods of statistical mechanics with those derived in the information theory literature and show how these methods can be generalized to include other channel types and related communication problems

replica symmetric solutions of the graph bipartitioning problem with fixed finite valence using the replica method the author presents the replica symmetric solutions of the graph bipartitioning problem with fixed finite valence with the constraint sigma ni 1 si 0 strictly enforced the author is able to find another solution which gives a lower cost function than that given by the spin glass solution

an improved column generation algorithm for minimum sum of squares clustering given a set of entities associated with points in euclidean space minimum sum of squares clustering mssc consists in partitioning this set into clusters such that the sum of squared distances from each point to the centroid of its cluster is minimized a column generation algorithm for mssc was given by du merle et al in siam journal scientific computing 21 1485 1505 the bottleneck of that algorithm is the resolution of the auxiliary problem of finding a column with negative reduced cost we propose a new way to solve this auxiliary problem based on geometric arguments this greatly improves the efficiency of the whole algorithm and leads to exact solution of instances with over 2 300 entities i e more than 10 times as much as previously done

the free energy at t 0 of the spin glass in the case of the continuous distribution of the effective field on the bethe lattice for z 4 5 6 abstract we evaluated the free energy at t 0 of the j random ising model on the bethe lattice of the coordination number z 4 5 6 in the case where the distribution function of the effective field has a continuous distribution by using the method of the integral equation of the effective field

the graph partitioning and the spin glass comments on papers by sherrington and his group the recent theory of graph partitioning proposed by sherrington and his group 1987 1988 used the integral equation to determine the field distribution this integral equation and its solution are pointed out to be the same as those used by katsura and co workers 1979 1980 for spin glasses

graph bipartitioning and spin glasses on a random network of fixed finite valence the authors study the problem of bipartitioning a random graph of fixed finite valence using a mean field replica symmetric theory of an ising ferromagnet with zero magnetisation constraint the thermodynamics is determined by the probability distribution of an auxiliary field the expression for the ground state energy agrees with that proposed by mezard and parisi 1986 using a cavity field method but their expression for the fraction of crazy spins is reinterpreted

optimal partitioning of a data set based on the p median model although the k means algorithm for minimizing the within cluster sums of squared deviations from cluster centroids is perhaps the most common method for applied cluster analyses a variety of other criteria are available the p median model is an especially well studied clustering problem that requires the selection of p objects to serve as cluster centers the objective is to choose the cluster centers such that the sum of the euclidean distances or some other dissimilarity measure of objects assigned to each center is minimized using 12 data sets from the literature we demonstrate that a three stage procedure consisting of a greedy heuristic lagrangian relaxation and a branch and bound algorithm can produce globally optimal solutions for p median problems of nontrivial size several hundred objects five or more variables and up to 10 clusters we also report the results of an application of the p median model to an empirical data set from the telecommunications industry

minimizing unsatisfaction in colourful neighbourhoods colouring sparse graphs under various restrictions is a theoretical problem of significant practical relevance here we consider the problem of maximizing the number of different colours available at the nodes and their neighbourhoods given a predetermined number of colours in the analytical framework of a tree approximation carried out at both zero and finite temperatures solutions obtained by population dynamics give rise to estimates of the threshold connectivity for the incomplete to complete transition which are consistent with those of existing algorithms the nature of the transition as well as the validity of the tree approximation are investigated

random graph coloring statistical physics approach the problem of vertex coloring in random graphs is studied using methods of statistical physics and probability our analytical results are compared to those obtained by exact enumeration and monte carlo simulations we critically discuss the merits and shortcomings of the various methods and interpret the results obtained we present an exact analytical expression for the two coloring problem as well as general replica symmetric approximated solutions for the thermodynamics of the graph coloring problem with p colors and k body edges

one step rsb scheme for the rate distortion function we apply statistical mechanics to an inverse problem of linear mapping to investigate the physics of the irreversible compression we use the replica symmetry breaking rsb technique with a toy model to demonstrate the shannon result the rate distortion function which is widely known as the theoretical limit of the compression with a fidelity criterion is derived using the parisi one step rsb scheme the bound cannot be achieved in the sparsely connected systems where suboptimal solutions dominate the capacity

the integration of an interior point cutting plane method within a branch and price algorithm this paper presents a novel integration of interior point cutting plane methods within branch and price algorithms unlike the classical method columns are generated at a central dual solution by applying the analytic centre cutting plane method accpm on the dual of the full master problem first we introduce some modifications to accpm we propose a new procedure to recover primal feasibility after adding cuts and use for the first time a dual newton s method to calculate the new analytic centre after branching second we discuss the integration of accpm within the branch and price algorithm we detail the use of accpm as the search goes deep in the branch and bound tree making full utilization of past information as a warm start we exploit dual information from accpm to generate incumbent feasible solutions and to guide branching finally the overall approach is implemented and tested for the bin packing problem and the capacitated facility location problem with single sourcing we compare against cplex mip 7 5 as well as a classical branch and price algorithm

zero temperature hysteresis in the random field ising model on a bethe lattice we consider the single spin flip dynamics of the random field ising model on a bethe lattice at zero temperature in the presence of a uniform external field we determine the average magnetization as the external field is varied from to by setting up the self consistent field equations which we show are exact in this case the qualitative behaviour of magnetization as a function of the external field unexpectedly depends on the coordination number z of the bethe lattice for z 3 with a gaussian distribution of the quenched random fields we find no jump in magnetization for any non zero strength of disorder for for weak disorder the magnetization shows a jump discontinuity as a function of the external uniform field which disappears for a larger variance of the quenched field we determine exactly the critical point separating smooth hysteresis curves from those with a jump we have checked our results by monte carlo simulations of the model on three and four coordinated random graphs which for large system sizes give the same results as on the bethe lattice but avoid surface effects altogether

graph bipartitioning and the bethe spin glass the problem of the equipartitioning of a random graph of fixed finite valence is studied by comparison with a ferromagnetic bethe lattice with random boundary conditions the simplicity of recursion relations for effective fields due to descendents on bethe lattices provides simple approximations for the optimal cost in quite good agreement with simulations

fuzzy j means a new heuristic for fuzzy clustering abstract a fuzzy clustering problem consists of assigning a set of patterns to a given number of clusters with respect to some criteria such that each of them may belong to more than one cluster with different degrees of membership in order to solve it we first propose a new local search heuristic called fuzzy j means where the neighbourhood is defined by all possible centroid to pattern relocations the integer solution is then moved to a continuous one by an alternate step i e by finding centroids and membership degrees for all patterns and clusters to alleviate the difficulty of being stuck in local minima of poor value this local search is then embedded into the variable neighbourhood search metaheuristic results on five standard test problems from the literature are reported and compared with those obtained with the well known fuzzy c means heuristic it appears that solutions of substantially better quality are obtained with the proposed methods than with this former one

q partitioning of graphs with finite coordination number the np complete optimisation problem of q partitioning of graphs with finite connectivity is discussed using the effective local field method the authors obtain the local field distributions with and without the continuous part and use them to calculate the ground state energies of the three state potts spin glass on lattices with finite coordination number equal to three this in turn gives analytic results for the optimal cost of the 3 partitioning of graphs they perform simulations of actual 3 partitions of random graphs to compare with their theoretical results and obtain very good agreement ways for further improvement of the estimates are discussed

sce a manifold regularized set covering method for data partitioning cluster analysis plays a very important role in data analysis in these years cluster ensemble as a cluster analysis tool has drawn much attention for its robustness stability and accuracy many efforts have been done to combine different initial clustering results into a single clustering solution with better performance however they neglect the structure information of the raw data in performing the cluster ensemble in this paper we propose a structural cluster ensemble sce algorithm for data partitioning formulated as a set covering problem in particular we construct a laplacian regularized objective function to capture the structure information among clusters moreover considering the importance of the discriminative information underlying in the initial clustering results we add a discriminative constraint into our proposed objective function finally we verify the performance of the sce algorithm on both synthetic and real data sets the experimental results show the effectiveness of our proposed method sce algorithm

pairwise clustering using a monte carlo markov chain abstract in this work an application of mcmc is proposed for unsupervised data classification in conjunction with a novel pairwise objective function which is shown to work well in situations where clusters to be identified have a strong overlap and the centroid oriented methods such as k means fail by construction in particular an exceptionally simple but difficult situation is addressed when cluster centroids coincide and one can differentiate between the clusters only on the basis of their variance performance of the proposed approach is tested on synthetic and real datasets

a comparison of heuristic procedures for minimum within cluster sums of squares partitioning perhaps the most common criterion for partitioning a data set is the minimization of the within cluster sums of squared deviation from cluster centroids although optimal solution procedures for within cluster sums of squares wcss partitioning are computationally feasible for small data sets heuristic procedures are required for most practical applications in the behavioral sciences we compared the performances of nine prominent heuristic procedures for wcss partitioning across 324 simulated data sets representative of a broad spectrum of test conditions performance comparisons focused on both percentage deviation from the best found wcss values as well as recovery of true cluster structure a real coded genetic algorithm and variable neighborhood search heuristic were the most effective methods however a straightforward two stage heuristic algorithm hk means also yielded exceptional performance a follow up experiment using 13 empirical data sets from the clustering literature generally supported the results of the experiment using simulated data our findings have important implications for behavioral science researchers whose theoretical conclusions could be adversely affected by poor algorithmic performances

graph bipartitioning and statistical mechanics the problem of bipartitioning a random graph of fixed finite local valence connectivity so as to minimise the number of cross edges is studied by the application of monte carlo simulation of thermodynamics and annealing this problem is np complete and is relevant as an idealisation of several practical organisation problems evidence is presented for the existence of a multiplicity of metastable states and for non self averaging and ultrametricity in the space of overlaps albeit with a possible critical valence an empirical formula is presented for the optimal cost in excellent accord with values obtained by simulation and those known exactly

graph optimization problems on a bethe lattice thep partitioning andp coloring problems on a bethe lattice of coordination numberz are analyzed it is shown that these two np complete optimization problems turn out to be equivalent to finding the ground state energy ofp state potts models with frustration numerical calculation of the cost function of both problems are carried out for several values ofz andp in the case ofp 2 the results are identical to those obtained by mezard and parisi for the case of the bipartitioning problem a numerical upper bound to the chromatic number is found for several values ofz

approximating k means type clustering via semidefinite programming one of the fundamental clustering problems is to assign n points into k clusters based on minimal sum of squared distances mssc which is known to be np hard in this paper by using matrix arguments we first model mssc as a so called 0 1 semidefinite programming sdp problem we show that our 0 1 sdp model provides a unified framework for several clustering approaches such as normalized k cut and spectral clustering moreover the 0 1 sdp model allows us to solve the underlying problem approximately via the linear programming and sdp relaxations second we consider the issue of how to extract a feasible solution of the original 0 1 sdp model from the optimal solution of the relaxed sdp problem by using principal component analysis we develop a rounding procedure to construct a feasible partitioning from a solution of the relaxed problem in our rounding procedure we need to solve a k means clustering problem in re k 1 which can be done in o n k 2 2k 2 time in case of biclustering the running time of our rounding procedure can be reduced to o n log n we show that our algorithm provides a 2 approximate solution to the original problem promising numerical results for biclustering based on our new method are reported

partitioning and modularity of graphs with arbitrary degree distribution we solve the graph bipartitioning problem in dense graphs with arbitrary degree distribution using the replica method we find the cut size to scale universally with k in contrast earlier results studying the problem in graphs with a poissonian degree distribution had found a scaling with kfu and anderson j phys a 19 1605 1986 our results also generalize to the problem of q partitioning they can be used to find the expected modularity q newman and girvan phys rev e 69 026113 2004 of random graphs and allow for the assessment of the statistical significance of the output of community detection algorithms given a graph or network gn m of n nodes and m edges the problem of graph partitioning is finding a partition of the nodes into q equal sized parts such that the number of edges connecting different parts the cut size is minimal the solution of this problem has many important practical applications in multiprocessor scheduling for parallel computing very large scale integrated chip design vlsi and data mining 1 due to the problem being np complete 2 i e no algorithm is known which is guaranteed to find an optimal solution in a number of steps that grows only polynomially with the size of the graph only heuristics exist to approximate a solution furthermore one cannot determine in polynomial time whether a solution found is indeed optimal the only way to assess the quality of a heuristic algorithm or a particular result is to compare with rigorous bounds on the cut size

statistical physics of regular low density parity check error correcting codes a variation of gallager error correcting codes is investigated using statistical mechanics in codes of this type a given message is encoded into a codeword that comprises boolean sums of message bits selected by two randomly constructed sparse matrices the similarity of these codes to ising spin systems with random interaction makes it possible to assess their typical performance by analytical methods developed in the study of disordered systems the typical case solutions obtained via the replica method are consistent with those obtained in simulations using belief propagation decoding we discuss the practical implications of the results obtained and suggest a computationally efficient construction for one of the more practical configurations

an interior point algorithm for minimum sum of squares clustering an exact algorithm is proposed for minimum sum of squares nonhierarchical clustering i e for partitioning a given set of points from a euclidean m space into a given number of clusters in order to minimize the sum of squared distances from all points to the centroid of the cluster to which they belong this problem is expressed as a constrained hyperbolic program in 0 1 variables the resolution method combines an interior point algorithm i e a weighted analytic center column generation method with branch and bound the auxiliary problem of determining the entering column i e the oracle is an unconstrained hyperbolic program in 0 1 variables with a quadratic numerator and linear denominator it is solved through a sequence of unconstrained quadratic programs in 0 1 variables to accelerate resolution variable neighborhood search heuristics are used both to get a good initial solution and to solve quickly the auxiliary problem as long as global optimality is not reached estimated bounds for the dual variables are deduced from the heuristic solution and used in the resolution process as a trust region proved minimum sum of squares partitions are determined for the first time for several fairly large data sets from the literature including fisher s 150 iris

extremal optimization of graph partitioning at the percolation threshold the benefits of a recently proposed method to approximate hard optimization problems are demonstrated on the graph partitioning problem the performance of this new method called extremal optimization eo is compared with simulated annealing sa in extensive numerical simulations while generally a complex np hard problem the optimization of the graph partitions is particularly difficult for sparse graphs with average connectivities near the percolation threshold at this threshold the relative error of sa for large graphs is found to diverge relative to eo at equalized runtime on the other hand eo based on the extremal dynamics of self organized critical systems reproduces known results about optimal partitions at this critical point quite well