logical analysis of numerical data logical analysis of data lad is a methodology developed since the late eighties aimed at discovering hidden structural information in data sets lad was originally developed for analyzing binary data by using the theory of partially defined boolean functions an extension of lad for the analysis of numerical data sets is achieved through the process of binarization consisting in the replacement of each numerical variable by binary indicator variables each showing whether the value of the original variable is above or below a certain level binarization was successfully applied to the analysis of a variety of real life data sets this paper develops the theoretical foundations of the binarization process studying the combinatorial optimization problems related to the minimization of the number of binary variables to provide an algorithmic framework for the practical solution of such problems we construct compact linear integer programming formulations of them we develop polynomial time algorithms for some of these minimization problems and prove np hardness of others

the lagrangian relaxation method for solving integer programming problems this article originally appeared in management science january 1981 volume 27 number 1 pp 1 18 published by the institute of management sciences one of the most computationally useful ideas of the 1970s is the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side constraints dualizing the side constraints produces a lagrangian problem that is easy to solve and whose optimal value is a lower bound for minimization problems on the optimal value of the original problem the lagrangian problem can thus be used in place of a linear programming relaxation to provide bounds in a branch and bound algorithm this approach has led to dramatically improved algorithms for a number of important problems in the areas of routing location scheduling assignment and set covering this paper is a review of lagrangian relaxation based on what has been learned in the last decade

the loewner equation maps and shapes an approach called schramm loewner evolution sle provides a new method for dealing with a wide variety of scale invariant problems in two dimensions this approach is based upon an older method called loewner evolution le which connects analytic and geometrical constructions in the complex plane in this paper the bases of le and sle are described and some simple applications are discussed in relatively non technical form a bibliography of the subject is presented

fractals in physics squig clusters diffusions fractal measures and the unicity of fractal dimensionality the three topics discussed in this paper are largely independent part 1 fractal squig clusters are introduced and it is shown that their properties can match to a remarkable extent those of percolation clusters at criticality physics on these new geometric shapes should prove tractable as background the author s theories of squig intervals and squig trees are reviewed and restated in more versatile form part 2 the notion of latent fractal dimensionality is introduced and motivated by the desire to simplify the algebra of dimensionality scaling noises are touched upon a common formalism is presented for three forms of anomalous diffusion the ant in the fractal labyrinth fractional brownian motion and levy stable motion the fractal dimensionalities common to diverse shapes generated by diffusion are given in table i as functions of the latent dimensionalities of the support of the motion and of the diffusion itself part 3 it is argued that every fractal point set has a unique fractal dimensionality but it is pointed out that many fractals involve diverse combinations of many fractal point sets such is in particular the case for fractal measures and for fractal graphs often called hierarchical lattices the fractal measures that the author had introduced in the early 1970s are described including new developments

nature of the largest cluster size distribution at the percolation threshold two distinct distribution functions psp m and pns m of the scaled largest cluster sizes m are obtained at the percolation threshold by numerical simulations depending on the condition whether the lattice is actually spanned or not with r pc the spanning probability the total distribution of the largest cluster is given by ptot m r pc psp m 1 r pc pns m the three distributions apparently have similar forms in three and four dimensions while in two dimensions ptot m does not exhibit a familiar form by studying the first and second cumulants of the distribution functions the different behaviour of ptot m in different dimensions may be quantified

an efficient tabu search approach for the 0 1 multidimensional knapsack problem abstract in this paper we describe a new approach to tabu search ts based on strategic oscillation and surrogate constraint information that provides a balance between intensification and diversification strategies new rules needed to control the oscillation process are given for the 0 1 multidimensional knapsack 0 1 mkp based on a portfolio of test problems from the literature our method obtains solutions whose quality is at least as good as the best solutions obtained by previous methods especially with large scale instances these encouraging results confirm the efficiency of the tunneling concept coupled with surrogate information when resource constraints are present

on the universality of distribution of ranked cluster masses at critical percolation the distribution of masses of clusters smaller than the infinite cluster is evaluated at the percolation threshold the clusters are ranked according to their masses and the distribution p m ld r of the scaled masses m for any rank r shows a universal behaviour for different lattice sizes l d is the fractal dimension for different ranks however there is a universal distribution function only in the large rank limit i e p m ld r r y g mry ld y and are defined in the text where the universal scaling function g is found to be gaussian in nature

enhancing an algorithm for set covering problems abstract in this note we present enhancements to a previously published algorithm for the optimal solution of set covering problems these enhancements relate to the use of a langrangean heuristic feasible solution exclusion constraints gomory f cuts and an improved branching strategy computational results for problems involving up to 400 rows and 4000 columns indicate that the enhanced algorithm gives better computational results than the original algorithm

min max and min max regret versions of combinatorial optimization problems a survey abstract min max and min max regret criteria are commonly used to define robust solutions after motivating the use of these criteria we present general results then we survey complexity results for the min max and min max regret versions of some combinatorial optimization problems shortest path spanning tree assignment min cut min s t cut knapsack since most of these problems are np hard we also investigate the approximability of these problems furthermore we present algorithms to solve these problems to optimality

scaling of the distribution of shortest paths in percolation we present a scaling hypothesis for the distribution function of the shortest paths connecting any two points on a percolating cluster which accounts for i the effect of the finite size of the system and ii the dependence of this distribution on the site occupancy probability p we test the hypothesis for the case of two dimensional percolation

on solving linear programs with the ordered weighted averaging objective abstract the problem of aggregating multiple criteria to form overall objective functions is of considerable importance in many disciplines the most commonly used aggregation is based on the weighted sum the ordered weighted averaging owa aggregation introduced by yager uses the weights assigned to the ordered values i e to the worst value the second worst and so on rather than to the specific criteria this allows to model various aggregation preferences preserving simultaneously the impartiality neutrality with respect to the individual criteria in this paper we analyze solution procedures for linear programs with the owa objective functions two alternative linear programming formulations are introduced and their computational efficiency is analyzed

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

stochastic renormalization group in percolation i fluctuations and crossover abstract a generalization of the renormalization group which describes order parameter fluctuations in finite systems is developed in the specific context of percolation this stochastic renormalization group srg expresses statistical self similarity through a non stationary branching process the srg provides a theoretical basis for analytical or numerical approximations both at and away from criticality whenever the correlation length is much larger than the lattice spacing regardless of the system size for example the srg predicts order parameter distributions and finite size scaling functions for the complete crossover between phases for percolation the simplest srg describes structural quantities conditional on spanning such as the total cluster mass or the minimum chemical distance between two boundaries in these cases the central limit theorem for independent random variables holds at the stable off critical fixed points while a fractal central limit theorem describing long range correlations holds at the unstable critical fixed point this first part of a series of articles explains these basic concepts and a general theory of crossover subsequent parts will focus on limit theorems and comparisons of small cell srg approximations with simulation results

asymptotic behaviour of disconnection and non intersection exponents we study the asymptotic behaviour of disconnection and non intersection exponents for planar brownian motionwhen the number of considered paths tends to infinity in particular if n respectively n p denotes the disconnection exponent for n paths respectively the non intersection exponent for n paths versus p paths then we show that limn n n 1 2 and that for a 0 and b 0 limn na nb n a b 2 2

golden mean arithmetic in the fractal branching of diffusion limited aggregates we use the wavelet transform microscope to explore the intricate fractal geometry of witten and sander s diffusion limited aggregates we report on the discovery of fibonacci sequences in the internal hierarchical structure of these clusters we discuss the relevance of the golden mean arithmetic to the numerically well established statistical self similarity of these aggregates

a heuristic method for the set covering problem we present a lagrangian based heuristic for the well known set covering problem scp the algorithm was initially designed for solving very large scale scp instances involving up to 5 000 rows an

copolymer networks and stars scaling exponents we explore and calculate the rich scaling behavior of copolymer networks in solution by renormalization group methods we establish a field theoretic description in terms of composite operators our third order resummation of the spectrum of scaling dimensions brings about remarkable features the special convexity properties of the spectra allow for a multifractal interpretation while preserving stability of the theory this behavior could not be found for power of field operators of usual ensuremath varphi 4 field theory the two dimensional 2d limit of the mutually avoiding walk star apparently corresponds to results of a conformal kac series such a classification seems not possible for the 2d limit of other copolymer stars we furthermore provide a consistency check of two complementary renormalization schemes ensuremath varepsilon expansion and renormalization at fixed dimension calculating a large collection of independent exponents in both approaches

combinatorial optimization problems with uncertain costs and the owa criterion in this paper a class of combinatorial optimization problems with uncertain costs is discussed the uncertainty is modeled by specifying a discrete scenario set containing k distinct cost scenarios the ordered weighted averaging owa for short aggregation operator is applied to choose a solution some well known criteria used in decision making under uncertainty such as the maximum minimum average hurwicz and median are special cases of owa furthermore by using owa the traditional robust min max approach to combinatorial optimization problems with uncertain costs can be generalized the computational complexity and approximability of the problem of minimizing owa for the considered class of problems are investigated and some new positive and negative results in this area are provided these results remain valid for many basic problems such as network or resource allocation problems

on the approximability of robust spanning tree problems in this paper the minimum spanning tree problem with uncertain edge costs is discussed in order to model the uncertainty a discrete scenario set is specified and a robust framework is adopted to choose a solution the min max min max regret and 2 stage min max versions of the problem are discussed the complexity and approximability of all these problems are explored it is proved that the min max and min max regret versions with nonnegative edge costs are hard to approximate within o log 1 en for any e 0 unless the problems in np have quasi polynomial time algorithms similarly the 2 stage min max problem cannot be approximated within o logn unless the problems in np have quasi polynomial time algorithms in this paper randomized lp based approximation algorithms with performance bound of o log 2n for min max and 2 stage min max problems are also proposed

ordered weighted average optimization in multiobjective spanning tree problem abstract multiobjective spanning tree problems are studied in this paper the ordered median objective function is used as an averaging operator to aggregate the vector of objective values of feasible solutions this leads to the ordered weighted average spanning tree problem a nonlinear combinatorial optimization problem different mixed integer linear programs are proposed based on the most relevant minimum cost spanning tree models in the literature these formulations are analyzed and several enhancements presented their empirical performance is tested over a set of randomly generated benchmark instances the results of the computational experiments show that the choice of an appropriate formulation allows to solve larger instances with more objectives than those previously solved in the literature

conformal invariance of voronoi percolation it is proved that in the voronoi model for percolation in dimension 2 and 3 the crossing probabilities are asymptotically invariant under conformal change of metric

a lagrangian based heuristic for large scale set covering problems we present a new lagrangian based heuristic for solving large scale set covering problems arising from crew scheduling at the italian railways ferrovie dello stato our heuristic obtained impressive results when compared to state of the art codes on a test bed provided by the company which includes instances with sizes ranging from 50 000 variables and 500 constraints to 1 000 000 variables and 5000 constraints 1998 the mathematical programming society inc published by elsevier science b v

intersections of random walks a direct renormalization approach various intersection probabilities of independent random walks ind dimensions are calculated analytically by a direct renormalization method adapted from polymer physics this heuristic approach based on edwards continuum model leads to a straightforward derivation and also to refinements of lawler s results for the simultaneous intersections of two walks in 4 or three walks in 3 these results are generalized top walks in d d frac 2p p 1 p 2 ford 4 an infinite set of universal critical exponents l l 1 are derived they govern the asymptotic probability z l sim s sigma l thatl star walks in d with a common origin do not intersect before times the l s are calculated up to ordero e2 whered 4 e this information is used to calculate the probability z g that a set of independent random walks in d or d d 4 respectivelyd 3 form a given topological networks g of multiple intersection points in the absence of any other double point respectively triple point this is generalized to a network in d leqq frac 2p p 1 dimension with exclusion ofp tuple points the method is quite general and can be used to calculate any critical intersection probability and provides the probabilist with a large variety of exact results yet to be proven rigorously

the thermodynamics of fractals revisited with wavelets abstract the multifractal formalism originally introduced to describe statistically the scaling properties of singular measures is revisited using the wavelet transform this new approach is based on the definition of partition functions from the wavelet transform modulus maxima we demonstrate that very much like thermodynamic functions the generalized fractal dimensions d q and the f singularity spectrum can be readily determined from the scaling behavior of these partition functions we show that this method provides a natural generalization of the classical box counting techniques to fractal signals the wavelets playing the role of generalized boxes we illustrate our theoretical considerations on pedagogical examples e g devil s staircases and fractional brownian motions we also report the results of some recent application of the wavelet transform modulus maxima method to fully developed turbulence data that we emphasize the wavelet transform as a mathematical microscope that can be further used to extract microscopic informations about the scaling properties of fractal objects in particular we show that a dynamical system which leaves invariant such an object can be uncovered form the space scale arrangement of its wavelet transform modulus maxima we elaborate on a wavelet based tree matching algorithm that provides a very promising tool for solving the inverse fractal problem this step towards a statistical mechanics of fractals is illustrated on discrete period doubling dynamical systems where the wavelet transform is shown to reveal the renormalization operation which is essential to the understanding of the universal properties of this transition to chaos finally we apply our technique to analyze the fractal hierarchy of dla azimuthal cantor sets defined by intersecting the inner frozen region of large mass off lattice diffusion limited aggregates dla wit a circle this study clearly lets out the existence of an underlying multiplicative process that is likely to account for the fibonacci structural ordering recently discovered in the apparently disordered arborescent dla morphology

critical exponents of manhattan hamiltonian walks in two dimensions from potts and o n models we consider a set of hamiltonian circuits filling a manhattan lattice i e a square lattice with alternating traffic regulation we show that the generating function with fugacityz of this set is identical to the critical partition function of aq state potts model on an unoriented square lattice withq1 2 z the set of critical exponents governing correlations of hamiltonian circuits is derived using a coulomb gas technique these exponents are also found to be those of an o n vector model in the low temperature phase withn q1 2 z the critical exponents in the limitz 0 are then those of spanning trees q 0 and of dense polymers n 0 t tc corresponding to a conformal theory with central chargec 2 this shows that the manhattan orientation and the hamiltonian constraint of filling all the lattice are irrelevant for the infrared critical properties of hamiltonian walks

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

distribution of shortest paths in percolation we present a scaling ansatz for the distribution function of the shortest paths connecting any two points on a percolating cluster which accounts for i the eect of the nite size of the system and ii the dependence of this distribution on the site occupancy probability p we present evidence supporting the scaling ansatz for the case of two dimensional percolation c 1999 elsevier science b v all rights reserved

quasiperiodicity in dissipative systems a renormalization group analysis abstract dynamical systems with quasiperiodic behavior i e two incommensurate frequencies may be studied via discrete maps which show smooth continuous invariant curves with irrational winding number in this paper these curves are followed using renormalization group techniques which are applied to a one dimensional system circle and also to an area contracting map of an annulus two fixed points are found representing different types of universal behavior a trivial fixed point for smooth motion and a nontrivial fixed point the latter representsthe incipient breakup of a quasiperiodic motion with frequency ratio the golden mean into a more chaotic flow fixed point functions are determined numerically and via an e expansion and eigenvalues are calculated

higher conformal multifractality we derive from conformal invariance and quantum gravity the multifractal spectrum f of the harmonic measure i e electrostatic potential or diffusion field near any conformally invariant fractal in two dimensions it gives the hausdorff dimension of the set of points where the potential varies with distance r to the fractal frontier as r first examples are a random walk i e a brownian motion a self avoiding walk or a critical percolation cluster the generalized dimensions d n as well as the multifractal functions f are derived and are all identical for these three cases the external frontiers of a brownian motion and of a percolation cluster are thus identical to a self avoiding walk in the scaling limit the multifractal mf function f c of the electrostatic potential near any conformally invariant fractal boundary like a critical o n loop or a q state potts cluster is given as a function of the central charge c of the associated conformal field theory the dimensions dep of the external perimeter and dh of the hull of a critical scaling curve or cluster obey the superuniversal duality equation d text ep 1 d text d 1 tfrac 1 4 finally for a conformally invariant scaling curve which is simple i e without double points we derive higher multifractal functions like the universal function f2 which gives the hausdorff dimension of the points where the potential varies jointly with distance r as r on one side of the curve and as r on the other the general case of the potential distribution between the branches of a star made of an arbitrary number of scaling paths is also treated the results apply to critical o n loops potts clusters and to the sle process we present a duality between external perimeters of potts clusters and o n loops at their critical point as well as the corresponding duality in the sle process for 16

a genetic algorithm for the set covering problem abstract in this paper we present a genetic algorithm based heuristic for non unicost set covering problems we propose several modifications to the basic genetic procedures including a new fitness based crossover operator fusion a variable mutation rate and a heuristic feasibility operator tailored specifically for the set covering problem the performance of our algorithm was evaluated on a large set of randomly generated problems computational results showed that the genetic algorithm based heuristic is capable of producing high quality solutions

scaling exponents and multifractal dimensions for independent random cascades this paper is concerned with mandelbrot s stochastic cascade measures the problems of i scaling exponents of structure functions of the measure q and ii multifractal dimensions are considered for cascades with a generator vector w1 wc of the general type these problems were previously studied for independent strongly bounded variableswi 0 a wi c consequently a broad class of models used in applications including kolmogorov s long normal model in turbulence long stable universal cascades in atmospheric dynamics has not been covered roughly speaking problems i ii are here solved under the condition that the scaling exists the function is calculated for all arguments previously this was done for positiveq and a new effect emerges the function can generally involve discontinuities in the first derivative as well as in the second

dimensions and entropies of strange attractors from a fluctuating dynamics approach abstract it is shown that the fluctuations in the divergence of near by trajectories on strictly deterministic strange attractors can be modelled by stochastic concepts in particular we propose kramers moyal type equations for correlation functions between points on the attractor the drift terms are the lyapunov exponents the diffusion terms depend on the above fluctuations from this we obtain bounds on generalized dimensions and entropies numerical results show that in nearly all studied cases henon map zaslavskii map mackey glass eq the attractors are fractal measures in the sense of farmer information dimension hausdorff dimension metric entropy topological entropy

wavelet transforms of self similar processes this paper deals with the wavelet transforms of self similar random processes of the kind assumed in the kolmogorov 1941 theory of turbulence it is shown that after suitable rescaling the wavelet transform at a given position becomes a stationary random function of the logarithm of the scale argument in the transform the rescaling depends on the scaling exponent the statistical properties of the resulting random fluctuations such as its correlation function depend on the choice of the analyzing wavelet some implications are i the presence of fluctuations in log log plots of the absolute value of wavelet transforms versus the scale ii an estimate of the small scale fluctuations generalizing the iterated logarithm law of brownian motion iii an ultraviolet ergodic formula giving the scaling exponents in terms of zoom averages some observations are made on the issue of fluctuations observed in wavelet transforms of turbulence data

the probability of intersection of independent random walks in four dimensions lets1 ands2 be independent simple random walks of lengthn inz4 starting at 0 andx0 respectively if x0 2 n it is shown that the probability that the paths intersect is of order logn 1 ifx0 0 it is shown that the probability of no intersection of the paths decays no faster than logn 1 and no slower than logn 1 2 it is conjectured that logn 1 2 is the actual decay rate

on the wavelet transformation of fractal objects the wavelet transformation is briefly presented it is shown how the analysis of the local scaling behavior of fractals can be transformed into the investigation of the scaling behavior of analytic functions over the half plane near the boundary of its domain of analyticity as an example a weierstrass like fractal function is considered for which the wavelet transform is related to a jacobi theta function some of the scalings of this theta function are analyzed and give some information about the scaling behavior of this fractal

non intersection exponents for brownian paths letx andy be independent 3 dimensional brownian motions x 0 0 0 0 y 0 1 0 0 and letp r p x 0 r y 0 r then the non intersection exponent mathop lim limits r to infty log p r mathord left vphantom log p r log r right kern nulldelimiterspace log r exists and is equal to a similar non intersection exponent for random walks analogous results hold inr2 and for more than 2 paths

simple lagrangian heuristic for the set covering problem abstract in this paper a simple lagrangian heuristic is proposed for the set covering problem it is based on simple and classical ideas lagrangian duality greedy heuristic for the set covering problem subgradient optimization and redundant covers the main interesting point of the paper lies in the fact that the proposed heuristic turns out to be efficient for low density set covering problems with a large number of variables

velocity probability density functions of high reynolds number turbulence abstract this paper deals with the probability density function pdf of velocity differences between two points separated by distance r measurements of pdfs were made for r lying in the inertial range for two different flows in a jet with r 852 and in a wind tunnel with r 2720 these pdfs have a characteristic non gaussian shape with exponential tails following kolmogorov s general ideas of log normality a new model for the pdf is developed which contains two parameters determined by experiments this empirical model agrees with the experimental results that the tails of the pdf deviate from a truly exponential behaviour in particular for small r in addition the model leads to the general scaling law ln e 2 r r 0 differenr from kolmogorov s third hypothesis ln e 2 ln r r 0 restricted to the inertial range only x is x x we develop also a formalism based on an extremum principle which is consistent with both the log normality of e and the above mentioned power law in this formalism can be interpreted as the codimension of dissipative structures and asymptotically varies as 1 ln r

optimal solution of set covering partitioning problems using dual heuristics we present an algorithm for a mixed set covering partitioning model that includes as special cases the well known set covering problem and set partitioning problem the novel feature of our algorithm is the use of continuous heuristics applied to the dual of the linear programming relaxation to provide lower bounds for a branch and bound algorithm the heuristics are continuous adaptations of the well known greedy and 3 opt methods that have been applied to a variety of combinatorial optimization problems our algorithm has outperformed the current best set covering algorithm of balas and ho 1980 by about a factor of 3 and appears to improve on the best existing set partitioning algorithm by more than an order of magnitude

nonintersection exponents for brownian paths ii estimates and applications to a random fractal let x and y be independent two dimensional brownian motions x 0 0 0 y 0 e 0 and let p e p x 0 1 y 0 1 o q e y 0 1 does not contain a closed loop around 0 asymptotic estimates when e 0 of p e q e and some related probabilities are given let f be the boundary of the unbounded connected component of r 2 z 0 1 where z t x t tx 1 for t 0 1 then f is a closed jordan arc and the hausdorff dimension of f is less or equal to 3 2 1 4 2

predicting oil recovery using percolation one particular practical problem in oil recovery is to predict the time to breakthrough of a uid injected in one well and the subsequent decay in the production rate of oil at another well because we only have a stochastic view of the distribution of rock properties we need to predict the uncertainty in the breakthrough time and post breakthrough behaviour in order to calculate the economic risk in this paper we use percolation theory to predict i the distribution of the chemical path shortest path between two points representing well pairs at a given euclidean separation and present a scaling hypothesis for this distribution which is conrmed by numerical simulation ii the distribution of breakthrough times which can be calculated algebraically rather than by very time consuming direct numerical simulation of large numbers of realisations c 1999 elsevier science b v all rights reserved

choquet based optimisation in multiobjective shortest path and spanning tree problems this paper is devoted to the search of choquet optimal solutions in finite graph problems with multiple objectives the choquet integral is one of the most sophisticated preference models used in decision theory for aggregating preferences on multiple objectives we first present a condition on preferences name hereafter preference for interior points that characterizes preferences favouring compromise solutions a natural attitude in various contexts such as multicriteria optimisation robust optimisation and optimisation with multiple agents within choquet expected utility theory this condition amounts to using a submodular capacity and a convex utility function under these assumptions we focus on the fast determination of choquet optimal paths and spanning trees after investigating the complexity of these problems we introduce a lower bound for the choquet integral computable in polynomial time then we propose different algorithms using this bound either based on a controlled enumeration of solutions ranking approach or an implicit enumeration scheme branch and bound finally we provide numerical experiments that show the actual efficiency of the algorithms on multiple instances of different sizes

robust discrete optimization and network flows we propose an approach to address data uncertainty for discrete optimization and network flow problems that allows controlling the degree of conservatism of the solution and is computationally tractable both practically and theoretically in particular when both the cost coefficients and the data in the constraints of an integer programming problem are subject to uncertainty we propose a robust integer programming problem of moderately larger size that allows controlling the degree of conservatism of the solution in terms of probabilistic bounds on constraint violation when only the cost coefficients are subject to uncertainty and the problem is a 0 1 discrete optimization problem on n variables then we solve the robust counterpart by solving at most n 1 instances of the original problem thus the robust counterpart of a polynomially solvable 0 1 discrete optimization problem remains polynomially solvable in particular robust matching spanning tree shortest path matroid intersection etc are polynomially solvable we also show that the robust counterpart of an np hard approximable 0 1 discrete optimization problem remains approximable finally we propose an algorithm for robust network flows that solves the robust counterpart by solving a polynomial number of nominal minimum cost flow problems in a modified network

inverse problems in fractal construction moment method solution abstract the inverse problem for several classes of fractals in one and two dimensions can be profitably solved via the application of the classical theory of moments this new perspective leads to important results on the reconstruction process when only a finite number of moments of the fractal measures are known while discussing the rigorous aspects of the theory we also implement it via efficient numerical algorithms

approximating combinatorial optimization problems with the ordered weighted averaging criterion abstract this paper deals with combinatorial optimization problems with k cost scenarios inducing k linear objectives the popular ordered weighted averaging owa criterion is used to aggregate the objectives and compute a solution it is well known that minimizing owa for most basic combinatorial problems is weakly np hard even if the number of scenarios k equals two and strongly np hard when k is a part of the input in this paper the problem with nonincreasing weights in the owa criterion and a large k is first considered a method of reducing the number of scenarios by appropriately aggregating the costs before solving the problem is proposed it is shown that an optimal solution to the reduced problem has a guaranteed worst case approximation ratio some new approximation results for the hurwicz criterion which is a special case of owa are also presented

limit lognormal multifractal as an exponential functional the limiting distribution of the limit lognormal multifractal first introduced by mandelbrot statistical models and turbulence m rosenblatt and c van atta eds lecture notes in physics 12 springer new york 1972 p 333 and constructed explicitly by bacry et al phys rev e 64 026103 2001 is investigated using its laplace transform a partial differential equation for the laplace transform is derived and it is shown that multifractality alone does not determine the limiting distribution the increments of the limit multifractal process are strongly stochastically dependent the precise nature of this stochastic dependence structure of increments sdsi is the determining characteristic of the limiting distribution the sdsi of the limit process is quantified by means of two integro differential relations obtained by renormalization in the sense of leipnik j aust math soc b 32 327 347 1991 one is interpreted as a counterpart of the star equation of mandelbrot and the other is shown to be an analogue of the classical girsanov theorem in the weak intermittency limit an approximate single variable equation for the laplace transform is obtained and successfully tested numerically by simulation

fractal structure of spin clusters and domain walls in the two dimensional ising model the fractal structure of spin clusters and their boundaries in the critical two dimensional ising model is investigated numerically the fractal dimensions of these geometrical objects are estimated by means of monte carlo simulations on relatively small lattices through standard finite size scaling the obtained results are in excellent agreement with theoretical predictions and partly provide significant improvements in precision over existing numerical estimates

multifractal formalism for fractal signals the structure function approach versus the wavelet transform modulus maxima method several attempts have been made recently to generalize the multifractal formalism originally introduced for singular measures to fractal signals we report on a systematic comparison between the structure function approach pioneered by parisi and frisch in 2 proceedings of the international school on turbulence and predictability in geophysical fluid dynamics and climate dynamics edited by m ghil r benzi and g parisi north holland amsterdam 1985 p 84 to account for the multifractal nature of fully developed turbulent signals and an alternative method we have developed within the framework of the wavelet transform analysis we comment on the intrinsic limitations of the structure function approach this technique has fundamental drawbacks and does not provide a full characterization of the singularities of a signal in many cases we demonstrate that our method based on the wavelet transform modulus maxima representation works in most situations and is likely to be the ground of a unified multifractal description of self affine distributions our theoretical considerations are both illustrated on pedagogical examples and supported by numerical simulations

the intersection of brownian paths as a case study of a renormalization group method for quantum field theory a new approach is presented for the study of the probability that the random paths generated by two independent brownian motions in d intersect or more generally are within a short distance a of each other the well known behavior of that function of a above below and at the critical dimension d 4 as well as further corrections are derived here by means of a single renormalization group equation the equation s derivation is expected to shed some light on the function of the o d 4 quantum field theory

multifractal stationary random measures and multifractal random walks with log infinitely divisible scaling laws we define a large class of continuous time multifractal random measures and processes with arbitrary log infinitely divisible exact or asymptotic scaling law these processes generalize within a unified framework both the recently defined log normal multifractal random walk j f muzy j delour and e bacry eur j phys b 17 537 2000 e bacry j delour and j f muzy phys rev e 64 026103 2001 and the log poisson product of cylindrical pulses j barral and b b mandelbrot cowles foundation discussion paper no 1287 2001 unpublished our construction is based on some continuous stochastic multiplication as introduced in f schmitt and d marsan eur j phys b 20 3 2001 from coarse to fine scales that can be seen as a continuous interpolation of discrete multiplicative cascades we prove the stochastic convergence of the defined processes and study their main statistical properties the question of genericity universality of limit multifractal processes is addressed within this new framework we finally provide a method for numerical simulations and discuss some specific examples

transport on percolation clusters with power law distributed bond strengths the simplest transport problem namely finding the maximum flow of current or maxflow is investigated on critical percolation clusters in two and three dimensions using a combination of extremal statistics arguments and exact numerical computations for power law distributed bond strengths of the type p ensuremath sigma ensuremath sim ensuremath sigma ensuremath ensuremath alpha assuming that only cutting bonds determine the flow the maxflow critical exponent v is found to be v ensuremath alpha d ensuremath 1 ensuremath nu 1 1 ensuremath ensuremath alpha this prediction is confirmed with excellent accuracy using large scale numerical simulation in two and three dimensions however in the region of anomalous bond capacity distributions 0l ensuremath alpha l 1 we demonstrate that due to cluster structure fluctuations it is not the cutting bonds but the blobs that set the transport properties of the backbone this blob dominance avoids a crossover to a regime where structural details the distribution of the number of red or cutting bonds would set the scaling the restored scaling exponents however still follow the simplistic red bond estimate this is argued to be due to the existence of a hierarchy of so called minimum cut configurations for which cutting bonds form the lowest level and whose transport properties scale all in the same way we point out the relevance of our findings to other scalar transport problems i e conductivity

wavelet transform of multifractals we present the wavelet transform as a mathematical microscope which is well suited for studying the local scaling properties of fractal objects we apply this technique to probability measures on self similar cantor sets and to the golden mean trajectories on two tori at the onset of chaos

the dimension spectrum of some dynamical systems we analyze the dimension spectrum previously introduced and measured experimentally by jensen kadanoff and libchaber using large deviation theory we prove for some invariant measures of expanding markov maps that the hausdorff dimensionf of the set on which the measure has a singularity is a well defined concave and regular function in particular we show that this is the case for the accumulation of period doubling and critical mappings of the circle with golden rotation number we also show in these particular cases that the functionf is universal

on the limit lognormal and other limit log infinitely divisible laws the limit log infinitely divisible multifractals of muzy and bacry phys rev e 66 056121 2002 are reviewed and shown to possess novel invariance relations that translate into functional feynman kac equations for the corresponding probability distributions in the special case of the limit lognormal process of mandelbrot in statistical models and turbulence m rosenblatt c van atta eds springer new york 1972 the limit distribution is represented exactly in an operator form using the technique of intermittency expansions a novel representation for the mellin transform of the limit distribution is derived and related to the hurwitz zeta function for application the cumulants of the logarithm of the limit lognormal distribution are computed explicitly

cluster analysis of the ising model and universal finite size scaling the recent progress in the study of finite size scaling fss properties of the ising model is briefly reviewed we calculate the universal fss functions for the binder parameter g and the magnetization distribution function p m for the ising model on l1 l2 two dimensional lattices with tilted boundary conditions we show that the fss functions are universal for fixed sets of the aspect ratio l1 l2 and the tilt parameter we also study the percolating properties of the ising model giving attention to the effects of the aspect ratio of finite systems we elucidate the origin of the complex structure of p m for the system with large aspect ratio by the multiple percolating cluster argument

a 3 flip neighborhood local search for the set covering problem the set covering problem scp calls for a minimum cost family of subsets from n given subsets which together covers the entire ground set in this paper we propose a local search algorithm for scp which has the following three characteristics 1 the use of 3 flip neighborhood which is the set of solutions obtainable from the current solution by exchanging at most three subsets as the size of 3 flip neighborhood is o n3 the neighborhood search becomes expensive if implemented naively to overcome this we propose an efficient implementation that reduces the number of candidates in the neighborhood without sacrificing the solution quality 2 we allow the search to visit the infeasible region and incorporate the strategic oscillation technique realized by adaptive control of penalty weights 3 the size reduction of the problem by using the information from the lagrangian relaxation is incorporated which is indispensable for solving very large instances according to computational comparisons on benchmark instances with other existing heuristic algorithms for scp our algorithm performs quite effectively for various types of problems especially for very large scale instances

wavelet analysis of a gaussian kolmogorov signal several attempts have been made recently to use wavelets transforms for extracting histograms of scaling exponents from experimental turbulence data similar techniques are here applied to a gaussian signal having a kolmogorov 5 3 energy spectrum this is an instance of the class of fractional brownian motions having scaling exponent 1 3 almost everywhere for the gaussian signal a spurious non trivial histogram is obtained by applying a mexican hat wavelet transform analysis on the other hand we show that the use of complex wavelets and even more so the application of an optimal wavelet transform method strongly reduce the spurious fluctuations observed in the processing of the gaussian signal

an algorithm for set covering problem in this paper we present an algorithm for the set covering problem that combines problem reduction tests with dual ascent subgradient optimisation and linear programming computational results are presented for problems involving up to 400 rows and 4000 columns

a dynamic subgradient based branch and bound procedure for set covering we discuss a branch and bound algorithm for set covering whose centerpiece is a new integrated upper bounding lower bounding procedure called dynamic subgradient optimization dynsgrad this new procedure applied to a lagrangean dual at every node of the search tree combines the standard subgradient method with primal and dual heuristics that interact to change the lagrange multipliers and tighten the upper and lower bounds fix variables and periodically restate the lagrangean itself extensive computational testing is reported as a stand alone heuristic dynsgrad performs significantly better than other procedures in terms of the quality of solutions obtainable with a certain computational effort when incorporated into a branch and bound algorithm dynsgrad considerably advances the state of the art in solving set covering problems

scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors the analysis of dynamical systems in terms of spectra of singularities is extended to higher dimensions and to nonhyperbolic systems prominent roles in our approach are played by the generalized partial dimensions of the invariant measure and by the distribution of effective liapunov exponents for hyperbolic attractors the latter determines the metric entropies and provides one constraint on the partial dimensions for nonhyperbolic attractors there are important modifications we discuss them for the examples of the logistic and henon map we show in particular that the generalized dimensions have singularities with noncontinuous derivative similar to first order phase transitions in statistical mechanics

functional feynman kac equations for limit lognormal multifractals a novel technique of functional feynman kac equations is developed for the probability distribution of the limit lognormal multifractal process introduced by mandelbrot in statistical models and turbulence m rosenblatt and c van atta eds springer new york 1972 and constructed explicitly by bacry delour and muzy phys rev e 64 026103 2001 the distribution of the process is known to be determined by the complicated stochastic dependence structure of its increments sdsi it is shown that the sdsi has two separate layers of complexity that can be captured in a precise way by a pair of functional feynman kac equations for the laplace transform exact solutions are obtained as power series expansions in the intermittency parameter using a novel intermittency differentiation rule the expansion of the moments gives a new representation of the selberg integral

multifractal random walk we introduce a class of multifractal processes referred to as multifractal random walks mrws to our knowledge it is the first multifractal process with continuous dilation invariance properties and stationary increments mrws are very attractive alternative processes to classical cascadelike multifractal models since they do not involve any particular scale ratio the mrws are indexed by four parameters that are shown to control in a very direct way the multifractal spectrum and the correlation structure of the increments we briefly explain how in the same way one can build stationary multifractal processes or positive random measures

an electromagnetism metaheuristic for the unicost set covering problem in this paper we propose a new heuristic algorithm to solve the unicost version of the well known set covering problem the method is based on the electromagnetism metaheuristic approach which after generating a pool of solutions to create the initial population applies a fixed number of local search and movement iterations based on the electromagnetism theory in addition to some random aspects used in the construction and local search phases we also apply mutation in order to further escape from local optima the proposed algorithm has been tested over 80 instances of the literature on the classical benchmark instances where the number of columns is larger than the number of rows the algorithm by using a fixed set of parameters always found the best known solution and for 12 instances it was able to improve the current best solution by using different parameter settings the algorithm improved 4 additional best known solutions moreover we proved the effectiveness of the electromagnetism metaheuristic approach for the unicost set covering problem by embedding the procedures of the proposed algorithm in a genetic algorithm scheme the worse results obtained by the genetic algorithm show the impact of the electromagnetism metaheuristic approach in conducting the search of the solution space by applying the movements based on the electromagnetism theory finally we report the results obtained by modifying the proposed electromagnetism metaheuristic algorithm for solving the non unicost set covering problem

structural five fold symmetry in the fractal morphology of diffusion limited aggregates abstract the statistical self similarity of the geometry of diffusion limited aggregates and the multifractal nature of the growth probability distribution on the surface of the growing clusters are investigated using the wavelet transform this study reveals the existence of a predominant structural five fold symmetry in the internal frozen region as well as in the active outer region of the interface this observation is corroborated by a statistical analysis of the screening effects that govern diffusion limited aggregation dla growth in linear and sector shaped cells the existence of this symmetry is likely to be a clue to a hierarchichal fractal ordering we report on the discovery of fibonacci sequences in the inner extinct region of large mass off lattice dla clusters with a branching ratio which converges asymptotically to the golden mean we suggest an interpretation of the dla morphology as a quasifractal counterpart of the well ordered snowflake fractal architecture

an efficient local search heuristic with row weighting for the unicost set covering problem abstract the set covering problem scp is np hard we propose a new row weighting local search rwls algorithm for solving the unicost variant of the scp i e uscps where the costs of all sets are identical rwls is a heuristic algorithm that has three major components united in its local search framework 1 a weighting scheme which updates the weights of uncovered elements to prevent convergence to local optima 2 tabu strategies to avoid possible cycles during the search and 3 a timestamp method to break ties when prioritizing sets rwls has been evaluated on a large number of problem instances from the or library and compared with other approaches it is able to find all the best known solutions bks and improve 14 of them although requiring a higher computational effort on several instances rwls is especially effective on the combinatorial or library instances and can improve the best known solution to the hardest instance cyc11 considerably rwls is conceptually simple and has no instance dependent parameters which makes it a practical and easy to use uscp solver

renormalization group calculation of distribution functions structural properties for percolation clusters distribution functions of properties of critical percolation clusters are calculated using the h cell real space renormalization group rg we consider structural properties which span two terminals on percolation clusters these include the lengths of minimal average edge to edge longest self avoiding walks the number of the singly connected bonds and the masses of percolation clusters as well as of the backbone we show that the rg corresponds to a galton watson branching process and apply theorems developed in the mathematical literature we derive recursion relations for the distribution functions and find exact functional forms for their asymptotic tail behavior at both small and large arguments the results for the minimal paths have implications on the multifractal distribution of wave functions while the singly connected bonds determine the moments of ising correlations on these clusters our results compare well with existing simulation data

mellin transform of the limit lognormal distribution the technique of intermittency expansions is applied to derive an exact formal power series representation for the mellin transform of the probability distribution of the limit lognormal multifractal process the negative integral moments are computed by a novel product formula of selberg type the power series is summed in general by means of its small intermittency asymptotic the resulting integral formula for the mellin transform is conjectured to be valid at all levels of intermittency the conjecture is verified partially by proving that the integral formula reproduces known results for the positive and negative integral moments of the limit lognormal distribution and gives a valid characteristic function of the levy khinchine type for the logarithm of the distribution the moment problem for the logarithm of the distribution is shown to be determinate whereas the moment problems for the distribution and its reciprocal are shown to be indeterminate the conjecture is used to represent the mellin transform as an infinite product of gamma factors generalizing selberg s finite product the conjectured probability density functions of the limit lognormal distribution and its logarithm are computed numerically by the inverse fourier transform

on the complexity of a class of combinatorial optimization problems with uncertainty we consider a robust minmax regret version of the problem of selecting p elements of minimum total weight out of a set of m elements with uncertainty in weights of the elements we present a polynomial algorithm with the order of complexity o min p m p 2m for the case where uncertainty is represented by means of interval estimates for the weights we show that the problem is np hard in the case of an arbitrary finite set of possible scenarios even if there are only two possible scenarios this is the first known example of a robust combinatorial optimization problem that is np hard in the case of scenario represented uncertainty but is polynomially solvable in the case of the interval representation of uncertainty