combinatorial characterization of sub hyperbolic rational maps in 1980 s thurston established a combinatorial characterization for post critically finite rational maps among post critically finite branched coverings of the two sphere to itself a completed proof was written by douady and hubbard in their paper a douady j h hubbard a proof of thurston s topological characterization of rational functions acta math 171 1993 263 297 this criterion was then extended by cui jiang and sullivan to sub hyperbolic rational maps among sub hyperbolic semi rational branched coverings of the two sphere to itself the goal of this paper is to present a new but simpler proof for the combinatorial characterization of sub hyperbolic rational maps by adapting some arguments in the proof in douady and hubbard s paper

linear programming in linear time when the dimension is fixed it is demonstrated that the linear programming problem in d variables and n constraints can be solved in o n time when d is fixed this bound follows from a multidimensional search technique which is applicable for quadratic programming as well there is also developed an algorithm that is polynomial in both n and d provided d is bounded by a certain slowly growing function of n

the sample average approximation method for stochastic discrete optimization in this paper we study a monte carlo simulation based approach to stochastic discrete optimization problems the basic idea of such methods is that a random sample is generated and the expected value function is approximated by the corresponding sample average function the obtained sample average optimization problem is solved and the procedure is repeated several times until a stopping criterion is satisfied we discuss convergence rates stopping rules and computational complexity of this procedure and present a numerical example for the stochastic knapsack problem

the hirsch conjecture for dual transportation polyhedra an algorithm is given that joins any pair of extreme points of a dual transportation polyhedron by a path of at most m 1 n 1 extreme edges

on the length of simplex paths the assignment case this paper reports on an experimental study of the distribution of the length of simplex paths for the optimal assignment problem we study the distribution of the pivot counts for a version of the simplex method that with essentially equal probabilities introduces any variable with negative reduced cost into the basis in this situation the distribution of the pivot counts turns out to be normally distributed and independent of the actual cost coefficients provided these are sufficiently spread out further the mean and standard deviation grow only moderately with the size of the problem namely asd1 8 andd1 5 respectively for ad d problem implying in particular that the pivot counts concentrate around the mean with growingd the usual simplex method on the other hand gives a growth ofd1 6 hence a large part of the favourable polynomial growth experienced on practical problems may be attributed to the fact that the simplex paths are rather short on the average at least for assignment problems

the combinatorial rigidity conjecture is false for cubic polynomials we show that there exist two cubic polynomials with connected julia sets which are combinatorially equivalent but not topologically conjugate on their julia sets this disproves a conjecture by mcmullen from 1995

on the average number of steps of the simplex method of linear programming the goal is to give some theoretical explanation for the efficiency of the simplex method of george dantzig fixing the number of constraints and using dantzig s self dual parametric algorithm we show that the number of pivots required to solve a linear programming problem grows in proportion to the number of variables on the average

fixed width sequential stopping rules for a class of stochastic programs monte carlo sampling based methods are frequently used in stochastic programming when exact solution is not possible a critical component of monte carlo sampling based methods is determining when to stop sampling to ensure the desired quality of the solutions in this paper we develop stopping rules for sequential sampling procedures that depend on the width of an optimality gap confidence interval estimator the procedures solve a sequence of sampling approximations with increasing sample size to generate solutions and stop when the width of the confidence interval on the current solution s optimality gap plus an inflation factor fall below a prespecified value epsilon we first present a method that takes the schedule of sample sizes as an input and provide guidelines on the growth of sample sizes then we present a method that increases the sample sizes according to the current estimates of the optimality gap the larger the estimates the larger the increases in sample sizes we provide conditi

on linear characterizations of combinatorial optimization problems we show that there can be no computationally tractable description by linear inequalities of the polyhedron associated with any np complete combinatorial optimization problem unless np co np a very unlikely event we also apply the ellipsoid method for linear programming to show that a combinatorial optimization problem is solvable in polynomial time if and only if it admits a small generator of violated inequalities

towards a genuinely polynomial algorithm for linear programming a linear programming algorithm is called genuinely polynomial if it requires no more than p m n arithmetic operations to solve problems of order m times n where p is a polynomial it is not known whether such an algorithm exists we present a genuinely polynomial algorithm for the simpler problem of solving linear inequalities with at most two variables per inequality the number of operations required is o mn 3 log text m the technique used was developed in a previous paper where a novel binary search idea was introduced

on locally constructible spheres and balls durhuus and jonsson 1995 introduced the class of locally constructible lc 3 spheres and showed that there are only exponentially many combinatorial types of simplicial lc 3 spheres such upper bounds are crucial for the convergence of models for 3d quantum gravity

new results on the average behavior of simplex algorithms it has been a challenge for mathematicians to theoreti cally confirm the extremely good performance of simplex algorithms for linear programming we have confirmed that a certain variant of the simplex method solves problems of order m x n in an expected num ber of steps which is bounded between two quadratic functions of the smaller dimension of the problem our probabilistic assumptions are rather weak 1 introduction we consider the linear programming problem of order m x n in the form maximize ctx over all x 0 in rn such that ax 5 6 where a e rmxn it has been observed that the simplex algorithms for linear programming developed by george dantzig dl work extremely well a central question in the field of analysis of algorithms is to estimate the expected number of steps that these algorithms perform relative to different probability distributions of inputs some background is given in 52 in am1 and t we consider a variant of the simplex method called the lexicographic self dual method we find that the expected number of steps of this variant relative to a rather weak probabilistic model is only o min m n 2 this is the first polynomial average case upper bound for a simplex algorithm which is capable of solving any linear programming problem the first two authors am21 have also determined a quadratic lower bound so the behavior of this variant is indeed quadratic whereas it has been conjectured that other variants take a linear expected number of steps with various choices of the starting point this variant can simulate different constraint by constraint and variable dimension algorithms as well as combinations thereof me2 adler karp and shamir aks provide a different proof of similar upper bound for a certain class of constraint by constraint algorithms it follows from me21 that their variant can also be simulated by the self dual algorithm with an appropriate starting point

a characterization of hyperbolic rational maps we give a topological characterization of rational maps with disconnected julia sets our results extend thurston s characterization of postcritically finite rational maps in place of iteration on teichmuller space we use quasiconformal surgery and thurston s original result

one hundred years of complex dynamics holomorphic dynamics has been a key area of study for 100 years in this survey we focus on the dynamics of rational functions

renormalizations and wandering jordan curves of rational maps we realize a dynamical decomposition for a post critically finite rational map which admits a combinatorial decomposition we split the riemann sphere into two completely invariant subsets one is a subset of the julia set consisting of uncountably many jordan curve components most of them are wandering the other consists of components that are pullbacks of finitely many renormalizations together with possibly uncountably many points the quotient action on the decomposed pieces is encoded by a dendrite dynamical system we also introduce a surgery procedure to produce post critically finite rational maps with wandering jordan curves and prescribed renormalizations

on the cycles of components of disconnected julia sets for any integers d ge 3 and n ge 1 we construct a hyperbolic rational map of degree d such that it has n cycles of the connected components of its julia set except single points and jordan curves

a decomposition theorem for herman maps abstract in 1980s thurston established a topological characterization theorem for postcritically finite rational maps in this paper a decomposition theorem for a class of postcritically infinite branched covering termed herman map is developed it s shown that every herman map can be decomposed along a stable multicurve into finitely many siegel maps and thurston maps such that the combinations and rational realizations of these resulting maps essentially dominate the original one this result is motivated by a non expanding version of mcmullen s problem and thurston s theory on characterization of rational maps it enables us to prove a thurston type theorem for rational maps with herman rings

real laminations and the topological dynamics of complex polynomials abstract we characterize the laminations associated to complex polynomials with connected julia sets and without irrationally neutral cycles

optimal allocation of simulation experiments in discrete stochastic optimization and approximative algorithms approximate solutions for discrete stochastic optimization problems are often obtained via simulation it is reasonable to complement these solutions by confidence regions for the argmin set we address the question how a certain total number of random draws should be distributed among the set of alternatives two goals are considered the minimization of the costs caused by using a statistical estimate of the true argmin and the minimization of the expected size of the confidence sets we show that an asymptotically optimal sampling strategy in the case of normal errors can be obtained by solving a convex optimization problem to reduce the computational effort we propose a regularization that leads to a simple one step allocation rule

unions of oriented matroids union or sum is an operation on ordinary matroids that yields great insight into matroid structure it has a wealth of applications and as pointed out by mirsky and others allows the treatment without difficulty of problems whose solution has previously required extremely complex arguments for all its power the operation is expressed simply in terms of independent sets in this paper the corresponding operation for oriented matroids is formulated in a constructive way by making use of the multiply ordered sets introduced by gutierrez novoa and algorithms are given for generating the union matroid for simple oriented matroids that is oriented matroids whose underlying matroids are uniform it is also shown that union yields another simple matroid and that the union operation may be carried out by concatenation of two set systems this construction is also useful in obtaining combinatorial types of some polytopes and convex polyhedral sets in particular concatenation is used to determine the polyhedral set found by klee and walkup as a counterexample to the polyhedral hirsch and d step conjectures

allocation of effort in monte carlo simulation for power of permutation tests abstract consider a randomization test in which an observed statistic is compared to a null distribution constructed by first drawing a random sample from the set of all possible permutations of the data and then calculating the statistic anew for each member of the random sample if the power of such a test is investigated by simulation the computer code must have two main loops in the outer loop which has o iterations data are generated under a specific alternative hypothesis for each outer iteration there are i inner iterations in each of which the data are permuted and the statistic calculated assume that the costs of inner and outer iterations are known and the total cost of the study is fixed so that i and o are constrained by o io for fixed this article considers what the optimal choice for i is in a mean squared sense a general expression is derived for the mean squared error of the distribution of p values if the alternative hypothesis is close to the null the optimal c

the efficiency of the simplex method a survey the linear programming problem is by far the most widely used optimization model its impact on economic and government modeling is immense the simplex method for solving the linear programming lp problem due to george dantzig has been an extremely efficient computational tool for almost four decades the method has been the subject of intense investigations for many years but some major aspects of its behavior are not fully understood yet the purpose of this paper is to survey the body of knowledge on the efficiency of the simplex method from both practical and theoretical points of view adopting the number of iterations pivot steps as the yardstick for efficiency we survey four aspects of the issue 1 reports on practical experience of the method s performance on real life lp problems 2 results on controlled monte carlo experiments solving lp problems which were randomly generated according to some predetermined distributions 3 complexity results including theoretical analyses on both upper and lower bounds for the performance of the simplex as well as non simplex algorithms for lp 4 results of recent theoretical studies using probabilistic analysis to derive bounds on the average behavior of the simplex method we discuss the consequences and limitations of the various studies special emphasis is given to open questions

is binary encoding appropriate for the problem language relationship abstract it is proved that there exist encoding schemes which are arbitrarily as efficient as the binary encoding in terms of compactness and arithmetic operations with respect to which khachiyan s algorithm for linear programming is exponential this constitutes an objection to the standard translation of problems into languages via the binary encoding

discrete morse theory for manifolds with boundary we introduce a version of discrete morse theory specific for manifolds with boundary the idea is to consider morse functions for which all boundary cells are critical we obtain relative morse inequalities relating the homology of the manifold to the number of interior critical cells we also derive a ball theorem in analogy to forman s sphere theorem the main corollaries of our work are 1 for each d 3 and for each k 0 there is a pl d sphere on which any discrete morse function has more than k critical d 1 cells this solves a problem by chari 2 for fixed d and k there are exponentially many combinatorial types of simplicial d manifolds counted with respect to the number of facets that admit discrete morse functions with at most k critical interior d 1 cells this connects discrete morse theory to enumerative combinatorics discrete quantum gravity 3 the barycentric subdivision of any simplicial constructible d ball is collapsible this almost solves a problem by hachimori 4 every constructible ball collapses onto its boundary minus a facet this improves a result by the author and ziegler 5 any 3 ball with a knotted spanning edge cannot collapse onto its boundary minus a facet this strengthens a classical result by bing and a recent result by the author and ziegler

geometrically finite and semi rational branched coverings of the two sphere in 1982 thurston gave a necessary and sufficient condition for a critically finite branched covering of the two sphere to itself to be combinatorially equivalent to a rational map we discuss extending this result to geometrically finite rational maps we give an example to show that thurston s original condition is not sufficient this example is topologically pathological near accumulation points of the postcritical set we give two conditions forbidding such pathology show that they are equivalent and in a sequel to the present paper will show that thurston s condition together with this tameness is both necessary and sufficient to characterize geometrically finite

global topology of hyperbolic components cantor circle case we prove that in the moduli space md of degree d 2 rational maps any hyperbolic component in the disconnectedness locus and of cantor circle type is a finite quotient of r4d 4 n tn where n is determined by dynamics the proof uses some ideas from riemann surface theory abel s theorem dynamical system and algebraic topology

decompositions of simplicial balls and spheres with knots consisting of few edges constructibility is a condition on pure simplicial complexes that is weaker than shellability in this paper we show that non constructible triangulations of the d dimensional sphere exist for every d geq 3 this answers a question of danaraj and klee 10 it also strengthens a result of lickorish 16 about non shellable spheres furthermore we provide a hierarchy of combinatorial decomposition properties that follow from the existence of a non trivial knot with few edges in a 3 sphere or 3 ball and a similar hierarchy for 3 balls with a knotted spanning arc that consists of few edges

a proof of thurston s topological characterization of rational functions the criterion proved in this paper was stated by thurston in november 1982 thurston lectured on its proof on several occasions notably at the nsf summer conference in duluth 1983 where one of the authors jhh was present using the notes of various attendants at these lectures we have reconstructed a proof that we have made as precise as we could since this required a certain amount of work on our part we thought it might be of some use to present this proof to the reader we thank dennis sullivan for useful conversations and fritz yon haesseler and especially ben wittner for help with the writing and valuable suggestions after the first version was written clifford earle pointed out that better estimates than what we had were to be found in b

the hirsch conjecture holds for normal flag complexes using an intuition from metric geometry we prove that any flag normal simplicial complex satisfies the nonrevisiting path conjecture as a consequence the diameter of its facet ridge graph is smaller than the number of vertices minus the dimension as in the hirsch conjecture this proves the hirsch conjecture for all flag polytopes and more generally for all connected flag homology manifolds

the d step conjecture and its relatives the d step conjecture arose from an attempt to understand the computational complexity of edge following algorithms for linear programming such as the simplex algorithm it can be stated in terms of diameters of graphs of convex polytopes in terms of the existence of nonrevisiting paths in such graphs in terms of an exchange process for simplicial bases of a vector space and in terms of matrix pivot operations first formulated by w m hirsch in 1957 the conjecture remains unsettled though it has been proved in many special cases and counterexamples have been found for slightly stronger conjectures if the conjecture is false as we believe to be the case then finding a counterexample will be merely a small first step in the line of investigation related to the conjecture this report summarizes what is known about the d step conjecture and its relatives a considerable amount of new material is included but it does not seem to come close to settling the conjecture of special interest is the first example of a polytope that is not vertex decomposable showing that a certain natural approach to the conjecture will not work also significant are the quantitative relations among the lengths of paths associated with various forms of the conjecture