fast approximation algorithms on maxcut k coloring and k color ordering for vlsi applications there are a number of vlsi problems that have a common structure we investigate such a structure that leads to a unified approach for three independent vlsi layout problems partitioning placement and via minimization along the line we first propose a linear time approximation algorithm on maxcut and two closely related problems k coloring and maximal k color ordering problem the k coloring is a generalization of the maxcut and the maximal k color ordering is a generalization of the k coloring for a graph g with e edges and n vertices our maxcut approximation algorithm runs in o e n sequential time yielding a nodebalanced maxcut with size at least w e w e n 2 improving the time complexity of o e log e known before building on the proposed maxcut technique and employing a height balanced binary decomposition we devise an o e n log k time algorithm for the k coloring problem which always finds a k partition of vertices such that the number of bad or defected edges does not exceed w e k n 1 n sup log k thus improving both the time complexity o enk and the bound e k known before the other related problem is the maximal k color ordering problem that has been an open problem we show the problem is np complete then present an approximation algorithm building on our k coloring structure a performance bound on maximal k color ordering cost 2kw e 3 is attained in o ek time the solution quality of this algorithm is also tested experimentally and found to be effective

oscillation results for linear hamiltonian systems for linear hamiltonian systems this paper established some new oscillation criteria which improve previous results even for self adjoint second order differential systems the main tool used is a generalized riccati transformation

nonlinear functionals and a theorem of sun we use a recently developed theory of nonlinear functionals in the study of oscillations of second order symmetric vector differential systems to extend a number of theorems of sun new kamenev type theorems for second order linear matrix differential systems appl math lett 2004 in press under a common theme the criteria presented here are of the form the integral of the coefficient matrix is bounded at infinity in a sense to be made explicit in the paper and bounded away from a positive absolute constant implies oscillation at infinity

the effect of variable change on oscillation and disconjugacy criteria with applications to spectral theory and asymptotic theory this study of the effect of variable change on differential operators was motivated by recent papers of tipler 131 and hinton and lewis 37 which appeared in the same issue of the journal of d erential equations tipler gave a geometric proof 13 p 1671 of the following result of hawking and penrose see 29 301 let f t be a real valued continuous nonnegative function on w co such that f b 0 for some point b then there exists a pair of conjugate points on co co for the differential equation

the fourth moment in luby s distribution luby 1988 proposed a way to derandomize randomized computations which is based on the construction of a small probability space whose elements are 3 wise independent in this paper we prove some new properties of luby s space more precisely we analyze the fourth moment and prove an interesting technical property which helps to understand better luby s distribution as an application we study the behavior of random edge cuts in a weighted graph

the maximal f dependent set problem for planar graphs is in nc the maximal f dependent set max f ds problem is the following problem given a graph g v e and a nonnegative integer valued function f defined on v find a maximal subset u of v such that no vertex u u has degree f u in the subgraph induced by u whether the problem is in nc or rnc or not is an open question concerning this question only a rather trivial result due to diks garrido and lingas is known up to now which says that the problem can be solved in nc if the maximum value of f is poly logarithmic in the input size proceedings of the 2nd international symposium on algorithms lncs 557 1991 385 395 in this paper we show a nontrivial interesting result that the max f ds problem for planar graphs can be solved in o log5n time with o n processors on a crcw pram where n is the input size

using maximal independent sets to solve problems in parallel by using an o log n 2 time erew pram algorithm for a maximal independent set problem mis we show the following two results 1 given a graph the maximal vertex induced subgraph satisfying a hereditary graph property can be found in time 0 t n log n 2 using a polynomial number of processors where is the maximum of diameters of minimal graphs violating and t n is the time needed to decide whether a graph with n vertices satisfies 2 given a family c c1 c m of subsets of a finite set s 1 n with s u i 1 m ci a minimal set cover for s can be computed on an erew pram in time o log n m 2 using a polynomial number of processors where max ci i 1 m and max dj j 1 n

tight bounds for some risk measures with applications to robust portfolio selection in this paper we develop tight bounds on the expected values of several risk measures that are of interest to us this work is motivated by the robust optimization models arising from portfolio selection problems indeed the whole paper is centered around robust portfolio models and solutions the basic setting is to find a portfolio that maximizes respectively minimizes the expected utility respectively disutility values in the midst of infinitely many possible ambiguous distributions of the investment returns fitting the given mean and variance estimations first we show that the single stage portfolio selection problem within this framework whenever the disutility function is in the form of lower partial moments lpm or conditional value at risk cvar or value at risk var can be solved analytically the results lead to the solutions for single stage robust portfolio selection models furthermore the results also lead to a multistage adjustable robust optimization aro solution when the disutility function is the second order lpm exploring beyond the confines of convex optimization we also consider the so called s shaped value function which plays a key role in the prospect theory of kahneman and tversky the nonrobust version of the problem is shown to be np hard in general however we present an efficient procedure for solving the robust counterpart of the same portfolio selection problem in this particular case the consideration of the robustness actually helps to reduce the computational complexity finally we consider the situation whereby we have some additional information about the chance that a quadratic function of the random distribution reaches a certain threshold that information helps to further reduce the ambiguity in the robust model we show that the robust optimization problem in that case can be solved by means of semidefinite programming sdp if no more than two additional chance inequalities are to be incorporated

convex envelopes for edge concave functions deterministic global optimization algorithms frequently rely on the convex underestimation of nonconvex functions in this paper we describe the structure of the polyhedral convex envelopes of edge concave functions over polyhedral domains using geometric arguments an algorithm for computing the facets of the convex envelope over hyperrectangles in 3 is described sufficient conditions are described under which the convex envelope of a sum of edge concave functions may be shown to be equivalent to the sum of the convex envelopes of these functions

concave extensions for nonlinear 0 1 maximization problems a well known linearization technique for nonlinear 0 1 maximization problems can be viewed as extending any polynomial in 0 1 variables to a concave function defined on 0 1 n some properties of this standard concave extension are investigated polynomials for which the standard extension coincides with the concave envelope are characterized in terms of integrality of a certain polyhedron or balancedness of a certain matrix the standard extension is proved to be identical to another type of concave extension defined as the lower envelope of a class of affine functions majorizing the given polynomial

new oscillation criteria for linear matrix hamiltonian systems abstract some new oscillation criteria are established for the matrix linear hamiltonian system x a t x b t y y c t x a t y under the hypothesis a t b t b t 0 and c t c t are n n real continuous matrix functions on the interval t 0 t 0 these results are sharper than some previous results even for self adjoint second order matrix differential systems

probe order backtracking the algorithm for constraint satisfaction problems probe order backtracking has an average running time much faster than any previously analyzed algorithm under conditions where solutions are common the algorithm uses a probing assignment a preselected test assignment to unset variables to help guide the search for a solution if the problem is not satisfied when the unset variables are temporarily set to the probing assignment the algorithm selects one of the relations which is not satisfied by the probing assignment and selects an unset variable which affects the value of that relation it then does a backtracking splitting step where it generates subproblems by setting the selected variable each possible way each subproblem is simplified and then solved recursively for random problems with v variables t clauses and probability p that a literal appears in a clause the average time for probe order backtracking is no more than v n when p geq ln t v plus lower order terms the best previous result was p geq sqrt ln t v when the algorithm is combined with an algorithm of franco that makes selective use of resolution the average time for solving random problems is no more than v n for all values of p when t leq o n 1 3 v ln v 2 3 the best previous result was t leq o n 1 3 v ln v 1 6 probe order backtracking also runs in polynomial average time when p leq 1 v compared with the best previous result of p leq 1 2v with probe order backtracking the range of p that leads to more than polynomial time is much smaller than that for previously analyzed algorithms

oscillation theorems for self adjoint matrix hamiltonian systems involving general means by use of monotone functionals and positive linear functionals a generalized riccati transformation and the general means technique some new oscillation criteria for the following self adjoint hamiltonian matrix system e x t a t x t b t y t y t c t x t a t y t are obtained the results obtained improve and complement that of kumari et al 2000 on kamenev type theorems moreover these results generalize and improve earlier results due to meng 2002 for e erbe et al 1993 meng et al 1998 and wang 2001 for p t x t q t x t 0 or its special cases and wong 2001 for the scalar system x t q t x t 0

low temperature stability of nonperiodic structures complex ground states appearing in classical lattice gas models and their behaviour in the presence of thermal motions are discussed our examples consist of a nearest neighbour model based on nonperiodic tilings of the plane and an ising type model with a fractal ground state

a semidefinite programming approach to optimal moment bounds for convex classes of distributions we provide an optimization framework for computing optimal upper and lower bounds on functional expectations of distributions with special properties given moment constraints bertsimas and popescu optimal inequalities in probability theory a convex optimization approach siam j optim 2004 forthcoming have already shown how to obtain optimal moment inequalities for arbitrary distributions via semidefinite programming these bounds are not sharp if the underlying distributions possess additional structural properties including symmetry unimodality convexity or smoothness for convex distribution classes that are in some sense generated by an appropriate parametric family we use conic duality to show how optimal moment bounds can be efficiently computed as semidefinite programs in particular we obtain generalizations of chebyshev s inequality for symmetric and unimodal distributions and provide numerical calculations to compare these bounds given higher order moments we also extend these results for multivariate distributions

approximation and intractability results for the maximum cut problem and its variants the maximum cut problem is known to be an important np complete problem with many applications the authors investigate this problem which they call the normal maximum cut problem and a variant of it which is referred to as the connected maximum cut problem they show that any n vertex e edge graph admits a cut with at least the fraction 1 2 1 2n of its edges thus improving the ratio 1 2 2 e known before it is shown that it is np complete to decide if a given graph has a normal maximum cut with at least a fraction 1 2 epsilon of its edges where the positive constant epsilon can be taken smaller than any value chosen the authors present an approximation algorithm for the normal maximum cut problem on any graph that runs in o e log e n log n p log p log n parallel time using p 1

the fourth moment method higher moment analysis has typically been used to upper bound certain functions in this paper we introduce a new combinatorial method to lower bound the expectation of the absolute value of a random variable x by the expectation of a quartic in x in the special case where we are looking at the absolute value of a weighted sum of 1 1 unbiased random variables we achieve tight bounds using only a fourth moment for the total discrepancy of a set system because the fourth moment depends only on 4 wise independence our bounds will hold over polynomially sized distributions and so these bounds will be directly applicable in removing randomness to obtain nc algorithms we obtain the first nc algorithms for the problems of total discrepancy maximum acyclic subgraph tournament ranking the gale berlekamp switching game and edge discrepancy we show that for most of these applications it is truly necessary to consider a fourth moment by exhibiting a 3 wise independent distribution which does not achieve the required bounds our method is strong enough to give a new combinatorial bound on tournament ranking

integral inequalities and second order linear oscillation on the interval 0 a u 0 below by a solution we mean a non trivial solution a solution is oscillatory if it has an infinite number of zeros numerous oscillation criteria are known and diversified approaches have proved fruitful among these is the use of the associated riccati equation although the theory of differential and integral inequalities has been used by some investigators see e g hartman 4 71 the authors believe that it has not been fully exploited the non oscillation of 1 1 is equivalent to the existence of a solution of the associated riccati equation on a half line thus one technique of proving oscillation is to show that any solution of the riccati equation diverges to i co at a finite point the theory of differential and integral inequalities comes in handy in the estimation of the growth of such a solution below we describe some known results that are relevant to results established in this paper although these are stated here for the case i co in the rest of the paper we consider both cases a 0 and a 00 it is well known that if q is large in the mean then 1 1 is oscillatory a typical example of such a criterion is due to fite leighton and wintner if

oscillation criteria for linear matrix hamiltonian systems abstract in this paper we present oscillation criteria in terms of the coefficient functions for the matrix linear hamiltonian system u a x u b x v v c x u a x v under the hypothesis h a x b x b x 0 and c x c x are n n matrices of real valued continuous functions on the interval i a a these criteria generalize earlier results due to etgen and pawlowski 1976 erbe qingkai kong and shigui ruan 1993 and fanwei meng jizhong wang and zhaowen zheng 1998 for the system p x u q x u 0 or its special cases

efficient graph based image segmentation this paper addresses the problem of segmenting an image into regions we define a predicate for measuring the evidence for a boundary between two regions using a graph based representation of the image we then develop an efficient segmentation algorithm based on this predicate and show that although this algorithm makes greedy decisions it produces segmentations that satisfy global properties we apply the algorithm to image segmentation using two different kinds of local neighborhoods in constructing the graph and illustrate the results with both real and synthetic images the algorithm runs in time nearly linear in the number of graph edges and is also fast in practice an important characteristic of the method is its ability to preserve detail in low variability image regions while ignoring detail in high variability regions

on convex envelopes for bivariate functions over polytopes in this paper we discuss convex envelopes for bivariate functions satisfying suitable assumptions over polytopes we first propose a technique to compute the value and a supporting hyperplane of the convex envelope over a general two dimensional polytope through the solution of a three dimensional convex subproblem with continuously differentiable constraint functions then for quadratic functions as well as for some polynomial and rational ones again satisfying suitable assumptions we show how the same computations can be carried out through the solution of a single semidefinite problem

generalized chebychev inequalities theory and applications in decision analysis in many decision analysis problems we have only limited information about the relevant probability distributions in problems like these it is natural to ask what conclusions can be drawn on the basis of this limited information for example in the early stages of analysis of a complex problem we may have only limited fractile information for the distributions in the problem what can we say about the optimal strategy or certainty equivalents given these few fractiles this paper describes a very general framework for analyzing these kinds of problems where given certain moments of a distribution we can compute bounds on the expected value of an arbitrary objective function by suitable choice of moment and objective functions we can formulate and solve many practical decision analysis problems we describe the general framework and theoretical results discuss computational strategies and provide specific results for examples in dynamic programming decision analysis with incomplete information bayesian statistics and option pricing

oscillation theorems for self adjoint matrix hamiltonian systems by using generalized riccati technique linear positive functional and the weighted averages technique some new oscillation criteria for self adjoint hamiltonian matrix system e u x a x u x b x v x v x c x u x a x v x are obtained these results generalize and improve earlier results due to kumari and umamaheswaram 9 for e parhi and praharaj 12 etgen and pawlowski 6 for p x u x q x u x 0 or its special cases and coles 3 for the scalar system y x q x y x 0

a fast parallel algorithm for the maximal independent set problem a parallel algorithm is presented that accepts as input a graph g and produces a maximal independent set of vertices in g on a p ram without the concurrent write or concurrent read features the algorithm executes in o log n 4 time and uses o n log n 3 processors where n is the number of vertices in g the algorithm has several novel features that may find other applications these include the use of balanced incomplete block designs to replace random sampling by deterministic sampling and the use of a dynamic pigeonhole principle that generalizes the conventional pigeonhole principle

on the dualization of hypergraphs with bounded edge intersections and other related classes of hypergraphs given a finite set v and integers k 1 and r 0 let us denote by the class of hypergraphs with k r bounded intersections i e in which the intersection of any k distinct hyperedges has size at most r we consider the problem given a hypergraph and a subfamily of its maximal independent sets mis either extend this subfamily by constructing a new mis or prove that there are no more mis that is it is known that for hypergraphs of bounded dimension as well as for hypergraphs of bounded degree where is a constant problem can be solved in incremental polynomial time in this paper we extend this result to any integers k r such that k r is a constant more precisely we show that for hypergraphs with k r const problem is nc reducible to the problem of generating a single mis for a partial subhypergraph of in particular this implies that is polynomial and we get an incremental polynomial algorithm for generating all mis furthermore combining this result with the currently known algorithms for finding a single maximally independent set of a hypergraph we obtain efficient parallel algorithms for incrementally generating all mis for hypergraphs in the classes and where is a constant we also show that for where k r const the problem of generating all mis of can be solved in incremental polynomial time and with space polynomial only in the size of

optimal inapproximability results for max cut and other 2 variable csps in this paper we show a reduction from the unique games problem to the problem of approximating max cut to within a factor of alpha text tiny gw epsilon for all epsilon 0 here alpha text tiny gw approx 878567 denotes the approximation ratio achieved by the algorithm of goemans and williamson in j assoc comput mach 42 1995 pp 1115 1145 this implies that if the unique games conjecture of khot in proceedings of the 34th annual acm symposium on theory of computing 2002 pp 767 775 holds then the goemans williamson approximation algorithm is optimal our result indicates that the geometric nature of the goemans williamson algorithm might be intrinsic to the max cut problem our reduction relies on a theorem we call majority is stablest this was introduced as a conjecture in the original version of this paper and was subsequently confirmed in e mossel r o donnell and k oleszkiewicz proceedings of the 46th annual ieee symposium on foundations of computer science 2005 pp 21 30 a stronger version of this conjecture called plurality is stablest is still open although e mossel r o donnell and k oleszkiewicz proceedings of the 46th annual ieee symposium on foundations of computer science 2005 pp 21 30 contains a proof of an asymptotic version of it our techniques extend to several other two variable constraint satisfaction problems in particular subject to the unique games conjecture we show tight or nearly tight hardness results for max 2sat max q cut and max 2lin q for max 2sat we show approximation hardness up to a factor of roughly 943 this nearly matches the 940 approximation algorithm of lewin livnat and zwick in proceedings of the 9th annual conference on integer programming and combinatorial optimization springer verlag berlin 2002 pp 67 82 furthermore we show that our 943 factor is actually tight for a slightly restricted version of max 2sat for max q cut we show a hardness factor which asymptotically for large q matches the approximation factor achieved by frieze and jerrum improved approximation algorithms for max k cut and max bisection in integer programming and combinatorial optimization springer verlag berlin pp 1 13 namely 1 1 q 2 rm ln q q 2 for max 2lin q we show hardness of distinguishing between instances which are 1 epsilon satisfiable and those which are not even roughly q epsilon 2 satisfiable these parameters almost match those achieved by the recent algorithm of charikar makarychev and makarychev proceedings of the 38th annual acm symposium on theory of computing 2006 pp 205 214 the hardness result holds even for instances in which all equations are of the form x i x j c at a more qualitative level this result also implies that 1 epsilon vs e hardness for max 2lin q is equivalent to the unique games conjecture

on distributional robust probability functions and their computations abstract consider a random vector and assume that a set of its moments information is known among all possible distributions obeying the given moments constraints the envelope of the probability distribution functions is introduced in this paper as distributional robust probability function we show that such a function is computable in the bi variate case under some conditions connections to the existing results in the literature and its applications in risk management are discussed as well

the boolean quadric polytope some characteristics facets and relatives we study unconstrained quadratic zero one programming problems havingn variables from a polyhedral point of view by considering the boolean quadric polytope qpn inn n 1 2 dimensions that results from the linearization of the quadratic form we show that qpn has a diameter of one descriptively identify three families of facets of qpn and show that qpn is symmetric in the sense that all facets of qpn can be obtained from those that contain the origin by way of a mapping the naive linear programming relaxation qpnlp of qpn is shown to possess the trubin property and its extreme points are shown to be 0 1 2 1 valued furthermore o n3 facet defining inequalities of qpn suffice to cut off all fractional vertices of qpnlp whereas the family of facets described by us has at least o 3n members the problem is also studied for sparse quadratic forms i e when several or many coefficients are zero and conditions are given for the previous results to carry over to this case polynomially solvable problem instances are discussed and it is shown that the known polynomiality result for the maximization of nonnegative quadratic forms can be obtained by simply rounding the solution to the linear programming relaxation in the case that the graph induced by the nonzero coefficients of the quadratic form is series parallel a complete linear description of the associated boolean quadric polytope is given the relationship of the boolean quadric polytope associated to sparse quadratic forms with the vertex packing polytope is discussed as well

a polyhedral branch and cut approach to global optimization a variety of nonlinear including semidefinite relaxations have been developed in recent years for nonconvex optimization problems their potential can be realized only if they can be solved with sufficient speed and reliability unfortunately state of the art nonlinear programming codes are significantly slower and numerically unstable compared to linear programming software in this paper we facilitate the reliable use of nonlinear convex relaxations in global optimization via a polyhedral branch and cut approach our algorithm exploits convexity either identified automatically or supplied through a suitable modeling language construct in order to generate polyhedral cutting planes and relaxations for multivariate nonconvex problems we prove that if the convexity of a univariate or multivariate function is apparent by decomposing it into convex subexpressions our relaxation constructor automatically exploits this convexity in a manner that is much superior to developing polyhedral outer approximators for the original function the convexity of functional expressions that are composed to form nonconvex expressions is also automatically exploited root node relaxations are computed for 87 problems from globallib and minlplib and detailed computational results are presented for globally solving 26 of these problems with baron 7 2 which implements the proposed techniques the use of cutting planes for these problems reduces root node relaxation gaps by up to 100 and expedites the solution process often by several orders of magnitude

oscillation criteria related to integral averaging technique for linear matrix hamiltonian systems by employing a generalized riccati technique and an integral averaging technique new oscillation criteria are established for the linear matrix hamiltonian system u a x u b x v v c x u a x v under the assumption that all a x b x b x 0 and c x c x are n n matrices of real valued continuous functions on the interval i x0 x0 these criteria extend improve and unify a number of existing results and also handle cases not covered by known criteria several interesting examples that illustrate the importance of our results are included

complexity of partial satisfaction a conjunctive normal form expression cnf is said to be 2 satisfiable if and only if any two of its clauses are simultaneously satisfiable it is shown that every 2 satisfiable cnf has a truth assignment that satisfies at least the fraction h of its clauses where h sqrt 5 1 2 sim 0 618 the reciprocal of the golden ratio the proof is constructive in that it provides a polynomial time algorithm that will find for any 2 satisfiable cnf a truth assignment satisfying at least the fraction h of its clauses furthermore this result is optimal in that the constant h is as large as possible it is shown that for any rational h h the set of all 2 satisfiable cnfs that have truth assignments satisfying at least the fraction h of their clauses is an np complete set

parallel algorithms for finding a near maximum independent set of a circle graph a parallel algorithm for finding a near maximum independent set in a circle graph is presented an independent set in a graph is a set of vertices no two of which are adjacent a maximum independent set is an independent set whose cardinality is the largest among all independent sets of a graph the algorithm is modified for predicting the secondary structure in ribonucleic acids rna the proposed system composed of an n neural network array where n is the number of edges in the circle graph of the number of possible base pairs not only generates a near maximum independent set but also predicts the secondary structure of ribonucleic acids within several hundred iteration steps the simulator discovered several solutions which are more stable structures in a sequence of 359 bases from the potato spindle tuber viroid than previously proposed structures

a new bound for 3 satisfiable maxsat and its algorithmic application let f be a cnf formula with n variables and m clauses f is 3 satisfiable if for any 3 clauses in f there is a truth assignment which satisfies all of them lieberherr and specker 1982 and later yannakakis 1994 proved that in each 3 satisfiable cnf formula at least 23 of its clauses can be satisfied by a truth assignment we improve this result by showing that every 3 satisfiable cnf formula f contains a subset of variables u such that some truth assignment t will satisfy at least 23m 13m u rn clauses where m is the number of clauses of f m u is the number of clauses of f containing a variable from u n is the total number of variables in clauses not containing a variable in u and r is a positive absolute constant both u and t can be found in polynomial time we use our result to show that the following parameterized problem is fixed parameter tractable and moreover has a kernel with a linear number of variables in 3 s maxsat ae we are given a 3 satisfiable cnf formula f with m clauses and asked to determine whether there is an assignment which satisfies at least 23m k clauses where k is the parameter

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

jointly constrained biconvex programming this paper presents a branch and bound algorithm for minimizing the sum of a convex function in x a convex function in y and a bilinear term in x and y over a closed set such an objective function is called biconvex with biconcave functions similarly defined the feasible region of this model permits joint constraints in x and y to be expressed the bilinear programming problem becomes a special case of the problem addressed in this paper we prove that the minimum of a biconcave function over a nonempty compact set occurs at a boundary point of the set and not necessarily an extreme point the algorithm is proven to converge to a global solution of the nonconvex program we discuss extensions of the general model and computational experience in solving jointly constrained bilinear programs for which the algorithm has been implemented

maximizing non monotone submodular functions submodular maximization generalizes many important problems including max cut in directed and undirected graphs and hypergraphs certain constraint satisfaction problems and maximum facility location problems unlike the problem of minimizing submodular functions the problem of maximizing submodular functions is np hard in this paper we design the first constant factor approximation algorithms for maximizing nonnegative non monotone submodular functions in particular we give a deterministic local search frac 1 3 approximation and a randomized frac 2 5 approximation algorithm for maximizing nonnegative submodular functions we also show that a uniformly random set gives a frac 1 4 approximation for symmetric submodular functions we show that a random set gives a frac 1 2 approximation which can also be achieved by deterministic local search these algorithms work in the value oracle model where the submodular function is accessible through a black box returning f s for a given set s we show that in this model a frac 1 2 epsilon approximation for symmetric submodular functions would require an exponential number of queries for any fixed epsilon 0 in the model where f is given explicitly as a sum of nonnegative submodular functions each depending only on a constant number of elements we prove np hardness of frac 5 6 epsilon approximation in the symmetric case and np hardness of frac 3 4 epsilon approximation in the general case

a microscopic model with quasicrystalline properties a classical lattice gas model with two body nearest neighbor interactions and without periodic ground state configurations is presented the main result is the existence of a decreasing sequence of temperatures for which the gibbs states have arbitrarily long periods it is possible that the sequence accumulates at nonzero temperature giving rise to a quasiperiodic equilibrium state

optimal inequalities in probability theory a convex optimization approach we propose a semidefinite optimization approach to the problem of deriving tight moment inequalities for p x in s for a set s defined by polynomial inequalities and a random vector x defined on omega subseteq cal r n that has a given collection of up to kth order moments in the univariate case we provide optimal bounds on p x in s when the first k moments of x are given as the solution of a semidefinite optimization problem in k 1 dimensions in the multivariate case if the sets s and omega are given by polynomial inequalities we obtain an improving sequence of bounds by solving semidefinite optimization problems of polynomial size in n for fixed k we characterize the complexity of the problem of deriving tight moment inequalities we show that it is np hard to find tight bounds for k geq 4 and omega cal r n and for k geq 2 and omega cal r n when the data in the problem is rational for k 1 and omega cal r n we show that we can find tight upper bounds by solving n convex optimization problems when the set s is convex and we provide a polynomial time algorithm when s and omega are unions of convex sets over which linear functions can be optimized efficiently for the case k 2 and omega cal r n we present an efficient algorithm for finding tight bounds when s is a union of convex sets over which convex quadratic functions can be optimized efficiently

classical lattice gas model with a unique nondegenerate but unstable periodic ground state configuration we construct a classical lattice gas model with a unique periodic ground state configuration such that the peierls condition is not satisfied the ground state configuration is nondegenerate which means that for any fixed energye and any integern the diameter of the support of alln connected local excitations with energy less thane is bounded nevertheless the configuration is not stable it does not give rise to a low temperature phase any translation invariant gibbs state of our model corresponds to quasiperiodic ground state configurations this requires the modification of a recent hypothesis of dobrushin and shlosman

tight bounds and 2 approximation algorithms for integer programs with two variables per inequality the problem of integer programming in bounded variables over constraints with no more than two variables in each constraint is np complete even when all variables are binary this paper deals with integer linear minimization problems inn variables subject tom linear constraints with at most two variables per inequality and with all variables bounded between 0 andu for such systems a 2 approximation algorithm is presented that runs in time o mnu2 log un2m so it is polynomial in the input size if the upper boundu is polynomially bounded the algorithm works by finding first a super optimal feasible solution that consists of integer multiples of 1 2 that solution gives a tight bound on the value of the minimum it furthermore has an identifiable subset of integer components that retain their value in an integer optimal solution of the problem these properties are a generalization of the properties of the vertex cover problem the algorithm described is in particular a 2 approximation algorithm for the problem of minimizing the total weight of true variables among all truth assignments to the 2 satisfiability problem

on sums of independent random variables with unbounded variance and estimating the average degree in a graph we prove the following inequality for every positive integer n and every collection x 1 ldots x n of nonnegative independent random variables each with expectation 1 the probability that their sum remains below n 1 is at least alpha 0 our proof produces a value of alpha 1 13 simeq 0 077 but we conjecture that the inequality also holds with alpha 1 e simeq 0 368 as an example for the use of the new inequality we consider the problem of estimating the average degree of a graph by querying the degrees of some of its vertices we show the following threshold behavior approximation factors above 2 require far fewer queries than approximation factors below 2 the new inequality is used in order to get tight up to multiplicative constant factors relations between the number of queries and the quality of the approximation we show how the degree approximation algorithm can be used in order to quickly find those edges in a network that belong to many shortest paths

some results on the strength of relaxations of multilinear functions we study approaches for obtaining convex relaxations of global optimization problems containing multilinear functions specifically we compare the concave and convex envelopes of these functions with the relaxations that are obtained with a standard relaxation approach due to mccormick the standard approach reformulates the problem to contain only bilinear terms and then relaxes each term independently we show that for a multilinear function having a single product term this approach yields the convex and concave envelopes if the bounds on all variables are symmetric around zero we then review and extend some results on conditions when the concave envelope of a multilinear function can be written as a sum of concave envelopes of its individual terms finally for bilinear functions we prove that the difference between the concave upper bounding and convex lower bounding functions obtained from the mccormick relaxation approach is always within a constant of the difference between the concave and convex envelopes these results along with numerical examples we provide give insight into how to construct strong relaxations of multilinear functions

bounding probability of small deviation a fourth moment approach in this paper we study the problem of upper bounding the probability that a random variable is above its expected value by a small amount relative to the expected value by means of the second and the fourth central moments of the random variable in this particular context many classical inequalities yield only trivial bounds we obtain tight upper bounds by studying the primal dual moments generating conic optimization problems as an application we demonstrate that given the new probability bounds a substantial sharpening and simplification of a recent result and its analysis by feige see feige u 2006 on sums of independent random variables with unbounded variances and estimating the average degree in a graph siam j comput 35 964 984 can be obtained also these bounds lead to new properties of the distribution of the cut values for the max cut problem we expect the new probability bounds to be useful in many other applications

identifying the minimal transversals of a hypergraph and related problems the paper considers two decision problems on hypergraphs hypergraph saturation and recognition of the transversal hypergraph and discusses their significance for several search problems in applied computer science hypergraph saturation i e given a hypergraph cal h decide if every subset of vertices is contained in or contains some edge of cal h is shown to be co np complete a certain subproblem of hypergraph saturation the saturation of simple hypergraphs i e sperner families is shown to be under polynomial transformation equivalent to transversal hypergraph recognition i e given two hypergraphs cal h 1 cal h 2 decide if the sets in cal h 2 are all the minimal transversals of cal h 1 the complexity of the search problem related to the recognition of the transversal hypergraph the computation of the transversal hypergraph is an open problem this task needs time exponential in the input size it is unknown whether an output polynomial algorithm exists for several important subcases for instance if an upper or lower bound is imposed on the edge size or for acyclic hypergraphs output polynomial algorithms are presented computing or recognizing the minimal transversals of a hypergraph is a frequent problem in practice which is pointed out by identifying important applications in database theory boolean switching theory logic and artificial intelligence ai particularly in model based diagnosis

riccati techniques and variational principles in oscillation theory for linear systems on etablit des conditions suffisantes pour que toutes les solutions soient oscillatoires on utilise les techniques de riccati et les principes variationnels

fast parallel constraint satisfaction a constraint satisfaction problem csp involves searching for an assignment of values to a given set of variables so that the values assigned satisfy a given set of constraints the general csp is np complete to confront the intractability of the general csp relaxation procedures have been devised instead of searching for a globally consistent assignment of values to the variables try to restrict the domain of values of each variable in a way that ensures local consistency only the relaxation procedures are efficient but have been proved to be inherently sequential in this paper we define a class of csps for which a global solution can be found by a fast parallel algorithm no relaxation preprocessing is needed for the parallel algorithm to work the result is motivated from the problem of labelling a 2 d line drawing of a 3 d object by the clowes huffman malik labelling scheme an important application of csp in computer vision for such a labelling csp the constraint graph can be general but the constraint relations are usually of the type we call implicational it is shown here that a csp with this type of constraint relations and no restrictions on its graph can be solved by an efficient i e with polynomial time complexity sequential algorithm also it is shown that it can be solved by a fast parallel algorithm that executes in time o log3n with o m n3 log n processors on a exlusive read exclusive write parallel random access machine n is the number of variables and m is the number of constrain relations the constraint relations may have arity more than two

breaking of periodicity at positive temperatures we discuss a classical lattice gas model without periodic or quasiperiodic ground states the only ground state configurations of our model are nonperiodic thue morse sequences we show that low temperature phases of such models can be ordered in fact we prove the existence of an ordered nonmixing low temperature translation invariant equilibrium state which has nonperiodic gibbs states in its extremal decomposition

classical lattice gas models of quasicrystals one of the fundamental problems of quasicrystals is to understand their occurrence in microscopic models of interacting particles we review here recent attempts to construct stable quasicrystalline phases in particular we compare two recently constructed classical lattice gas models with translation invariant interactions and without periodic ground state configurations the models are based on nonperiodic tilings of the plane by square like tiles in the first model all interactions can be minimized simultaneously the second model is frustrated its nonperiodic ground state can arise only by the minimization of the energy of competing interactions we put forward some hypotheses concerning stabilities of nonperiodic ground states in particular we introduce two criteria the so called strict boundary conditions and prove their equivalence to the zero temperature stability of ground states against small perturbations of potentials of interacting particles we discuss the relevance of these conditions for the low temperature stability i e for the existence of thermodynamically stable nonperiodic equilibrium states

implications of forbidden structures for extremal algorithmic problems abstract we develop and analyze p optimal approximation algorithms for various generalized satisfiability problems and determine the corresponding p optimal thresholds the most novel aspect of the paper is that we develop new p optimal algorithms for restricted generalized satisfiability problems which are defined by forbidden subformulas all our algorithms run in linear time on a ram

semidefinite relaxation bounds for indefinite homogeneous quadratic optimization this paper studies the relationship between the optimal value of a homogeneous quadratic optimization problem and its semidefinite programming sdp relaxation we consider two quadratic optimization models 1 min x c x mid x a k x ge 1 k 0 1 ldots m x in mathbb f n and 2 max x c x mid x a k x le 1 k 0 1 ldots m x in mathbb f n where mathbb f is either the real field mathbb r or the complex field mathbb c and a k c are symmetric matrices for the minimization model 1 we prove that if the matrix c and all but one of the a k s are positive semidefinite then the ratio between the optimal value of 1 and its sdp relaxation is upper bounded by o m 2 when mathbb f mathbb r and by o m when mathbb f mathbb c moreover when two or more of the a k s are indefinite this ratio can be arbitrarily large for the maximization model 2 we show that if c and at most one of the a k s are indefinite while other a k s are positive semidefinite then the ratio between the optimal value of 2 and its sdp relaxation is bounded from below by o 1 log m for both the real and the complex case this result improves the bound based on the so called approximate s lemma of ben tal nemirovski and roos siam j optim 13 2002 pp 535 560 when two or more of the a k s in 2 are indefinite we derive a general bound in terms of the problem data and the sdp solution for both optimization models we present examples to show that the derived approximation bounds are essentially tight

simulating log c n wise independence in nc abstract we develop a general framework for removing randomness from randomized nc algorithms whose analysis uses only polyalgorithmic independence previously no techniques were known to determine those rnc algorithms depending on more than constant independence one application of our techniques is an nc algorithm for the set discrepancy problem which can be used to obtain many other nc algorithms including a better nc edge coloring algorithm as another application of our techniques we provide an nc algorithm for the hypergraph coloring problem

oscillation of linear hamiltonian systems abstract in this paper sufficient conditions have been obtained for the oscillation of a class of linear hamiltonian systems

a unified approach to approximation algorithms for bottleneck problems in this paper a powerful and yet simple technique for devising approximation algorithms for a wide variety of np complete problems in routing location and communication network design is investigated each of the algorithms presented here delivers an approximate solution guaranteed to be within a constant factor of the optimal solution in addition for several of these problems we can show that unless p np there does not exist a polynomial time algorithm that has a better performance guarantee

linear transformation and oscillation criteria for hamiltonian systems abstract using a linear transformation similar to the kummer transformation some new oscillation criteria for linear hamiltonian systems are established these results generalize and improve the oscillation criteria due to i s kumari and s umanaheswaram i sowjaya kumari s umanaheswaram oscillation criteria for linear matrix hamiltonian systems j differential equations 165 2000 174 198 q yang et al q yang r mathsen s zhu oscillation theorems for self adjoint matrix hamiltonian systems j differential equations 190 2003 306 329 and s chen and z zheng shaozhu chen zhaowen zheng oscillation criteria of yan type for linear hamiltonian systems comput math appl 46 2003 855 862 these criteria also unify many of known criteria in literature and simplify the proofs

probability bounds for polynomial functions in random variables random sampling is a simple but powerful method in statistics and in the design of randomized algorithms in a typical application random sampling can be applied to estimate an extreme value say maximum of a function f over a set s n to do so one may select a simpler even finite subset s0 s randomly take some samples over s0 for a number of times and pick the best sample the hope is to find a good approximate solution with reasonable chance this paper sets out to present a number of scenarios for f s and s0 where certain probability bounds can be established leading to a quality assurance of the procedure in our setting f is a multivariate polynomial function we prove that if f is a d th order homogeneous polynomial in n variables and f is its corresponding super symmetric tensor and i i 1 2 n are i i d bernoulli random variables taking 1 or 1 with equal probability then prob f 1 2 n n d 2 f 1 where 0 are two universal constants and 1 denotes

approximation algorithms for connected maximum cut and related problems an instance of the connected maximum cut problem consists of an undirected graph g v e and the goal is to find a subset of vertices s v that maximizes the number of edges in the cut s such that the induced graph g s is connected we present the first non trivial omega frac 1 log n approximation algorithm for the connected maximum cut problem in general graphs using novel techniques we then extend our algorithm to an edge weighted case and obtain a poly logarithmic approximation algorithm interestingly in stark contrast to the classical max cut problem we show that the connected maximum cut problem remains np hard even on unweighted planar graphs on the positive side we obtain a polynomial time approximation scheme for the connected maximum cut problem on planar graphs and more generally on graphs with bounded genus

oscillation theorems for linear matrix hamiltonian systems using a matrix linear transformation which preserves oscillatory property some new oscillation criteria for the following linear matrix hamiltonian system u a t t i u b t v v c t u t i a t v t t 0 are obtained which improve and generalize some recent results in literature

quantifying double mccormick when using the standard mccormick inequalities twice to convexify trilinear monomials as is often the practice in modeling and software there is a choice of which variables to group first for the important case in which the domain is a nonnegative box we calculate the volume of the resulting relaxation as a function of the bounds defining the box in this manner we precisely quantify the strength of the different possible relaxations defined by all three groupings in addition to the trilinear hull itself as a by product we characterize the best double mccormick relaxation we wish to emphasize that in the context of spatial branch and bound for factorable formulations our results do not only apply to variables in the input formulation our results apply to monomials that involve auxiliary variables as well so our results apply to the product of any three possibly complicated expressions in a formulation

riccati techniques and oscillation for self adjoint matrix hamiltonian systems the purpose of this paper is to develop a generalized matrix riccati technique for the self adjoint matrix hamiltonian system u a x u b x v v c x u a x v together with the integral averaging technique and monotone functionals new oscillation and interval oscillation criteria are established for the system these criteria extend improve complement a number of existing results and handle some cases not covered by known criteria in particular two interesting examples are included to illustrate the versatility of our results