a polynomial time approximation scheme for the multiple knapsack problem the multiple knapsack problem mkp is a natural and well known generalization of the single knapsack problem and is defined as follows we are given a set of n items and m bins knapsacks such that each item i has a profit p i and a size s i and each bin j has a capacity c j the goal is to find a subset of items of maximum profit such that they have a feasible packing in the bins mkp is a special case of the generalized assignment problem gap where the profit and the size of an item can vary based on the specific bin that it is assigned to gap is apx hard and a 2 approximation for it is implicit in the work of shmoys and tardos math program a 62 1993 pp 461 474 and thus far this was also the best known approximation for mkp the main result of this paper is a polynomial time approximation scheme ptas for mkp apart from its inherent theoretical interest as a common generalization of the well studied knapsack and bin packing problems it appears to be the strongest special case of gap that is not apx hard we substantiate this by showing that slight generalizations of mkp are apx hard thus our results help demarcate the boundary at which instances of gap become apx hard an interesting aspect of our approach is a ptas preserving reduction from an arbitrary instance of mkp to an instance with o log n distinct sizes and profits

observer design for nonlinear control systems abstract this paper deals with the observer design problem of a wide class of nonlinear systems subjected to bounded nonlinearities a sufficient liapunovlike condition is provided and the proposed dynamic observer is a direct extension of the one in linear case

multiple phase changes induced by frustration in randomly connected cellular automata frustration is introduced in randomly connected totalistic cellular automata via mixing rules leading to incompatible periods as the respective concentration of rules is varied these cellular automata go through eight phases many of which with symmetries different from the two rules mixed in the complex phase diagram so created is similar to those seen in frustrated systems in static equilibrium it shows that a minimized free energy is not necessary for frustration to produce this rich behaviour

global stabilization and convergence of nonlinear systems with uncertain exogenous dynamics in this note a class of nonlinear uncertain systems are considered and uncertainties in the systems are assumed to be generated by exogenous dynamics robust control is designed by employing nonlinear observers to estimate the uncertainties it is shown that if a partial knowledge of the exogenous system is available and its known dynamics meet certain conditions or if input channel of the plant has certain properties global stability and global estimation convergence can be achieved in the latter case the results on stability and convergence hold even if exogenous dynamics are completely unknown but bounded by some known function

on the phase diagram of frustrated quasi periodic cellular automata we introduce and study frustrated cellular automata ca obtained by quenching competing chate manneville rules a period two p2 rule and a quasi periodic one with period close to three qp3 are frozen at random on the lattice sites we find that the periodic and quasi periodic cycles are resilient to internal frustration as well as to external unbounded noise a low concentration of impurities improves the quasi periodicity of the ca damping the chaotic background noise significantly starting from pure qp3 ca a first phase transition happens at a concentration of rule p2 leading to a macroscopic fixed point a second phase transition at brings the p2 phase although macroscopically stable the central phase displays a stretched exponential relaxation of the site site autocorrelations indicating the presence of a new type of glass with slow dynamics superimposed on the natural cyclic dynamics of the ca rules these results appear to be quite general and are found for many pairs of rules

phase transition from periodic to quasiperiodic behaviour in 4d cellular automata we have found a phase transition from periodic to quasiperiodic behaviour in four dimensional cellular automata by mixing rules as far as we know this kind of phase transition has not been studied before and may thus serve as a pilot case

global stabilization of nonlinear systems with a class of unmatched uncertainties abstract global and asymptotic stabilization of uncertain dynamical systems which violate the matching conditions is investigated a class of unmatched uncertainties called equivalently matched uncertainties is defined a robust controller is proposed to stabilize the uncertain system asymptotically in the large provided that the nominal system is uniformly asymptotically stable and that the unmatched uncertainties are equivalently matched the required information about uncertain dynamics in the system is merely that the uncertainties are bounded in euclidean norm by known functions of the system state

coexistence and critical behavior in a lattice model of competing species in the present paper we study a lattice model of two species competing for the same resources monte carlo simulations for d 1 2 and 3 show that when resources are easily available both species coexist however when the supply of resources is on an intermediate level the species with slower metabolism becomes extinct on the other hand when resources are scarce it is the species with faster metabolism that becomes extinct the range of coexistence of the two species increases with dimension we suggest that our model might describe some aspects of the competition between normal and tumor cells with such an interpretation examples of tumor remission recurrence and different morphologies are presented in the d 1 and d 2 models we analyze the nature of phase transitions they are either discontinuous or belong to the directed percolation universality class and in some cases they have an active subcritical phase in the d 2 case one of the transitions seems to be characterized by critical exponents that differ from directed percolation ones but this transition could be also weakly discontinuous in the d 3 version monte carlo simulations are in a good agreement with the solution of the mean field approximation this approximation predicts that oscillatory behavior occurs in the present model but only for d 2 for d 2 a steady state depends on the initial configuration in some cases

noisy collective behaviour in deterministic cellular automata we investigate cellular automata in four and five dimensions for which chate and manneville recently have found nontrivial collective behaviour more precisely though being fully deterministic the average magnetization seems to be periodic respectively quasiperiodic with superimposed noise whose amplitude decreases with system size we confirm this behaviour on very large systems and over very large times we analyse in detail the statistical properties of the noise systems on small lattices and or subject to additional external noise are metastable arguments by grinstein et al suggest that in the periodic case the infinite deterministic systems should be metastable too these arguments are generalized to quasiperiodic systems we find evidence that they do indeed apply but we find no direct evidence for metastability of large systems

collective behaviour in a cellular automaton and isotropy of the neighbourhood abstract a three dimensional deterministic cellular automata rule 33 which exhibits a nontrivial quasiperiodic behaviour is studied we show that the way in which the neighbours are selected causes the appearance or disappearance of that behaviour

a totalistic three dimensional cellular automaton with quasiperiodic behaviour abstract a three dimensional deterministic totalistic cellular automation with non trivial quasiperiodic behaviour is presented different properties as intrinsic noise and influence of external noise are investigated an approach to an explanation of the phenomenon is discussed

a threshold of ln n for approximating set cover given a collection f of subsets of s 1 n setcover is the problem of selecting as few as possiblesubsets from f such that their union covers s and maxk cover is the problem of selecting k subsets from f such that their union has maximum cardinality both these problems arenp hard we prove that 1 o 1 ln n is a threshold below which setcover cannot be approximated efficiently unless np has slightlysuperpolynomial time algorithms this closes the gap up to low orderterms between the ratio of approximation achievable by the greedyalogorithm which is 1 o 1 lnn and provious results of lund and yanakakis that showed hardness ofapproximation within a ratio of log 2 n 2s0 72 ln n for max k cover we show an approximationthreshold of 1 1 e up tolow order terms under assumption that p np

oscillatory patterns in an autocatalytic reaction ring network simulated by a probabilistic cellular automaton abstract a two dimensional probabilistic cellular automaton is presented it simulates a reaction diffusion process occurring in biomembranes during active transport described by an autocatalytic ring network the automaton suffers a transition from a completely disordered state to a state with well defined oscillations and spatial inhomogeneous patterns by changing the parameters the different dynamic states have been characterized by means of fourier transforms of concentration time evolution and through the definition of an agglomeration parameter vacancy concentration and lattice size effects are analyzed

cascaded feedback linearization and its application to stabilization of nonholonomic systems in this paper a new cascaded feedback linearization problem is formulated and a set of conditions on the cascaded feedback linearizability are established for a class of two input affine nonlinear systems the proposed cascaded feedback linearization method enlarges the classes of nonlinear systems which can be dealt with using the feedback linearization technique in particular the proposed design can be applied to address the feedback stabilization problem for a few classes of nonlinear systems which have uncontrollable linearization and do not satisfy the standard feedback linearization conditions as an illustrative application the proposed cascade feedback linearization concept is used to solve the feedback stabilization problem of nonholonomic systems within the framework of continuously differentiable state feedback control simulation results are provided to illustrate the proposed method

some optimal inapproximability results we prove optimal up to an arbitrary e 0 inapproximability results for max e k sat for k 3 maximizing the number of satisfied linear equations in an over determined system of linear equations modulo a prime p and set splitting as a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously in particular for max e2 sat max cut max di cut and vertex cover

symmetry and approximability of submodular maximization problems a number of recent results on optimization problems involving submodular functions have made use of the multilinear relaxation of the problem these results hold typically in the value oracle model where the objective function is accessible via a black box returning f s for a given s we present a general approach to deriving inapproximability results in the value oracle model based on the notion of symmetry gap our main result is that for any fixed instance that exhibits a certain symmetry gap in its multilinear relaxation there is a naturally related class of instances for which a better approximation factor than the symmetry gap would require exponentially many oracle queries this unifies several known hardness results for submodular maximization e g the optimality of 1 1 e approximation for monotone submodular maximization under a cardinality constraint and the impossibility of frac12 epsilon approximation for unconstrained nonmonotone submodular maximization as a new applica

gadgets approximation and linear programming we present a linear programming based method for finding gadgets i e combinatorial structures reducing constraints of one optimization problem to constraints of another a key step in this method is a simple observation which limits the search space to a finite one using this new method we present a number of new computer constructed gadgets for several different reductions this method also answers a question posed by bellare goldreich and sudan siam j comput 27 1998 pp 804 915 of how to prove the optimality of gadgets linear programming duality gives such proofs the new gadgets when combined with recent results of ha stad proceedings of the 29th acm symposium on theory of computing 1997 pp 1 10 improve the known inapproximability results for max cut and max dicut showing that approximating these problems to within factors of 16 17 epsilon and 12 13 epsilon respectively is np hard for every epsilon 0 prior to this work the best known inapproximability thresholds for both problems were 71 72 m bellare o goldreich and m sudan siam j comput 27 1998 pp 804 915 without using the gadgets from this paper the best possible hardness that would follow from bellare goldreich and sudan and ha s tad is 18 19 we also use the gadgets to obtain an improved approximation algorithm for max3 sat which guarantees an approximation ratio of 801 this improves upon the previous best bound implicit from m x goemans and d p williamson j acm 42 1995 pp 1115 1145 u feige and m x goemans proceedings of the third israel symposium on theory of computing and systems 1995 pp 182 189 of 7704

continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems we consider a dynamic system containing uncertain elements only the set of possible values of these uncertainties is known based on this information a class of state feedback controls is proposed in order to guarantee uniform ultimate boundedness of every system response within an arbitrarily small neighborhood of the zero state these feedback controls are continuous functions of the state

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

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

near optimal sensor placements in gaussian processes theory efficient algorithms and empirical studies when monitoring spatial phenomena which can often be modeled as gaussian processes gps choosing sensor locations is a fundamental task there are several common strategies to address this task for example geometry or disk models placing sensors at the points of highest entropy variance in the gp model and a d or e optimal design in this paper we tackle the combinatorial optimization problem of maximizing the mutual information between the chosen locations and the locations which are not selected we prove that the problem of finding the configuration that maximizes mutual information is np complete to address this issue we describe a polynomial time approximation that is within 1 1 e of the optimum by exploiting the submodularity of mutual information we also show how submodularity can be used to obtain online bounds and design branch and bound search procedures we then extend our algorithm to exploit lazy evaluations and local structure in the gp yielding significant speedups we also extend our approach to find placements which are robust against node failures and uncertainties in the model these extensions are again associated with rigorous theoretical approximation guarantees exploiting the submodularity of the objective function we demonstrate the advantages of our approach towards optimizing mutual information in a very extensive empirical study on two real world data sets

on the analysis of spatiotemporally chaotic data abstract common problems encountered when dealing with spatially extended systems in extensive chaos regimes are discussed from the point of view of the experimentalist it is argued that both local and global probes are insufficient and potentially misleading and that the space time evolution of at least one variable even with low local resolution should be accessible in order to estimate the possibly large scale coherence in spatiotemporal chaos regimes

comparing apples and oranges query trade off in submodular maximization fast algorithms for submodular maximization problems have a vast potential use in applicative settings such as machine learning social networks and economics though fast algorithms were known for some special cases only recently such algorithms were considered in the general case of maximizing a monotone submodular function subject to a matroid independence constraint the known fast algorithm matches the best possible approximation guarantee while trying to reduce the number of value oracle queries the algorithm performs our main result is a new algorithm for this general case that establishes a surprising trade off between two seemingly unrelated quantities the number of value oracle queries and the number of matroid independence queries performed by the algorithm specifically one can decrease the former by increasing the latter and vice versa while maintaining the best possible approximation guarantee such a trade off is very useful since various applications might incur significantly diff

collective behaviour in one dimensional locally coupled map lattices the authors observe true and quasiperiod 4 collective evolution of average values in one dimensional lattices of logistic maps with variable coupling range the phenomenon appears to be quite prevalent for local coupling beyond nearest neighbours they find that in one dimension it is easy to visualize the structure of the four quasi states these appear to be quite stable to disturbances such as domain reversals

maximizing a monotone submodular function subject to a matroid constraint let f 2 x rightarrow cal r be a monotone submodular set function and let x cal i be a matroid we consider the problem rm max s in cal i f s it is known that the greedy algorithm yields a 1 2 approximation m l fisher g l nemhauser and l a wolsey math programming stud no 8 1978 pp 73 87 for this problem for certain special cases e g rm max s leq k f s the greedy algorithm yields a 1 1 e approximation it is known that this is optimal both in the value oracle model where the only access to f is through a black box returning f s for a given set s g l nemhauser and l a wolsey math oper res 3 1978 pp 177 188 and for explicitly posed instances assuming p neq np u feige j acm 45 1998 pp 634 652 in this paper we provide a randomized 1 1 e approximation for any monotone submodular function and an arbitrary matroid the algorithm works in the value oracle model our main tools are a variant of the pipage rounding technique of ageev and sviridenko j combin optim 8 2004 pp 307 328 and a continuous greedy process that may be of independent interest as a special case our algorithm implies an optimal approximation for the submodular welfare problem in the value oracle model j vondrak proceedings of the 38 th acm symposium on theory of computing 2008 pp 67 74 as a second application we show that the generalized assignment problem gap is also a special case although the reduction requires x to be exponential in the original problem size we are able to achieve a 1 1 e o 1 approximation for gap simplifying previously known algorithms additionally the reduction enables us to obtain approximation algorithms for variants of gap with more general constraints