multi constrained matroidal knapsack problems we consider multi constrained knapsack problems where the sets of elements to be selected are subject to combinatorial constraints of matroidal nature for this important class of np hard combinatorial optimization problems we prove that lagrangean relaxation techniques not only provide good bounds to the value of the optimum but also yield approximate solutions which are asymptotically optimal under mild probabilistic assumptions

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

probabilistic analysis of the multidimensional knapsack problem we analyse the multi constraint zero one knapsack problem under the assumption that all coefficients are drawn from a uniform 0 1 distribution and there are m 01 constraints as the number of variables grows we show that results on the single constraint problem can be extended to this situation chiefly we generalise a result of lueker on the duality gap and a result of goldberg marchetti spaccamela on exact solvability in the latter case our methods differ markedly from those for the single constraint result

fluctuation theorems and entropy production with odd parity variables we show that the total entropy production in stochastic processes with odd parity variables under time reversal is separated into three parts only two of which satisfy the integral fluctuation theorems in general one is the usual excess contribution that can appear only transiently and is called nonadiabatic another one is attributed solely to the breakage of detailed balance the last part that does not satisfy the fluctuation theorem comes from the steady state distribution asymmetry for odd parity variables that is activated in a nontransient manner the latter two parts combine together as the housekeeping adiabatic contribution whose positivity is not guaranteed except when the excess contribution completely vanishes our finding reveals that the equilibrium requires the steady state distribution symmetry for odd parity variables independently in addition to the usual detailed balance

asymptotically periodic behaviour in the dynamics of chaotic mappings we will prove that for a chaotic mapping f belonging to a suitable class of c 1 alpha functions the set a infty f has lebesgue measure zero with a infty f a nonempty set consisting of points whose orbits do not converge to an asymptotically stable periodic orbit of f or to the absorbing boundary moreover almost every point is asymptotically periodic with period p for some positive integer p further we will show that the same conclusions hold for maps with nonpositive schwarzian derivative under some additional assumptions

the online knapsack problem advice and randomization we study the advice complexity and the random bit complexity of the online knapsack problem given a knapsack of unit capacity and n items that arrive in successive time steps an online algorithm has to decide for every item whether it gets packed into the knapsack or not the goal is to maximize the value of the items in the knapsack without exceeding its capacity in the model of advice complexity of online problems one asks how many bits of advice about the unknown parts of the input are both necessary and sufficient to achieve a specific competitive ratio it is well known that even the unweighted online knapsack problem does not admit any competitive deterministic online algorithm for this problem we show that a single bit of advice helps a deterministic online algorithm to become 2 competitive but that w logn advice bits are necessary to further improve the deterministic competitive ratio this is the first time that such a phase transition for the number of advice bits has been observed for any problem additionally we show that surprisingly instead of an advice bit a single random bit allows for a competitive ratio of 2 and any further amount of randomness does not improve this moreover we prove that in a resource augmentation model i e when allowing the online algorithm to overpack the knapsack by some small amount a constant number of advice bits suffices to achieve a near optimal competitive ratio we also study the weighted version of the problem proving that with o logn bits of advice we can get arbitrarily close to an optimal solution and using asymptotically fewer bits we are not competitive furthermore we show that an arbitrary number of random bits does not permit a constant competitive ratio

probabilistic properties of the dual structure of the multidimensional knapsack problem and fast statistically efficient algorithms properties of several dual characteristics of the multidimensional knapsack problem such as the probability of existence of optimal and optimal feasible lagrange function generalized saddle points magnitude of relative duality gap etc are investigated for different probabilistic models sufficient conditions of good asymptotic behavior of the dual characteristics are given a fast statistically efficient approximate algorithm with linear running time complexity for problems with random coefficients is presented

nonequilibrium thermodynamics of stochastic systems with odd and even variables the total entropy production of stochastic systems can be divided into three quantities the first corresponds to the excess heat while the second two comprise the housekeeping heat we denote these two components the transient and generalized housekeeping heat and we obtain an integral fluctuation theorem for the latter valid for all markovian stochastic dynamics a previously reported formalism is obtained when the stationary probability distribution is symmetric for all variables that are odd under time reversal which restricts consideration of directional variables such as velocity

qualitative analysis of the dynamics and stability properties for axiom a maps abstract assume that f c1 x x is an axiom a mapping from a nontrivial interval x into itself i e f satisfies the conditions a the nonwandering set f can be split up into at most two but at least one nonempty positively f invariant compact sets say f a f e f such that the restriction of f to a f is contracting and the restriction of f to e f is expanding b the periodic points for f are dense in f then we have 1 a the nonwandering set of f decomposes into finitely many positively f invariant sets and f is one sided topologically transitive on each of these basic sets 2 b the map f is c1 stable 3 c there exists a positive integer p such that almost every point in x is asymptotically periodic with period p provided that f x is bounded and that f is of class c1 for some positive real number 4 d the map f is c2 structurally stable provided that f satisfies some suitable conditions finally it will turn out that maps with nonpositive schwarzian derivative are under reasonable assumptions axiom a maps

2d knapsack packing squares in this paper we study a two dimensional knapsack problem packing squares as many as possible into a unit square our results are the following i we propose an algorithm called ihs increasing height shelf and prove that the packing is optimal if in an optimal packing there are at most 5 squares and this upper bound is sharp ii if all the squares have side length at most 1k we propose a simple and fast algorithm with an approximation ratio k 2 3k 2k 2 in time o nlogn iii we give an eptas for the problem where the previous result in jansen and solis oba 2008 16 is a ptas not an eptas however our approach does not work on the previous model of jansen and solis oba 2008 16 where each square has an arbitrary weight

brownian colloidal particles ito stratonovich or a different stochastic interpretation recent experiments on brownian colloidal particles have been studied theoretically in terms of overdamped langevin equations with multiplicative white noise using an unconventional stochastic interpretation complementary numerical simulations of the same system are well described using the conventional stratonovich interpretation here we address this dichotomy from a more generic starting point the underdamped langevin equation and its corresponding fokker planck equation

on rates of convergence and asymptotic normality in the multiknapsack problem in meanti et al 1990 an almost sure asymptotic characterization has been derived for the optimal solution value as function of the knapsack capacities when the profit and requirement coefficients of items to be selected from are random variables in this paper we establish a rate of convergence for this process using results from the theory of empirical processes

is every approximate trajectory of some process near an exact trajectory of a nearby process this paper deals with the problem can a noisy orbit be tracked by a real orbit in particular we will study the one parameter family of tent maps and the one parameter family of quadratic maps we writeg for eitherf orf withf x x forx 1 2 andf x 1 x forx 1 2 andf x x 1 x for a given we will say g permits increased parameter shadowing if for each x 0 there exists some 0 and some f 0 such that every f pseudog orbit starting in some invariant interval can be x shadowed by a realg orbit with we show thatg typically permits increased parameter shadowing

absolutely continuous invariant measures for one parameter families of one dimensional maps given a one parameter familyf x of maps of the interval 0 1 we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to lebesgue measure we show that this set has positive measure for two classes of maps i f x f x where 0 4 andf x is a functionc3 near the quadratic mapx 1 x and ii f x f x mod 1 wheref isc3 f 0 f 1 0 andf has a unique nondegenerate critical point in 0 1

online removable knapsack problem under convex function abstract in this paper we address an online knapsack problem under a convex size profit function we first give a greedy online algorithm with a competitive ratio 2 then we propose an improved online algorithm with a competitive ratio 5 3 we also prove that when the convex function has a specific property our improved online algorithm is 1 5 2 competitive which is optimal finally we prove that the lower bound of this problem is 1 5 2

online minimization knapsack problem in this paper we address the online minimization knapsack problem i e the items are given one by one over time and the goal is to minimize the total cost of items that covers a knapsack we study the removable model where it is allowed to remove old items from the knapsack in order to accept a new item we obtain the following results i we propose a 8 competitive deterministic and memoryless algorithm for the problem which contrasts with the result for the online maximization knapsack problem that no online algorithm has a bounded competitive ratio 14 ii we propose a 2e competitive randomized algorithm for the problem iii we derive a lower bound of 2 for deterministic algorithms for the problem iv we propose a 1 618 competitive deterministic algorithm for the case in which each item has size equal to its cost and show that this is best possible

an optimal time algorithm for slope selection given n points in the plane and an integer k the problem of selecting that pair of points that determines the line with the kth smallest or largest slope is considered in the restricted case where k is o n line sweeping gives an optimal o n log n time algorithm for general k the parametric search technique of megiddo is used to describe an o n log n 2 time algorithm this is modified to produce a new optimal o n log n time selection algorithm by incorporating an approximation idea

adiabatic elimination for systems of brownian particles with nonconstant damping coefficients we discuss the problem of eliminating the momentum variable in the phase space langevin equations for a system of brownian particles in two related situations i position dependent damping and ii existence of hydrodynamic interactions we discuss the problems associated with the conventional elimination and we develop an alternative elimination procedure in the lagevin framework which leads to the correct smoluchowski equation we give a heuristic argument on the basis of stochastic differential equations for the smoluchowski limit and establish rigorously the limit for the general case of position dependent friction and diffusion coefficents

decomposable multi parameter matroid optimization problems a framework for solving certain multi dimensional parametric matroid optimization problems in randomized linear time by prune and search is presented the common feature of these problems which include the multi parameter minimum spanning tree problem on planar and dense graphs is that their fixed parameter versions are solvable by tournament like algorithms whose structure is represented by a balanced decomposition tree

approximation algorithms for the m dimensional 0 1 knapsack problem worst case and probabilistic analyses abstract we describe a polynomial approximation scheme for an m constraint 0 1 integer programming problem m fixed based on the use of the dual simplex algorithm for linear programming we also analyse the asymptotic properties of a particular random model

extended forms of the second law for general time dependent stochastic processes the second law of thermodynamics represents a universal principle applicable to all natural processes physical systems and engineering devices hatano and sasa have recently put forward an extended form of the second law for transitions between nonequilibrium stationary states phys rev lett 86 3463 2001 in this paper we further extend this form to an instantaneous interpretation which is satisfied by quite general time dependent stochastic processes including master equation models and langevin dynamics without the requirements of the stationarity for the initial and final states the theory is applied to several thermodynamic processes and its consistence with the classical thermodynamics is shown

fluctuation theorem for hidden entropy production elimination of seemingly unnecessary variables in markovian models may cause a difference in the value of irreversible entropy production between the original and reduced dynamics we show that such difference which we call the hidden entropy production obeys an integral fluctuation theorem if all variables are time reversal invariant or if the density function is symmetric with respect to the change of sign of the time reversal antisymmetric variables the theorem has wide applicability since the proposed condition is mostly satisfied in the case where the hidden fast variables are equilibrated the main consequence of this theorem is that the entropy production decreases by the coarse graining procedure by contrast in the case where a stochastic process is obtained by coarse graining a deterministic and reversible dynamics the entropy production may increase implying that the integral fluctuation theorem should not hold for such reductions we reveal with an explicit example that the nonequilibrated time reversal antisymmetric variables play a crucial role in distinguishing these two cases thus guaranteeing the consistency of the presented theorem

time reversibility and nonequilibrium thermodynamics of second order stochastic processes nonequilibrium thermodynamics of a general second order stochastic system is investigated we prove that at steady state under inversion of velocities the condition of time reversibility over the phase space is equivalent to the antisymmetry of spatial flux and the symmetry of velocity flux then we show that the condition of time reversibility alone cannot always guarantee the maxwell boltzmann distribution comparing the two conditions together we find that the frictional force naturally emerges as the unique odd term of the total force at thermodynamic equilibrium and is followed by the einstein relation the two conditions respectively correspond to two previously reported different entropy production rates in the case where the external force is only position dependent the two entropy production rates become one we prove that such an entropy production rate can be decomposed into two non negative terms expressed respectively by the conditional mean and variance of the thermodynamic force associated with the irreversible velocity flux at any given spatial coordinate in the small inertia limit the former term becomes the entropy production rate of the corresponding overdamped dynamics while the anomalous entropy production rate originates from the latter term furthermore regarding the connection between the first law and second law we find that in the steady state of such a limit the anomalous entropy production rate is also the leading order of the boltzmann factor weighted difference between the spatial heat dissipation densities of the underdamped and overdamped dynamics while their unweighted difference always tends to vanish

linear time algorithms for parametric minimum spanning tree problems on planar graphs abstract a linear time algorithm for the minimum ratio spanning tree problem on planar graphs is presented the algorithm is based on a new planar minimum spanning tree algorithm the approach extends to other parametric minimum spanning tree problems on planar graphs and to other families of graphs having small separators

physical origins of entropy production free energy dissipation and their mathematical representations a unifying mathematical theory of nonequilibrium thermodynamics of stochastic systems in terms of master equations is presented as generalizations of isothermal entropy and free energy two functions of state play central roles the gibbs entropy s and the relative entropy f which are related via the stationary distribution of the stochastic dynamics s satisfies the fundamental entropy balance equation ds dt e p ensuremath h d t with entropy production rate e p ensuremath ge 0 and heat dissipation rate h d while df dt ensuremath f d ensuremath le 0 for closed systems that satisfy detailed balance t e p t f d t for open systems one has t e p t f d t q hk t where the housekeeping heat q hk ensuremath ge 0 was first introduced in the phenomenological nonequilibrium steady state thermodynamics put forward by oono and paniconi q hk represents the irreversible work done by the surrounding to the system that is kept away from reaching equilibrium hence entropy production e p consists of free energy dissipation associated with spontaneous relaxation i e self organization f d and active energy pumping that sustains the open system q hk the amount of excess heat involved in the relaxation q ex h d ensuremath q hk f d ensuremath t ds dt two kinds of irreversibility and the meaning of the arrow of time emerge quasistationary processes adiabaticity and maximum principle for entropy are also generalized to nonequilibrium settings

stochastic on line knapsack problems different classes of on line algorithms are developed and analyzed for the solution of 0 1 and relaxed stochastic knapsack problems in which both profit and size coefficients are random variables in particular a linear time on line algorithm is proposed for which the expected difference between the optimum and the approximate solution value iso log3 2n an 1 lower bound on the expected difference between the optimum and the solution found by any on line algorithm is also shown to hold

the equality of fractal dimension and uncertainty dimension for certain dynamical systems mgoy introduced the uncertainty dimension as a quantative measure for final state sensitivity in a system in mgoy and p it was conjectured that the box counting dimension equals the uncertainty dimension for basin boundaries in typical dynamical systems in this paper our main result is that the box counting dimension the uncertainty dimension and the hausdorff dimension are all equal for the basin boundaries of one and two dimensional systems which are uniformly hyperbolic on their basin boundary when the box counting dimension of the basin boundary is large that is near the dimension of the phase space this result implies that even a large decrease in the uncertainty of the position of the initial condition yields only a relatively small decrease in the uncertainty of which basin that initial point is in

weighted multidimensional search and itsapplication to convex optimization we present a weighted version of megiddo s multidimensional search technique and use it to obtain faster algorithms for certain convex optimization problems in reals d for fixed d this leads to speed ups by a factor of log d n for applications such as solving the lagrangian duals of matroidal knapsack problems and of constrained optimum subgraph problems on graphs of bounded tree width

the multidimensional 0 1 knapsack problem an overview abstract the multidimensional 0 1 knapsack problem is one of the most well known integer programming problems and has received wide attention from the operational research community during the last four decades although recent advances have made possible the solution of medium size instances solving this np hard problem remains a very interesting challenge especially when the number of constraints increases this paper surveys the main results published in the literature the focus is on the theoretical properties as well as approximate or exact solutions of this special 0 1 program

fast algorithms for shortest paths in planar graphs with applications graph decomposition and data structures techniques are presented that exploit the structure of planar graphs to yield faster algorithms for a number of shortest path and related problems improved

on the creation of wada basins in interval maps through fixed point tangent bifurcation basin boundaries play an important role in the study of dynamics of nonlinear models in a variety of disciplines such as biology chemistry economics engineering and physics one of the goals of nonlinear dynamics is to determine the global structure of the system such as boundaries of basins a basin having the strange property that every point which is on the boundary of that basin is on the boundary of at least three different basins is called a wada basin and its boundary is called a wada basin boundary here we consider maps on the interval we present a sufficient and necessary condition guaranteeing that three wada basins are emerging from a tangent bifurcation for certain one dimensional maps having negative schwarzian derivative two fixed point attractors on one side of the tangent bifurcation and three fixed point attractors on the other side of the tangent bifurcation all the conditions involved are numerically verifiable

a randomized linear time algorithm to find minimum spanning trees we present a randomized linear time algorithm to find a minimum spanning tree in a connected graph with edge weights the algorithm uses random sampling in combination with a recently discovered linear time algorithm for verifying a minimum spanning tree our computational model is a unit cost random access machine with the restriction that the only operations allowed on edge weights are binary comparisons