a dynamic programming based reduction procedure for the multidimensional 0 1 knapsack problem abstract this paper presents a preprocessing procedure for the 0 1 multidimensional knapsack problem first a non increasing sequence of upper bounds is generated by solving lp relaxations then a non decreasing sequence of lower bounds is built using dynamic programming the comparison of the two sequences allows either to prove that the best feasible solution obtained is optimal or to fix a subset of variables to their optimal values in addition a heuristic solution is obtained computational experiments with a set of large scale instances show the efficiency of our reduction scheme particularly it is shown that our approach allows to reduce the cpu time of a leading commercial software

capacity constraints across nests in assortment optimization under the nested logit model we consider assortment optimization problems when customers choose according to the nested logit model and there is a capacity constraint limiting the total capacity consumption of all products offered in all nests when each product consumes one unit of capacity our capacity constraint limits the cardinality of the offered assortment for the cardinality constrained case we develop an efficient algorithm to compute the optimal assortment when the capacity consumption of each product is arbitrary we give an algorithm to obtain a 4 approximate solution we show that we can compute an upper bound on the optimal expected revenue for an individual problem instance by solving a linear program in our numerical experiments we consider problem instances involving products with arbitrary capacity consumptions comparing the expected revenues from the assortments obtained by our 4 approximation algorithm with the upper bounds on the optimal expected revenues our numerical results indicate that the 4 approxim

models of social networks based on social distance attachment we propose a class of models of social network formation based on a mathematical abstraction of the concept of social distance social distance attachment is represented by the tendency of peers to establish acquaintances via a decreasing function of the relative distance in a representative social space we derive analytical results corroborated by extensive numerical simulations showing that the model reproduces the main statistical characteristics of real social networks large clustering coefficient positive degree correlations and the emergence of a hierarchy of communities the model is confronted with the social network formed by people that shares confidential information using the pretty good privacy pgp encryption algorithm the so called web of trust of pgp

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

space complexity in propositional calculus we study space complexity in the framework of propositional proofs we consider a natural model analogous to turing machines with a read only input tape and such popular propositional proof systems as resolution polynomial calculus and frege systems we propose two different space measures corresponding to the maximal number of bits and clauses monomials that need to be kept in the memory simultaneously we prove a number of lower and upper bounds in these models as well as some structural results concerning the clause space for resolution and frege systems

local and global lifted cover inequalities for the 0 1 multidimensional knapsack problem abstract the 0 1 multidimensional knapsack problem 0 1 mkp is a well known and strongly np hard combinatorial optimization problem with many applications up to now the majority of upper bounding techniques for the 0 1 mkp have been based on lagrangian or surrogate relaxation we show that good upper bounds can be obtained by a cutting plane method based on lifted cover inequalities lcis as well as using traditional lcis we use some new global lcis which take the whole constraint matrix into account

clustering in complex networks i general formalism we develop a full theoretical approach to clustering in complex networks a key concept is introduced the edge multiplicity that measures the number of triangles passing through an edge this quantity extends the clustering coefficient in that it involves the properties of two and not just one vertices the formalism is completed with the definition of a three vertex correlation function which is the fundamental quantity describing the properties of clustered networks the formalism suggests different metrics that are able to thoroughly characterize transitive relations a rigorous analysis of several real networks which makes use of this formalism and the metrics is also provided it is also found that clustered networks can be classified into two main groups the weak and the strong transitivity classes in the first class edge multiplicity is small with triangles being disjoint in the second class edge multiplicity is high and so triangles share many edges as we shall see in the following paper the class a network belongs to has strong implications in its percolation properties

random sequencing jobs with deadlines problem growth of the optimal solution values in the paper an asymptotic value of the sequencing jobs with deadlines sjd problem is computed for the case of the random sjd problems it is assumed that problem coefficients are realizations of independent uniformly distributed over 0 1 random variables n with deadlines remaining deterministic

the asymptotic value to capacity ratio for the multi class stochastic knapsack problem we consider a stochastic knapsack problem that packs multiple classes of random items the pairs of profit and resource requirement for items of the same class are independent and identically distributed however such pairs for different classes of items are independent but not necessarily identically distributed we investigate convergence and the asymptotic value of the ratio of the knapsack value function to the capacity when the capacity increases we consider two growth models for the multi class stochastic knapsack problem one where the number of available items grows proportionally to the capacity and the other one where the available items are sampled until their total resource requirement does not exceed the capacity by extending the results of meanti et al on a single class of random items we show that for each growth model the ratio converges almost surely to a computable constant for special cases where resource requirements or profit coefficients are deterministic we present simple interpretations on the asymptotic values finally we present an exponential order upperbound on the tail probability of the ratio

analysis of a multiobjective evolutionary algorithm on the 0 1 knapsack problem multiobjective evolutionary algorithms moeas are increasingly being used for effectively solving many real world problems and many empirical results are available however theoretical analysis is limited to a few simple toy functions in this work we select the well known knapsack problem for the analysis the multiobjective knapsack problem in its general form is np complete moreover the size of the set of pareto optimal solutions can grow exponentially with the number of items in the knapsack thus we formalize a 1 e approximate set of the knapsack problem and attempt to present a rigorous running time analysis of a moea to obtain the formalized set the algorithm used in the paper is based on a restricted mating pool with a separate archive to store the remaining population we call the algorithm a restricted evolutionary multiobjective optimizer remo we also analyze the running time of remo on a special bi objective linear function known as lotz leading ones trailing zeros whose pareto set is shown to be a subset of the knapsack an extension of the analysis to the simple evolutionary multiobjective optimizer semo is also presented a strategy based on partitioning of the decision space into fitness layers is used for the analysis

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

pp is as hard as the polynomial time hierarchy in this paper two interesting complexity classes pp and oplus text p are compared with ph the polynomial time hierarchy it is shown that every set in ph is polynomial time turing reducible to a set in pp and ph is included in text bp cdot oplus text p as a consequence of the results it follows that text pp subseteq text ph or oplus text p subseteq text ph implies a collapse of ph a stronger result is also shown every set in pp ph is polynomial time turing reducible to a set in pp

a non markovian sir network model with fixed infectious period and preventive rewiring abstract in the face of disease occurrence susceptible individuals tend to protect themselves by rewiring their links i e cutting off connection with infected person and switching to contact with healthy ones therefore the adaptive rewiring mechanism is considered in the network epidemic model moreover the infection periods for different diseases do not always follow exponential distributions in the process of disease transmission which may be of fixed length using the idea of age structure we establish the susceptible infected recovered epidemic model for each node and each link resulting in a delayed non markovian sir pairwise model with fixed infectious period and preventive rewiring then give the pairwise reproduction number r 0 p and provide the formula for the final epidemic size by rigorous analysis and tedious computation the simulation results show that adaptive rewiring inhibits the spread of disease and decreases the size of disease outbreak while the extension of the infectious period promotes disease transmission in addition the numerical simulation results are in good agreement with the stochastic simulation to our best knowledge the approach in building our model is novel the results may provide new insights into the study of the network disease transmission

random self reducibility of complete sets this paper generalizes the previous formal definitions of random self reducibility it is shown that even under a very general definition sets that are complete for any level of the polynomial hierarchy are not nonadaptively random self reducible unless the hierarchy collapses in particular np complete sets are not nonadaptively random self reducible unless the hierarchy collapses at the third level by contrast we show that sets complete for the classes pp and text mod m text p are random self reducible

the complexity of partial derivatives abstract let l denote the nonscalar complexity in k x 1 x n we prove l x 1 x n 3l using this we determine the complexity of single power sums single elementary symmetric functions the resultant and the discriminant as root functions up to order of magnitude also we linearly reduce matrix inversion to computing the determinant

cascades on clique based graphs we present an analytical approach to determining the expected cascade size in a broad range of dynamical models on the class of highly clustered random graphs introduced by gleeson j p gleeson phys rev e 80 036107 2009 a condition for the existence of global cascades is also derived applications of this approach include analyses of percolation and watts s model we show how our techniques can be used to study the effects of in group bias in cascades on social networks

on the effective nullstellensatz let mathbb k be an algebraically closed field and let x subset mathbb k m be an n dimensional affine variety assume that f1 fk are polynomials which have no common zeros on x we estimate the degrees of polynomials a i in mathbb k x such that 1 ki 1aifi on x our estimate is sharp for k n and nearly sharp for k n now assume that f1 fk are polynomials on x let i f 1 dots f k subset mathbb k x be the ideal generated by fi it is well known that there is a number e i the noether exponent such that ie i i we give a sharp estimate of e i in terms of n deg x and deg fi we also give similar estimates in the projective case finally we obtain a result from the elimination theory if f 1 dots f n in mathbb k x 1 dots x n is a system of polynomials with a finite number of common zeros then we have the following optimal elimination phi i x i sum n j 1 f jg ij quad i 1 dots n where deg f jg ij le prod n i 1 deg f i

approximation schemes for deal splitting and covering integer programs with multiplicity constraints we consider the problem of splitting an order for r goods r 1 among a set of sellers each having bounded amounts of the goods so as to minimize the total cost of the deal in deal splitting with packages dsp the sellers offer packages containing combinations of the goods in deal splitting with price tables dst the buyer can generate such combinations using price tables our problems which often occur in online reverse auctions generalize covering integer programs with multiplicity constraints cip where we must fill up an r dimensional bin by selecting with a bounded number of repetitions from a set of r dimensional items such that the overall cost is minimized thus both dsp and dst are np hard already for a single good and hard to approximate for arbitrary number of goods in this paper we focus on finding efficient approximations for dsp and dst instances where the number of goods is some fixed constant in particular we develop polynomial time approximation schemes ptas for several subclasses of instances of practical interest our results include a ptas for cip in fixed dimension and a more efficient combinatorial scheme for cip where the multiplicity constraints are omitted our approximation scheme for cip is based on a non trivial application of the fast scheme for the fractional covering problem proposed by fleischer l fleischer a fast approximation scheme for fractional covering problems with variable upper bounds in proc of the 15th acm siam symposium on discrete algorithm 2004 pp 994 1003

coupled and k sided placements generalizing generalized assignment in modern data centers and cloud computing systems jobs often require resources distributed across nodes providing a wide variety of services motivated by this we study the coupled placement problem in which we place jobs into computation and storage nodes with capacity constraints so as to optimize some costs or profits associated with the placement the coupled placement problem is a natural generalization of the widely studied generalized assignment problem which concerns the placement of jobs into single nodes providing one kind of service we also study a further generalization the k sided placement problem in which we place jobs into k tuples of nodes each node in a tuple offering one of k services for both the coupled and k sided placement problems we consider minimization and maximization versions in the minimization versions mincpand min k ksp the goal is to achieve minimum placement cost while incurring a minimum blowup in the capacity of the individual nodes our first main result is an algorithm for min k ksp that achieves optimal cost while increasing capacities by at most a factor of k 1 k 1 also yielding the first constant factor approximation for mincp in the maximization versions maxcp and max k ksp the goal is to maximize the total weight of the jobs that are placed under hard capacity constraints max k ksp can be expressed as a k column sparse integer program and can be approximated to within a factor of o k factor using randomized rounding of a linear program relaxation we consider alternative combinatorial algorithms that are much more efficient in practice our second main result is a local search based combinatorial algorithm that yields a 15 approximation and o k 2 o k2 approximation for maxcp and max k ksp respectively finally we consider an online version of max k ksp and present algorithms that achieve logarithmic competitive ratio under certain necessary technical assumptions

the unreasonable effectiveness of tree based theory for networks with clustering we demonstrate that a tree based theory for various dynamical processes operating on static undirected networks yields extremely accurate results for several networks with high levels of clustering we find that such a theory works well as long as the mean intervertex distance l is sufficiently small that is as long as it is close to the value of l in a random network with negligible clustering and the same degree degree correlations we support this hypothesis numerically using both real world networks from various domains and several classes of synthetic clustered networks we present analytical calculations that further support our claim that tree based theories can be accurate for clustered networks provided that the networks are sufficiently small worlds

percolation on random networks with arbitrary k core structure the k core decomposition of a network has thus far mainly served as a powerful tool for the empirical study of complex networks we now propose its explicit integration in a theoretical model we introduce a hard core random network hrn model that generates maximally random networks with arbitrary degree distribution and arbitrary k core structure we then solve exactly the bond percolation problem on the hrn model and produce fast and precise analytical estimates for the corresponding real networks extensive comparison with real databases reveals that our approach performs better than existing models while requiring less input information

approximation algorithms for knapsack problems with cardinality constraints abstract we address a variant of the classical knapsack problem in which an upper bound is imposed on the number of items that can be selected this problem arises in the solution of real life cutting stock problems by column generation and may be used to separate cover inequalities with small support within cutting plane approaches to integer linear programs we focus our attention on approximation algorithms for the problem describing a linear storage polynomial time approximation scheme ptas and a dynamic programming based fully polynomial time approximation scheme fptas the main ideas contained in our ptas are used to derive ptas s for the knapsack problem and its multi dimensional generalization which improve on the previously proposed ptas s we finally illustrate better ptas s and fptas s for the subset sum case of the problem in which profits and weights coincide

cascades on a class of clustered random networks we present an analytical approach to determining the expected cascade size in a broad range of dynamical models on the class of random networks with arbitrary degree distribution and nonzero clustering introduced previously in m e j newman phys rev lett 103 058701 2009 a condition for the existence of global cascades is derived as well as a general criterion that determines whether increasing the level of clustering will increase or decrease the expected cascade size applications examples of which are provided include site percolation bond percolation and watts threshold model in all cases analytical results give excellent agreement with numerical simulations

a geometric effective nullstellensatz the purpose of this paper is to present a geometric theorem which clarifies and extends in several directions work of brownawell kollar and others on the effective nullstellensatz specifically we work on an arbitrary smooth complex projective variety x with the previous classical results corresponding to the case when x is projective space in this setting we prove a local effective nullstellensatz for ideal sheaves and a corresponding global division theorem for adjoint type bundles we also make explicit the connection with the intersection theory of fulton and macpherson finally constructions involving products of prime ideals that appear in earlier work are replaced by geometrically more natural conditions involving orders of vanishing along subvarieties much of the previous activity in this area has been algebraic in nature and seems perhaps not well known in detail among geometers therefore we have felt it worthwhile to include here a rather extended introduction we start with an overview of the questions and earlier work on them then we present the set up and statement of our main theorem we conclude with a series of examples which can be read before the general result of what it yields in special cases

a note on approximation schemes for multidimensional knapsack problems polynomial and fully polynomial approximation algorithms for single dimensional knapsack problems have been extensively studied and a number of such algorithms constructed this note shows that the problem of finding a fully polynomial approximation algorithm for multidimensional knapsack problems is np hard

approximation algorithms for combinatorial fractional programming problems we are concerned with a combinatorial optimization problem which has the ratio of two linear functions as the objective function this type of problems can be solved by an algorithm that uses an auxiliary problem with a parametrized linear objective function because of its combinatorial nature however it is often difficult to solve the auxiliary problem exactly in this paper we propose an algorithm which assumes that the auxiliary problems are solved only approximately and prove that it gives an approximate solution to the original problem of which the accuracy is at least as good as that of approximate solutions to the auxiliary problems it is also shown that the time complexity is bounded by the square of the computation time of the approximate algorithm for the auxiliary problem as an example of the proposed algorithm we present a fully polynomial time approximation scheme for the fractional 0 1 knapsack problem

approximate strong separation with application in fractional graph coloring and preemptive scheduling in this paper we show that approximation algorithms for the weighted independent set and s dimensional knapsack problem with ratio a can be turned into approximation algorithms with the same ratio for fractional weighted graph coloring and preemptive resource constrained scheduling in order to obtain these results we generalize known results by grotschel lovasz and schrijver on separation non emptiness test optimization and violation in the direction of approximability

resilience to damage of graphs with degree correlations the existence or nonexistence of a percolation threshold on power law correlated graphs is a fundamental question for which a general criterion is lacking in this work we investigate the problems of site and bond percolation on graphs with degree correlations and their connection with spreading phenomena we obtain some general expressions that allow the computation of the transition thresholds or their bounds using these results we study the effects of assortative and disassortative correlations on the resilience to damage of networks

network cloning unfolds the effect of clustering on dynamical processes we introduce network l cloning a technique for creating ensembles of random networks from any given real world or artificial network each member of the ensemble is an l cloned network constructed from l copies of the original network the degree distribution of an l cloned network and more importantly the degree degree correlation between and beyond nearest neighbors are identical to those of the original network the density of triangles in an l cloned network and hence its clustering coefficient is reduced by a factor of l compared to those of the original network furthermore the density of loops of any fixed length approaches zero for sufficiently large values of l other variants of l cloning allow us to keep intact the short loops of certain lengths as an application we employ these network cloning methods to investigate the effect of short loops on dynamical processes running on networks and to inspect the accuracy of corresponding tree based theories we demonstrate that dynamics on l cloned networks with sufficiently large l are accurately described by the so called adjacency tree based theories examples of which include the message passing technique some pair approximation methods and the belief propagation algorithm used respectively to study bond percolation si epidemics and the ising model

properties of solutions to a class of differential models incorporating preisach hysteresis operator abstract we consider a class of equations which have been recently proposed as a mathematical tool for modelling dynamics of hydrological economic and biological systems exhibiting hysteresis and related memory effects the detail of the modelling approach is illustrated by an example from the hydrological context where a balance equation is coupled with a hysteretic constitutive relationship between the water content in the soil and the matric potential m of the soil matrix and where the preisach hysteresis operator is used as a model of this constitutive relationship in particular we present assumptions which eliminate spatial variation and lead to balance equations in the form of odes two examples of such hydrological models are considered followed by a less detailed discussion of applications of similar modelling approach and equations in economics and biology in the proposed formalism the closed system is described by an operator differential equation where the rate of change of the output of the preisach operator is a function of its input and time in the main part of the paper for such operator differential equations we study the initial value problem uniqueness existence extendability of solutions their dependence on initial data and the structure of the projection of the phase portrait onto the t m plane solutions are characterised by jumps of the derivative induced by either of the two reasons one is the memory of past extremum values of the solution the other is a singularity at the lines of zero flow we analyse the singularity and calculate the value of the jumps thus providing an important input to numerical solution of the equation furthermore we identify possible non uniqueness points and points of sensitivity to small perturbations of initial data as well as conditions that ensure uniqueness and stability to such perturbations regularisation of the equation and natural monotonicity conditions ensuring global stability of a periodic solution for the equation with periodic input are discussed rigorous analysis of the well posedness of the models is presented for the first time

circuit complexity proof complexity and polynomial identity testing the ideal proof system we introduce a new and natural algebraic proof system whose complexity measure is essentially the algebraic circuit size of nullstellensatz certificates this enables us to exhibit close connections between effective nullstellensatze proof complexity and algebraic circuit complexity in particular we show that any super polynomial lower bound on any boolean tautology in our proof system implies that the permanent does not have polynomial size algebraic circuits vnp vp we also show that super polynomial lower bounds on the number of lines in polynomial calculus proofs imply the permanent versus determinant conjecture note that there was no proof system prior to ours for which lower bounds on an arbitrary tautology implied any complexity class lower bound our proof system helps clarify the relationships between previous algebraic proof systems in doing so we highlight the importance of polynomial identity testing pit in proof complexity in particular we use pit to illuminate ac0 p frege lower bounds which have been open for nearly 30 years with no satisfactory explanation as to their apparent difficulty finally we explain the obstacles that must be overcome in any attempt to extend techniques from algebraic circuit complexity to prove lower bounds in proof complexity using the algebraic structure of our proof system we propose a novel route to such lower bounds although such lower bounds remain elusive this proposal should be contrasted with the difficulty of extending ac0 p circuit lower bounds to ac0 p frege lower bounds

clique percolation in random graphs as a generation of the classical percolation clique percolation focuses on the connection of cliques in a graph where the connection of two k cliques means that they share at least l 1 makes a step function like discontinuous phase transition in the thermodynamic limit and a continuous phase transition for l 1 more interesting our analysis shows that at the critical point the order parameter c for l 1 is neither 0 nor 1 but a constant depending on k and l all these theoretical findings are in agreement with the simulation results which give theoretical support and clarification for previous simulation studies of clique percolation

sharp effective nullstellensatz the usual proofs of this result however give no information about the g1 s for instance they give no bound on their degrees this question was first considered by g hermann h she used elimination theory to get a bound on the degree of the gi s which was doubly exponential in the number of variables her results were later improved in mw and in th all these produce bounds that are doubly exponential in the number of variables a major breakthrough was achieved by brownawell b1 who proved the following result

beyond the locally treelike approximation for percolation on real networks theoretical attempts proposed so far to describe ordinary percolation processes on real world networks rely on the locally treelike ansatz such an approximation however holds only to a limited extent because real graphs are often characterized by high frequencies of short loops we present here a theoretical framework able to overcome such a limitation for the case of site percolation our method is based on a message passing algorithm that discounts redundant paths along triangles in the graph we systematically test the approach on 98 real world graphs and on synthetic networks we find excellent accuracy in the prediction of the whole percolation diagram with significant improvement with respect to the prediction obtained under the locally treelike approximation residual discrepancies between theory and simulations do not depend on clustering and can be attributed to the presence of loops longer than three edges we present also a method to account for clustering in bond percolation but the improvement with respect to the method based on the treelike approximation is much less apparent

approximation schemes for knapsack problems with shelf divisions given a knapsack of size k non negative values d and and a set s of items each item e s with size se and value ve we define a shelf as a subset of items packed inside a bin with total items size at most two subsequent shelves must be separated by a shelf divisor of size d the size of a shelf is the total size of its items plus the size of the shelf divisor the shelf knapsack problem sk is to find a subset s s partitioned into shelves with total shelves size at most k and maximum value the class constrained shelf knapsack ccsk is a generalization of the problem sk where each item in s has a class and each shelf in the solution must have only items of the same class we present approximation schemes for the sk and the ccsk problems to our knowledge these are the first approximation results where shelves of non null size are used in knapsack problems

highly clustered scale free networks we propose a model for growing networks based on a finite memory of the nodes the model shows stylized features of real world networks power law distribution of degree linear preferential attachment of new links and a negative correlation between the age of a node and its link attachment rate notably the degree distribution is conserved even though only the most recently grown part of the network is considered as the network grows the clustering reaches an asymptotic value larger than that for regular lattices of the same average connectivity and similar to the one observed in the networks of movie actors coauthorship in science and word synonyms these highly clustered scale free networks indicate that memory effects are crucial for a correct description of the dynamics of growing networks

persistency model and its applications in choice modeling given a discrete maximization problem with a linear objective function where the coefficients are chosen randomly from a distribution we would like to evaluate the expected optimal value and the marginal distribution of the optimal solution we call this the persistency problem for a discrete optimization problem under uncertain objective and the marginal probability mass function of the optimal solution is named the persistence value in general this is a difficult problem to solve even if the distribution of the objective coefficient is well specified in this paper we solve a subclass of this problem when the distribution is assumed to belong to the class of distributions defined by given marginal distributions or given marginal moment conditions under this model we show that the persistency problem maximizing the expected objective value over the set of distributions can be solved via a concave maximization model the persistency model solved using this formulation can be used to obtain important qualitative insights to the behavior of stochastic discrete optimization problems we demonstrate how the approach can be used to obtain insights to problems in discrete choice modeling using a set of survey data from a transport choice modeling study we calibrate the random utility model with choice probabilities obtained from the persistency model numerical results suggest that our persistency model is capable of obtaining estimates that perform as well if not better than classical methods such as logit and cross nested logit models we can also use the persistency model to obtain choice probability estimates for more complex choice problems we illustrate this on a stochastic knapsack problem which is essentially a discrete choice problem under budget constraint

analytical results for bond percolation and k core sizes on clustered networks an analytical approach to calculating bond percolation thresholds sizes of k cores and sizes of giant connected components on structured random networks with nonzero clustering is presented the networks are generated using a generalization of trapman s p trapman theor popul biol 71 160 2007 model of cliques embedded in treelike random graphs the resulting networks have arbitrary degree distributions and tunable degree dependent clustering the effect of clustering on the bond percolation thresholds for networks of this type is examined and contrasted with some recent results in the literature for very high levels of clustering the percolation threshold in these generalized trapman networks is increased above the value it takes in a randomly wired unclustered network of the same degree distribution in assortative scale free networks where the variance of the degree distribution is infinite this clustering effect can lead to a nonzero percolation epidemic threshold

lower bounds in a parallel model without bit operations we define a natural and realistic model of parallel computation called the pram model without bit operations it is like the usual pram model the main difference being that no bit operations are provided it encompasses virtually all known parallel algorithms for weighted combinatorial optimization and algebraic problems in this model we prove that for some large enough constant b the mincost flow problem for graphs with n vertices cannot be solved deterministically or with randomization in sqrt n b expected time using 2 sqrt n b processors this is so even if we restrict every cost and capacity to be an integer nonnegative if it is a capacity of bitlength at most an for some large enough constant a a similar lower bound is also proved for the max flow problem it follows that these problems cannot be solved in our model deterministically or with randomization in omega n c expected time with 2 omega n c processors where c is an appropriate positive constant and n is the total bitlength of the input since these problems were known to be p complete this provides concrete support for the belief that p completeness implies high parallel complexity and for the p not nc conjecture our lower bounds also extend to the pram model with limited bit operations which provides instructions for parity and left or right shift by one bit our proof is based on basic algebraic geometry so we investigate if the algebrogeometric approach could also work for the p versus nc problem our results support this possibility and a close analysis of the limitation of our technique in this context suggests that such a proof of p not nc should somehow use geometric invariant theory in a deep way

np is as easy as detecting unique solutions abstract for every known np complete problem the number of solutions of its instances varies over a large range from zero to exponentially many it is therefore natural to ask if the inherent intractability of np complete problem is caused by this wide variation we give a negative answer to this question using the notion of randomized polynomial time reducibility we show that the problems of distinguishing between instances of sat having zero or one solution or of finding solutions to instances of sat having a unique solution are as hard as sat under randomized reductions several corollaries about the difficulty of specific problems follow for example computing the parity of the number of solutions of a sat formula is shown to be np hard and deciding if a sat formula has a unique solution is shown to be d p hard under randomized reduction central to the study of cryptography is the question as to whether there exist np problems whose instances have solutions that are unique but are hard to find our result can be interpreted as strengthening the belief that such problems exist

on the compatibility of nested logit models with utility maximization abstract the paper examines the relationship between the nested multinomial logit nmnl model and rational consumer behavior we show that the daly zachary mcfadden dzm condition of the validity of stochastic utility maximization in nmnl models is unnecessarily strong and leads too often to rejection of nmnl models with large dissimilarity parameters in many cases the distribution of the stochastic utility components in the nmnl model can be modified in a way that preserves the nmnl choice probabilities on a subset of ri where i is the number of alternatives in the choice set on this subset the dzm condition can be violated without violating the restrictions of stochastic utility maximization

how clustering affects the bond percolation threshold in complex networks the question of how clustering nonzero density of triangles in networks affects their bond percolation threshold has important applications in a variety of disciplines recent advances in modeling highly clustered networks are employed here to analytically study the bond percolation threshold in comparison to the threshold in an unclustered network with the same degree distribution and correlation structure the presence of triangles in these model networks is shown to lead to a larger bond percolation threshold i e clustering increases the epidemic threshold or decreases resilience of the network to random edge deletion

an approximate dynamic programming approach to multidimensional knapsack problems we present an approximate dynamic programming adp approach for the multidimensional knapsack problem mkp we approximate the value function a using parametric and nonparametric methods and busing a base heuristic we propose a new heuristic which adaptively rounds the solution of the linear programming relaxation our computational study suggests athe new heuristic produces high quality solutions fast and robustly bstate of the art commercial packages like cplex require significantly larger computational time to achieve the same quality of solutions c the adp approach using the new heuristic competes successfully with alternative heuristic methods such as genetic algorithms dthe adp approach based on parametric and nonparametric approximations while producing reasonable solutions is not competitive overall this research illustrates that the base heuristic approach is a promising computational approach for mkps worthy of further investigation

large scale topological and dynamical properties of the internet we study the large scale topological and dynamical properties of real internet maps at the autonomous system level collected in a 3 yr time interval we find that the connectivity structure of the internet presents statistical distributions settled in a well defined stationary state the large scale properties are characterized by a scale free topology consistent with previous observations correlation functions and clustering coefficients exhibit a remarkable structure due to the underlying hierarchical organization of the internet the study of the internet time evolution shows a growth dynamics with aging features typical of recently proposed growing network models we compare the properties of growing network models with the present real internet data analysis

a new approximation method for set covering problems with applications to multidimensional bin packing in this paper we introduce a new general approximation method for set covering problems based on the combination of randomized rounding of the near optimal solution of the linear programming lp relaxation leading to a partial integer solution and the application of a well behaved approximation algorithm to complete this solution if the value of the solution returned by the latter can be bounded in a suitable way as is the case for the most relevant generalizations of bin packing the method leads to improved approximation guarantees along with a proof of tighter integrality gaps for the lp relaxation for d dimensional vector packing we obtain a polynomial time randomized algorithm with asymptotic approximation guarantee arbitrarily close to ln d 1 for d 2 this value is 1 693 dots i e we break the natural 2 barrier for this case moreover for small values of d this is a notable improvement over the previously known o ln d guarantee by chekuri and khanna siam j comput 33 2004 pp 837 851 for two dimensional bin packing with and without rotations we obtain polynomial time randomized algorithms with asymptotic approximation guarantee 1 525 dots improving upon previous algorithms with asymptotic performance guarantees arbitrarily close to 2 by jansen and van stee on strip packing with rotations in proceedings of the 37th annual acm symposium on the theory of computing 2005 pp 755 761 for the problem with rotations and 1 691 ldots by caprara math oper res 33 2008 pp 203 215 for the problem without rotations the previously unknown key property used in our proofs follows from a retrospective analysis of the implications of the landmark bin packing approximation scheme by fernandez de la vega and lueker combinatorica 1 1981 pp 349 355 we prove that their approximation scheme is subset oblivious which leads to numerous applications

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

lower bounds on time accuracy trade offs for the 0 1 knapsack problem the 0 1 knapsack optimization problem is well known to be np hard the best known approximation algorithms for this problem are approximation schemes which exhibit trade offs between the running time of the algorithm and the guaranteed accuracy of the solution this paper establishes lower bounds on the time required to find approximate solutions of guaranteed accuracy for a class of knapsack algorithms which must use oracles to perform feasibility tests and dominance tests

cook s versus valiant s hypothesis we first investigate the dependency of our complexity concepts on the field k we prove that for algebraically closed fields k the truth of valiant s hypothesis vp vnp depends at most on the characteristic of k

random networks with tunable degree distribution and clustering we present an algorithm for generating random networks with arbitrary degree distribution and clustering frequency of triadic closure we use this algorithm to generate networks with exponential power law and poisson degree distributions with variable levels of clustering such networks may be used as models of social networks and as a testable null hypothesis about network structure finally we explore the effects of clustering on the point of the phase transition where a giant component forms in a random network and on the size of the giant component some analysis of these effects is presented

a theoretical and empirical investigation on the lagrangian capacities of the 0 1 multidimensional knapsack problem abstract we present a novel lagrangian method to find good feasible solutions in theoretical and empirical aspects after investigating the concept of lagrangian capacity which is the value of the capacity constraint that lagrangian relaxation can find an optimal solution we formally reintroduce lagrangian capacity suitable to the 0 1 multidimensional knapsack problem and present its new geometric equivalent condition including a new associated property based on the property we propose a new lagrangian heuristic that finds high quality feasible solutions of the 0 1 multidimensional knapsack problem we verify the advantage of the proposed heuristic by experiments we make comparisons with existing lagrangian approaches on benchmark data and show that the proposed method performs well on large scale data

generating random graphs with tunable clustering coefficients abstract most real world networks exhibit a high clustering coefficient the probability that two neighbors of a node are also neighbors of each other we propose two algorithms conf and throw that take triangle and single edge degree sequences as input and generate a random graph with a target clustering coefficient we analyze them theoretically for the case of a regular graph conf generates a random graph with the input degree sequence and the clustering coefficient anticipated from the input experimental results match quite well with the anticipated clustering coefficient except for highly dense graphs in which case the experimental clustering coefficient is higher than the anticipated value for throw the degree sequence and the clustering coefficient of the generated graph varies from the input however it maintains the expected degree distribution and the clustering coefficient of the generated graph can also be predicted using analytical results experiments show that for throw the results match quite well with the analytical results typically only information about degree distribution is available we also propose an algorithm deg that takes degree sequence and clustering coefficient as input and generates a graph with the same properties experiments show results for deg that are quite similar to those for conf

geometric complexity theory i an approach to the p vs np and related problems we suggest an approach based on geometric invariant theory to the fundamental lower bound problems in complexity theory concerning formula and circuit size specifically we introduce the notion of a partially stable point in a reductive group representation which generalizes the notion of stability in geometric invariant theory due to mumford geometric invariant theory springer verlag berlin 1965 then we reduce fundamental lower bound problems in complexity theory to problems concerning infinitesimal neighborhoods of the orbits of partially stable points we also suggest an approach to tackle the latter class of problems via construction of explicit obstructions

properties of highly clustered networks we propose and solve exactly a model of a network that has both a tunable degree distribution and a tunable clustering coefficient among other things our results indicate that increased clustering leads to a decrease in the size of the giant component of the network we also study susceptible infective recovered type epidemic processes within the model and find that clustering decreases the size of epidemics but also decreases the epidemic threshold making it easier for diseases to spread in addition clustering causes epidemics to saturate sooner meaning that they infect a near maximal fraction of the network for quite low transmission rates

the growth of m constraint random knapsacks the author computes the asymptotic value of a particular m constraint n variable 0 1 random integer programming problem as n increases m remaining fixed this solves a problem of frieze and clarke 1984

sharp estimates for the arithmetic nullstellensatz we present sharp estimates for the degree and the height of the polynomials in the nullstellensatz over the integer ring zz the result improves previous work of philippon berenstein yger and krick pardo we also present degree and height estimates of intrinsic type which depend mainly on the degree and the height of the input polynomial system as an application we derive an effective arithmetic nullstellensatz for sparse polynomial systems the proof of these results relies heavily on the notion of local height of an affine variety defined over a number field we introduce this notion and study its basic properties

zero temperature hysteresis in the random field ising model on a bethe lattice we consider the single spin flip dynamics of the random field ising model on a bethe lattice at zero temperature in the presence of a uniform external field we determine the average magnetization as the external field is varied from to by setting up the self consistent field equations which we show are exact in this case the qualitative behaviour of magnetization as a function of the external field unexpectedly depends on the coordination number z of the bethe lattice for z 3 with a gaussian distribution of the quenched random fields we find no jump in magnetization for any non zero strength of disorder for for weak disorder the magnetization shows a jump discontinuity as a function of the external uniform field which disappears for a larger variance of the quenched field we determine exactly the critical point separating smooth hysteresis curves from those with a jump we have checked our results by monte carlo simulations of the model on three and four coordinated random graphs which for large system sizes give the same results as on the bethe lattice but avoid surface effects altogether

tuning clustering in random networks with arbitrary degree distributions we present a generator of random networks where both the degree dependent clustering coefficient and the degree distribution are tunable following the same philosophy as in the configuration model the degree distribution and the clustering coefficient for each class of nodes of degree k are fixed ad hoc and a priori the algorithm generates corresponding topologies by applying first a closure of triangles and second the classical closure of remaining free stubs the procedure unveils an universal relation among clustering and degree degree correlations for all networks where the level of assortativity establishes an upper limit to the level of clustering maximum assortativity ensures no restriction on the decay of the clustering coefficient whereas disassortativity sets a stronger constraint on its behavior correlation measures in real networks are seen to observe this structural bound

the effects of coefficient correlation structure in two dimensional knapsack problems on solution procedure performance this paper presents the results of an empirical study of the effects of coefficient correlation structure and constraint slackness settings on the performance of solution procedures on synthetic two dimensional knapsack problems 2kp the population correlation structure among 2kp coefficients the level of constraint slackness and the type of correlation product moment or rank are varied in this study representative branch and bound and heuristic solution procedures are used to investigate the influence of these problem parameters on solution procedure performance population correlation structure and in particular the interconstraint component of the correlation structure is found to be a significant factor influencing the performance of both the algorithm and the heuristic in addition the interaction between constraint slackness and population correlation structure is found to influence solution procedure performance

the intractability of resolution abstract we prove that for infinitely many disjunctive normal form propositional calculus tautologies the length of the shortest resolution proof of cannot be bounded by any polynomial of the length of the tautologies we use were introduced by cook and reckhow 1979 and encode the pigeonhole principle extended resolution can furnish polynomial length proofs of these formulas

the growth of multi constraint random knapsacks with various right hand sides of the constraints abstract the asymptotic value of the m constraint n variable 0 1 random knapsack problem with b j n the right hand side of the j th constraint m fixed n is computed previously known results have been generalized

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

bipartite graphs as models of complex networks abstract it appeared recently that the classical random graph model used to represent real world complex networks does not capture their main properties since then various attempts have been made to provide accurate models we study here a model which achieves the following challenges it produces graphs which have the three main wanted properties clustering degree distribution average distance it is based on some real world observations and it is sufficiently simple to make it possible to prove its main properties this model consists in sampling a random bipartite graph with prescribed degree distribution indeed we show that any complex network may be viewed as a bipartite graph with some specific characteristics and that its main properties may be viewed as consequences of this underlying structure we also propose a growing model based on this observation

geometric complexity theory ii towards explicit obstructions for embeddings among class varieties in k d mulmuley and m sohoni siam j comput 31 2001 pp 496 526 henceforth referred to as part i we suggested an approach to the p vs np and related lower bound problems in complexity theory through geometric invariant theory in particular it reduces the arithmetic characteristic zero version of the np not subseteq p conjecture to the problem of showing that a variety associated with the complexity class np cannot be embedded in a variety associated with the complexity class p we shall call these class varieties associated with the complexity classes p and np this paper develops this approach further reducing these lower bound problems which are all nonexistence problems to some existence problems specifically to proving the existence of obstructions to such embeddings among class varieties it gives two results towards explicit construction of such obstructions the first result is a generalization of the borel weil theorem to a class of orbit closures which include class varieties the second result is a weaker form of a conjectured analogue of the second fundamental theorem of invariant theory for the class variety associated with the complexity class nc these results indicate that the fundamental lower bound problems in complexity theory are in turn intimately linked with explicit construction problems in algebraic geometry and representation theory the results here were announced in k d mulmuley and m sohoni in advances in algebra and geometry hyderabad 2001 hindustan book agency new delhi india 2003 pp 239 261

arithmetic circuits the chasm at depth four gets wider in their paper on the chasm at depth four agrawal and vinay have shown that polynomials in m variables of degree o m which admit arithmetic circuits of size 2 o m also admit arithmetic circuits of depth four and size 2 o m this theorem shows that for problems such as arithmetic circuit lower bounds or black box derandomization of identity testing the case of depth four circuits is in a certain sense the general case in this paper we show that smaller depth four circuits can be obtained if we start from polynomial size arithmetic circuits for instance we show that if the permanent of nxn matrices has circuits of size polynomial in n then it also has depth 4 circuits of size n o n l o g n if the original circuit uses only integer constants of polynomial size then the same is true for the resulting depth four circuit these results have potential applications to lower bounds and deterministic identity testing in particular for sums of products of sparse univariate polynomials we also use our techniques to reprove two results on the existence of nontrivial boolean circuits of constant depth for languages in logcfl reduction to polylogarithmic depth for arithmetic circuits of polynomial size and polynomially bounded degree

on interpolation and automatization for frege systems the interpolation method has been one of the main tools for proving lower bounds for propositional proof systems loosely speaking if one can prove that a particular proof system has the feasible interpolation property then a generic reduction can usually be applied to prove lower bounds for the proof system sometimes assuming a usually modest complexity theoretic assumption in this paper we show that this method cannot be used to obtain lower bounds for frege systems or even for tc0 frege systems more specifically we show that unless factoring of blum integers is feasible neither frege nor tc0 frege has the feasible interpolation property in order to carry out our argument we show how to carry out proofs of many elementary axioms theorems of arithmetic in polynomial sized tc0 frege as a corollary we obtain that tc0 frege as well as any proof system that polynomially simulates it is not automatizable under the assumption that factoring of blum integers is hard we also show under the same hardness assumption that the k provability problem for frege systems is hard

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

clustering in complex networks ii percolation properties the percolation properties of clustered networks are analyzed in detail in the case of weak clustering we present an analytical approach that allows us to find the critical threshold and the size of the giant component numerical simulations confirm the accuracy of our results in more general terms we show that weak clustering hinders the onset of the giant component whereas strong clustering favors its appearance this is a direct consequence of the differences in the k core structure of the networks which are found to be totally different depending on the level of clustering an empirical analysis of a real social network confirms our predictions

approximating multiobjective knapsack problems for multiobjective optimization problems it is meaningful to compute a set of solutions covering all possible trade offs between the different objectives the multiobjective knapsack problem is a generalization of the classical knapsack problem in which each item has several profit values for this problem efficient algorithms for computing a provably good approximation to the set of all nondominated feasible solutions the pareto frontier are studied for the multiobjective one dimensional knapsack problem a practical fully polynomial time approximation scheme fptas is derived it is based on a new approach to the single objective knapsack problem using a partition of the profit space into intervals of exponentially increasing length for the multiobjectivem dimensional knapsack problem the first known polynomial time approximation scheme ptas based on linear programming is presented

bond percolation on a class of clustered random networks analytical results are derived for the bond percolation threshold and the size of the giant connected component in a class of random networks with nonzero clustering the network s degree distribution and clustering spectrum may be prescribed and theoretical results match well with numerical simulations on both synthetic and real world networks

fast parallel computation of polynomials using few processors it is shown that any multivariate polynomial of degree d that can be computed sequentially in c steps can be computed in parallel in o log d log c log d steps using only cd o 1 processors

on multidimensional packing problems we study the approximability of multidimensional generalizations of three classical packing problems multiprocessor scheduling bin packing and the knapsack problem specifically we study the vector scheduling problem its dual problem namely the vector bin packing problem and a class of packing integer programs the vector scheduling problem is to schedule n d dimensional tasks on m machines such that the maximum load over all dimensions and all machines is minimized the vector bin packing problem on the other hand seeks to minimize the number of bins needed to schedule all n tasks such that the maximum load on any dimension across all bins is bounded by a fixed quantity say 1 such problems naturally arise when scheduling tasks that have multiple resource requirements finally packing integer programs capture a core problem that directly relates to both vector scheduling and vector bin packing namely the problem of packing a maximum number of vectors in a single bin of unit height we obtain a variety of new algorithmic as well as inapproximability results for these three problems

general and exact approach to percolation on random graphs we present a comprehensive and versatile theoretical framework to study site and bond percolation on clustered and correlated random graphs our contribution can be summarized in three main points i we introduce a set of iterative equations that solve the exact distribution of the size and composition of components in finite size quenched or random multitype graphs ii we define a very general random graph ensemble that encompasses most of the models published to this day and also makes it possible to model structural properties not yet included in a theoretical framework site and bond percolation on this ensemble is solved exactly in the infinite size limit using probability generating functions i e the percolation threshold the size and the composition of the giant extensive and small components several examples and applications are also provided iii our approach can be adapted to model interdependent graphs whose most striking feature is the emergence of an extensive component via a discontinuous phase transition in an equally general fashion we show how a graph can successively undergo a continuous then a discontinuous phase transition and preliminary results suggest that clustering increases the amplitude of the discontinuity at the transition

tightening simple mixed integer sets with guaranteed bounds in this paper we study how to reformulate knapsack sets and simple mixed integer sets in order to obtain provably tight polynomially large formulations

analysis of complex contagions in random multiplex networks we study the diffusion of influence in random multiplex networks where links can be of r different types and for a given content e g rumor product or political view each link type is associated with a content dependent parameter ci in 0 that measures the relative bias type i links have in spreading this content in this setting we propose a linear threshold model of contagion where nodes switch state if their perceived proportion of active neighbors exceeds a threshold namely a node connected to mi active neighbors and ki mi inactive neighbors via type i links will turn active if cimi ciki exceeds its threshold under this model we obtain the condition probability and expected size of global spreading events our results extend the existing work on complex contagions in several directions by i providing solutions for coupled random networks whose vertices are neither identical nor disjoint ii highlighting the effect of content on the dynamics of complex contagions and iii showing that content dependent propagation over a multiplex network leads to a subtle relation between the giant vulnerable component of the graph and the global cascade condition that is not seen in the existing models in the literature

the complexity of the word problems for commutative semigroups and polynomial ideals abstract any decision procedure for the word problems for commutative semigroups and polynomial deals inherently requires computational storage space growing exponentially with the size of the problem instance to which the procedure is applied this bound is achieved by a simple procedure for the semigroup problem