moderately exponential approximation for makespan minimization on related machines we consider in this paper the classical q c m a x scheduling problem the objective is to minimize the maximum completion time called makespan while scheduling independent jobs in parallel on machines that have different speeds while several approximation schemes has been proposed and in particular a recent eptas jansen 2010 12 the current best direct algorithm i e an algorithm specifically designed for reaching a given approximation ratio is still due to chen 1991 4 with a 1 382 ratio our objective in this work is not to provide yet another improvement of the asymptotic dependencies in email protected ensuring a email protected ratio but to design faster direct algorithms by targeting respectively 43 and 54 ratios indeed instantiating any of the existing approximation scheme for e 13 respectively 14 leads to polynomial complexities but not to practical algorithms because of the hidden large constants in the computational complexity thus our approach focuses on a moderately exponential algorithm and provides a email protected dual approximation algorithm running in o m 1 1 3 e l o g 3 b 1 m n where m is the number of machines n the number of jobs b an integer lower than m depending on the instance this result is obtained through an oracle framework where the algorithm guesses possible answers from an oracle the terseness of the answers points out the critical information needed while solving any instance such an approach leads to a better comprehension of the problem similarly we obtain the same kind of results for a email protected ratio moreover the proposed techniques seem promising for tackling classical specific cases like scheduling on identical machines as the complexity becomes a low degree polynomial when the speeds are non arbitrary

using dual approximation algorithms for scheduling problems theoretical and practical results the problem of scheduling a set of n jobs on m identical machines so as to minimize the makespan time is perhaps the most well studied problem in the theory of approximation algorithms for np hard optimization problems in this paper the strongest possible type of result for this problem a polynomial approximation scheme is presented more precisely for each e an algorithm that runs in time o n e 1 e 2 and has relative error at most e is given in addition more practical algorithms for e 1 5 2 k and e 1 6 2 k which have running times o n k log n and o n km 4 log n are presented the techniques of analysis used in proving these results are extremely simple especially in comparison with the baroque weighting techniques used previously the scheme is based on a new approach to constructing approximation algorithms which is called dual approximation algorithms where the aim is to find superoptimal but infeasible solutions and the performance is measured by the degree of infeasibility allowed this notion should find wide applicability in its own right and should be considered for any optimization problem where traditional approximation algorithms have been particularly elusive

continuity of the radius of convergence of differential equations on p adic analytic curves this paper deals with connections on non archimedean especially p adic analytic curves in the sense of berkovich the curves must be compact but the connections are allowed to have a finite number of meromorphic singularities on them for any choice of a semistable formal model of the curve we define a geometric intrinsic notion of normalized radius of convergence of a full set of local solutions as a function on the curve with values in 0 1 for a sufficiently refined choice of the semistable model we prove continuity logarithmic concavity and logarithmic piece wise linearity of that function we introduce and characterize robba connections that is connections whose sheaf of solutions is constant on any open disk contained in the curve precisely as it happens in the classical case

scheduling and fixed parameter tractability fixed parameter tractability analysis and scheduling are two core domains of combinatorial optimization which led to deep understanding of many important algorithmic questions however even though fixed parameter algorithms are appealing for many reasons no such algorithms are known for many fundamental scheduling problems in this paper we present the first fixed parameter algorithms for classical scheduling problems such as makespan minimization scheduling with job dependent cost functions one important example being weighted flow time and scheduling with rejection to this end we identify crucial parameters that determine the problems complexity in particular we manage to cope with the problem complexity stemming from numeric input values such as job processing times which is usually a core bottleneck in the design of fixed parameter algorithms we complement our algorithms with mathsf w 1 w 1 hardness results showing that for smaller sets of parameters the respective problems do not allow fixed parameter algorithms in particular our positive and negative results for scheduling with rejection explore a research direction proposed by daniel marx

solving the degree concentrated fault tolerant spanning subgraph problem by dc programming in this paper we consider the maximum and minimum versions of degree concentrated fault tolerant spanning subgraph problem which has many applications in network communications we prove that both this two problems are np hard for the maximum version we use dc programming relaxation to propose a heuristic algorithm numerical tests indicate that the proposed algorithm is efficient and effective for the minimum version we also formulate it as a dc program and show that the dc algorithm does not perform well for this problem

approximation schemes for scheduling and covering on unrelated machines we examine the problem of assigning n independent jobs to m unrelated parallel machines so that each job is processed without interruption on one of the machines and at any time every machine processes at most one job we focus on the case where m is a fixed constant and present a new rounding approach that yields approximation schemes for multi objective minimum makespan scheduling with a fixed number of linear cost constraints the same approach gives approximation schemes for covering problems like maximizing the minimum load on any machine and for assigning specific or equal loads to the machines

approximation algorithms for scheduling unrelated parallel machines we consider the following scheduling problem there arem parallel machines andn independent jobs each job is to be assigned to one of the machines the processing of jobj on machinei requires timep ij the objective is to find a schedule that minimizes the makespan our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum we also present a polynomial approximation scheme for the case that the number of machines is fixed both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations in particular we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints in contrast to our main result we prove that no polynomial algorithm can achieve a worst case ratio less than 3 2 unlessp np we finally obtain a complexity classification for all special cases with a fixed number of processing times

approximation algorithms for single source unsplittable flow in the single source unsplittable flow problem we are given a network g a source vertex s and k commodities with sinks ti and real valued demands rho i 1 leq i leq k we seek to route the demand rho i of each commodity i along a single s ti flow path so that the total flow routed across any edge e is bounded by the edge capacity ue the conceptual difficulty of this np hard problem arises from combining packing constraints due to the existence of capacities with path selection in a graph of arbitrary topology in this paper we give a generic framework which yields approximation algorithms that are simpler than those previously known and achieve significant improvements upon the approximation ratios our framework with appropriate subroutines applies to all optimization versions previously considered and unlike previous work treats in a unified manner directed and undirected graphs we provide extensions of our algorithms which yield the best possible approximation guarantees for restricted sets of demand values and an associated scheduling problem

existence of correlated equilibria an elementary proof based on linear duality is provided for the existence of correlated equilibria in finite games the existence result is then extended to infinite games including some that possess no nash equilibria

a note on the prize collecting traveling salesman problem we study the version of the prize collecting traveling salesman problem where the objective is to find a tour that visits a subset of vertices such that the length of the tour plus the sum of penalties associated with vertices not in the tour is as small as possible we present an approximation algorithm with constant bound the algorithm is based on christofides algorithm for the traveling salesman problem as well as a method to round fractional solutions of a linear programming relaxation to integers feasible for the original problem

a cutting plane algorithm for the unrelated parallel machine scheduling problem this paper deals with the polyhedral structure of the scheduling problem r cmax strong valid inequalities are identified for fixed values of the maximum completion time and are used to build a cutting plane scheme from which an exact algorithm and an approximation algorithm are developed the main algorithm includes a preprocessing phase to compute an upper bound with the list scheduling algorithm of davis and jaffe and a lower bound from the preemptive version of the problem computational results show that the proposed exact algorithm gives an optimal solution for almost all tested cases within the fixed time and memory limits for the few cases where the limit has been exceeded rather good solutions are obtained computational requirements of both algorithms are reported for a collection of test problems and the quality of the schedules provided by the approximation algorithm is measured

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

a linear time algorithm for trust region problems we consider the fundamental problem of minimizing a general quadratic function over an ellipsoidal domain also known as the trust region sub problem we give the first provable linear time in the number of non zero entries of the input algorithm for approximately solving this problem specifically our algorithm returns an epsilon approximate solution in runtime of order n sqrt epsilon where n is the number of non zero entries in the input this matches the runtime of nesterov s accelerated gradient descent suitable for the special case in which the quadratic objective is convex and the runtime of the lanczos method which is applicable when the problem is purely quadratic

existence of stationary correlated equilibria with symmetric information for discounted stochastic games the following theorem is proved every nonzero sum discounted stochastic game in countably generated measurable state space with compact metric action spaces admits a stationary correlated equilibrium point with symmetric public information whenever the immediate rewards and transition densities are measurable with respect to the state variable and continuous with respect to joint actions

query complexity of approximate nash equilibria we study the query complexity of approximate notions of nash equilibrium in games with a large number of players n our main result states that for n player binary action games and for constant e the query complexity of an e well supported nash equilibrium is exponential in n as a consequence of this result we get an exponential lower bound on the rate of convergence of adaptive dynamics to approximate nash equilibria

a simple adaptive procedure leading to correlated equilibrium we propose a new and simple adaptive procedure for playing a game regret matching in this procedure players may depart from their current play with probabilities that are proportional to measures of regret for not having used other strategies in the past it is shown that our adaptive procedure guarantees that with probability one the empirical distributions of play converge to the set of correlated equilibria of the game

analogue p adique du theoreme de turrittin et le theoreme de la monodromie p adique 1 rappel de quelques resultats sur la structure p adique d un point singulier d une equation differentielle 322 2 cas d une pente entiere 327 3 cas d une pente fractionnaire premiere a p 333 4 cas d un rang premier a p 335 5 cas general 339 6 l exemple mx 1 3 343 7 la theorie des modules et l implication dr pst 346

classification of integrable equations on quad graphs the consistency approach a classification of discrete integrable systems on quad graphs i e on surface cell decompositions with quadrilateral faces is given the notion of integrability laid in the basis of the classification is the three dimensional consistency this property yields among other features the existence of the discrete zero curvature representation with a spectral parameter for all integrable systems of the obtained exhaustive list the so called three leg forms are found this establishes lagrangian and symplectic structures for these systems and the connection to discrete systems of the toda type on arbitrary graphs generalizations of these ideas to the three dimensional integrable systems and to the quantum context are also discussed

explicit auto transformations of integrable chains a construction scheme for explicit auto transformations of integrable discrete differential equations chains is presented these transformations are rather convenient to obtain the exact solutions for chains as well as associated partial differential systems on the other hand they exemplify new integrable discrete mappings their group properties are also of great interest the scheme is illustrated by several examples of integrable systems which contains the nonlinear schrodinger system and the landau lifshits model

filtrations de type hasse arf et monodromie p adique one proves the crew tsuzuki p adic local monodromy conjecture for local fields of characteristic p 0

approximation algorithms for scheduling jobs with release times and arbitrary sizes on batch machines with non identical capacities abstract we consider the problem of scheduling jobs with release times and arbitrary sizes on batch machines with non identical capacities to minimize makespan a job can only be assigned to a machine whose capacity is not less than the size of the job a machine can process a group of jobs as a batch simultaneously as long as the total size of the jobs in this batch does not exceed the capacity of the machine the processing time of a batch is equal to the longest processing time of all the jobs in the batch for the case of equal release times we obtain a fast 5 approximation algorithm as well as an algorithm with approximation ratio arbitrarily close to 2 for the case of unequal release times and a fixed number of different batch capacities we obtain an algorithm with approximation ratio arbitrarily close to 2 we also study the problem of scheduling jobs with equal release times arbitrary sizes inclusive processing set restrictions and incompatible families on batch machines with identical capacities to minimize makespan we obtain a 8 3 approximation algorithm as well as an algorithm with approximation ratio arbitrarily close to 2 if only a single family is considered a 2 approximation algorithm is presented it is likely that these results cannot be too substantially improved upon since all of the problems studied cannot be approximated to within a ratio smaller than 2 unless p np

truthful mechanism design via correlated tree rounding a powerful algorithmic technique for truthful mechanism design is the maximal in distributional range midr paradigm unfortunately many such algorithms use heavy algorithmic machinery e g the ellipsoid method and approximate solution of convex programs in this paper we present a correlated rounding technique for designing mechanisms that are truthful in expectation it is elementary and can be implemented quickly the main property we rely on is that the domain offers fractional optimum solutions with a tree structure in auctions based on the generalized assignment problem each bidder has a publicly known knapsack constraint that captures the subsets of items that are of value to him he has a private valuation for each item and strives to maximize the value of assigned items minus payment for this domain we design a truthful 2 approximate midr mechanism for social welfare maximization it avoids using the ellipsoid method or convex programming in contrast to some previous work our mechanism achieves exact truthfulness in restricted related scheduling with selfish machines each job comes with a public weight and it must be assigned to a machine from a public job specific subset each machine has a private speed and strives to maximize payments minus workload of jobs assigned to it here we design a mechanism for makespan minimization this is a single parameter domain but the approximation status of the optimization problem is similar to unrelated machine scheduling the best known algorithm obtains a non truthful 2 approximation for unrelated machines and there is 1 5 hardness our mechanism matches this bound with a truthful 2 approximation

rank one solvable p adic differential equations and finite abelian characters via lubin tate groups we introduce a new class of exponentials of artin hasse type called exponentials these exponentials depend on the choice of a generator of the tate module of a lubin tate group mathfrak g over mathbb z p they arise naturally as solutions of solvable differential modules over the robba ring if mathfrak g is isomorphic to widehat mathbb g m over mathbb z p we develop methods to test their over convergence and get in this way a stronger version of the frobenius structure theorem for differential equations we define a natural transformation of the artin schreier complex into the kummer complex this provides an explicit generator of the kummer unramified extension of epsilon dagger k infty whose residue field is a given artin schreier extension of k t where k is the residue field of k we then compute explicitly the group under tensor product of isomorphism classes of rank one solvable differential equations moreover we get a canonical way to compute the rank one module over epsilon dagger k infty attached to a rank one representation of mathrm gal k t mathrm sep k t defined by an artin schreier character

a p adic local monodromy theorem we produce a canonical filtration for locally free sheaves on an open p adic annulus equipped with a frobenius structure using this filtration we deduce a conjecture of crew on p adic differential equations analogous to grothendieck s local monodromy theorem also a consequence of results of andre and of mebkhout namely given a finite locally free sheaf on an open p adic annulus with a connection and a compatible frobenius structure the module admits a basis over a finite cover of the annulus on which the connection acts via a nilpotent matrix

convergent learning algorithms for unknown reward games in this paper we address the problem of convergence to nash equilibria in games with rewards that are initially unknown and must be estimated over time from noisy observations these games arise in many real world applications whenever rewards for actions cannot be prespecified and must be learned online but standard results in game theory do not consider such settings for this problem we derive a multiagent version of mathcal q learning to estimate the reward functions using novel forms of the epsilon greedy learning policy using these mathcal q learning schemes to estimate reward functions we then provide conditions guaranteeing the convergence of adaptive play and the better reply processes to nash equilibria in potential games and games with more general forms of acyclicity and of regret matching to the set of correlated equilibria in generic games a secondary result is that we prove the strong ergoditicity of stochastic adaptive play and stochastic better reply processes in the ca

approximation algorithms for general packing problems and their application to the multicast congestion problem in this paper we present approximation algorithms based on a lagrangian decomposition via a logarithmic potential reduction to solve a general packing or min max resource sharing problem with m non negative convex constraints on a convex set b we generalize a method by grigoriadis et al to the case with weak approximate block solvers i e with only constant logarithmic or even worse approximation ratios given an accuracy varepsilon in 0 1 we show that our algorithm needs o m ln m varepsilon 2 ln varepsilon 1 calls to the block solver a bound independent of the data and the approximation ratio of the block solver for small approximation ratios the algorithm needs o m ln m varepsilon 2 calls to the block solver as an application we study the problem of minimizing the maximum edge congestion in a multicast communication network interestingly the block problem here is the classical steiner tree problem that can be solved only approximately we show how to use approximation algorithms for the steiner tree problem to solve the multicast congestion problem approximately

a general framework for designing approximation schemes for combinatorial optimization problems with many objectives combined into one in this paper we present a general framework for designing approximation schemes for combinatorial optimization problems in which the objective function is a combination of more than one function examples of such problems include those in which the objective function is a product or ratio of two linear functions parallel machine scheduling problems with the makespan objective robust versions of weighted multiobjective optimization problems and assortment optimization problems with logit choice models the main idea behind our approximation schemes is the construction of an approximate pareto optimal frontier of the functions that constitute the given objective using this idea we give the first fully polynomial time approximation schemes for the max min resource allocation problem with a fixed number of agents combinatorial optimization problems in which the objective function is the sum of a fixed number of ratios of linear functions or the product of a fixed number of linear functions and assor

new constructive aspects of the lovasz local lemma the lovasz local lemma lll is a powerful tool that gives sufficient conditions for avoiding all of a given set of bad events with positive probability a series of results have provided algorithms to efficiently construct structures whose existence is non constructively guaranteed by the lll culminating in the recent breakthrough of moser and tardos 2010 for the full asymmetric lll we show that the output distribution of the moser tardos algorithm well approximates the conditional lll distribution the distribution obtained by conditioning on all bad events being avoided we show how a known bound on the probabilities of events in this distribution can be used for further probabilistic analysis and give new constructive and nonconstructive results we also show that when a lll application provides a small amount of slack the number of resamplings of the moser tardos algorithm is nearly linear in the number of underlying independent variables not events and can thus be used to give efficient constructions in cases where the underlying proof applies the lll to super polynomially many events even in cases where finding a bad event that holds is computationally hard we show that applying the algorithm to avoid a polynomial sized core subset of bad events leads to a desired outcome with high probability this is shown via a simple union bound over the probabilities of non core events in the conditional lll distribution and automatically leads to simple and efficient monte carlo and in most cases rnc algorithms we demonstrate this idea on several applications we give the first constant factor approximation algorithm for the santa claus problem by making a lll based proof of feige constructive we provide monte carlo algorithms for acyclic edge coloring nonrepetitive graph colorings and ramsey type graphs in all these applications the algorithm falls directly out of the non constructive lll based proof our algorithms are very simple often provide better bounds than previous algorithms and are in several cases the first efficient algorithms known as a second type of application we show that the properties of the conditional lll distribution can be used in cases beyond the critical dependency threshold of the lll avoiding all bad events is impossible in these cases as the first even nonconstructive result of this kind we show that by sampling a selected smaller core from the lll distribution we can avoid a fraction of bad events that is higher than the expectation max k sat is an illustrative example of this

the complexity of uniform nash equilibria and related regular subgraph problems we investigate the complexity of finding nash equilibria in which the strategy of each player is uniform on its support set we show that even for a restricted class of win lose bimatrix games deciding the existence of such uniform equilibria is an np complete problem our proof is graph theoretical motivated by this result we also give np completeness results for the problems of finding regular induced subgraphs of large size or regularity which can be of independent interest

computing pure bayesian nash equilibria in games with finite actions and continuous types we extend the well known fictitious play fp algorithm to compute pure strategy bayesian nash equilibria in private value games of incomplete information with finite actions and continuous types g facts we prove that if the frequency distribution of actions fictitious play beliefs converges then there exists a pure strategy equilibrium strategy that is consistent with it we furthermore develop an algorithm to convert the converged distribution of actions into an equilibrium strategy for a wide class of games where utility functions are linear in type this algorithm can also be used to compute pure e nash equilibria when distributions are not fully converged we then apply our algorithm to find equilibria in an important and previously unsolved game simultaneous sealed bid second price auctions where various types of items e g substitutes or complements are sold finally we provide an analytical characterisation of equilibria in games with linear utilities specifically we show how equilibria can be found by solving a system of polynomial equations for a special case of simultaneous auctions we also solve the equations confirming the results obtained numerically

finite local monodromy of overconvergent unit root f isocrystals on a curve we prove that the category of p adic continuous representations of a fundamental group of a curve over a perfect field of a positive characteristic p with finite local monodromy is equivalent to that of overconvergent unit root f isocrystals on the curve to show the equivalence of categories we study the local theory and apply it to the global case

correlated equilibria in two player repeated games with nonobservable actions four kinds of correlated equilibrium payoff sets in undiscounted repeated games with nonobservable actions are studied three of them the upper the uniform and banach lead to the same payoff set whereas the lower one in general is associated with a larger set the extensive form correlated equilibrium is also explored it turns out that both the regular and extensive form correlated equilibria yield the same sets of payoffs

nonlinear differential difference equations as backlund transformations shows that the best known nonlinear differential difference equations associated with the discrete schrodinger spectral problem and also with the discrete zakharov shabat spectral problem can be interpreted as backlund transformations for some continuous nonlinear evolution equations

concurrent particle filtering and data association using game theory for tracking multiple maneuvering targets we propose a particle filtering technique to track multiple maneuvering targets in the presence of clutter we treat data association and state estimation which are the two important sub problems in tracking as separate problems we develop a game theoretic framework to solve the data association in which we model each tracker as a player and the set of measurements as strategies we develop utility functions for each player and then use a regret based learning algorithm to find the equilibrium of this game the game theoretic approach allows us to associate measurements to all the targets simultaneously further in contrast to the traditional monte carlo data association algorithms that use samples of the association vector obtained from a proposal distribution our method finds the association in a deterministic fashion we then use monte carlo sampling on the reduced dimensional state of each target independently and thereby mitigate the curse of dimensionality problem that is known to occur in particle filtering we provide a number of numerical results to demonstrate the performance of our proposed filtering algorithm

a finite genus solution of the hirota equation via integrable symplectic maps two integrable symplectic maps are constructed through nonlinearization of the discrete linear spectral problems in the lax pair of the hirota equation i e the lattice sine gordon equation as an application these maps are used to calculate the finite genus solutions of the hirota equation and the closely related lattice potential mkdv equation i e the special h3 model in the adler bobenko suris list

arithmetic and differential swan conductors of rank one representations with finite local monodromy we consider a complete discrete valuation field of characteristic p with possibly nonperfect residue field let rm v be a rank one continuous representation of its absolute galois group with finite local monodromy we will prove that the it arithmetic swan conductor of rm v defined after k kato which fits in the more general theory of abbes saito coincides with the it differential swan conductor of the associated differential module rm d dag rm v defined by k kedlaya this construction is a generalization to the nonperfect residue case of the fontaine s formalism as presented in the work of n tsuzuki our method of proof will allow us to give a new interpretation of the refined swan conductor

a polynomial approximation scheme for scheduling on uniform processors using the dual approximation approach in this paper we present a polynomial approximation scheme for the minimum makespan problem on uniform parallel processors more specifically the problem is to find a schedule for a set of independent jobs on a collection of machines of different speeds so that the last job to finish is completed as quickly as possible we give a family of polynomial time algorithms a such that a delivers a solution that is within a relative error of e of the optimum the technique employed is the dual approximation approach where infeasible but superoptimal solutions for a related dual problem are converted to the desired feasible but possibly suboptimal solution

parallel machine scheduling with nested processing set restrictions abstract we consider the problem of scheduling a set of n independent jobs on m parallel machines where each job can only be scheduled on a subset of machines called its processing set the machines are linearly ordered and the processing set of job j is given by two machine indexes a j and b j i e job j can only be scheduled on machines a j a j 1 b j two distinct processing sets are either nested or disjoint preemption is not allowed our goal is to minimize the makespan it is known that the problem is strongly np hard and that there is a list type algorithm with a worst case bound of 2 1 m in this paper we give an improved algorithm with a worst case bound of 7 4 for two and three machines the algorithm gives a better worst case bound of 5 4 and 3 2 respectively

extensive form correlated equilibrium definition and computational complexity this paper defines the extensive form correlated equilibrium efce for extensive games with perfect recall the efce concept extends aumann s strategic form correlated equilibrium ce before the game starts a correlation device generates a move for each information set this move is recommended to the player only when the player reaches the information set in two player perfect recall extensive games without chance moves the set of efce can be described by a polynomial number of consistency and incentive constraints assuming p is not equal to np this is not possible for the set of ce or if the game has chance moves

strong lp formulations for scheduling splittable jobs on unrelated machines a natural extension of the makespan minimization problem on unrelated machines is to allow jobs to be partially processed by different machines while incurring an arbitrary setup time in this paper we present increasingly stronger lp relaxations for this problem and their implications on the approximability of the problem first we show that the straightforward lp extending the approach for the original problem has an integrality gap of 3 and yields an approximation algorithm of the same factor by applying a lift and project procedure we are able to improve both the integrality gap and the implied approximation factor to 1 phi 1 where phi is the golden ratio since this bound remains tight for the seemingly stronger machine configuration lp we propose a new job configuration lp that is based on an infinite continuum of fractional assignments of each job to the machines we prove that this lp has a finite representation and can be solved in polynomial time up to any accuracy interestingly we show that our problem cannot be approximated within a factor better than frac e e 1 approx 1 582 hbox unless mathcal p mathcal np ee 1 1 582 unlessp np which is larger than the inapproximability bound of 1 5 for the original problem

an approximation algorithm for the generalized assignment problem the generalized assignment problem can be viewed as the following problem of scheduling parallel machines with costs each job is to be processed by exactly one machine processing jobj on machinei requires timep ij and incurs a cost ofc ij each machinei is available fort i time units and the objective is to minimize the total cost incurred our main result is as follows there is a polynomial time algorithm that given a valuec either proves that no feasible schedule of costc exists or else finds a schedule of cost at mostc where each machinei is used for at most 2t i time units we also extend this result to a variant of the problem where instead of a fixed processing timep ij there is a range of possible processing times for each machine job pair and the cost linearly increases as the processing time decreases we show that these results imply a polynomial time 2 approximation algorithm to minimize a weighted sum of the cost and the makespan i e the maximum job completion time we also consider the objective of minimizing the mean job completion time we show that there is a polynomial time algorithm that given valuesm andt either proves that no schedule of mean job completion timem and makespant exists or else finds a schedule of mean job completion time at mostm and makespan at most 2t

integrable symplectic maps associated with the zs akns spectral problem the darboux transformations dts associated with the well known zs akns spectral problem are regarded as discrete spectral problems two kinds of integrable symplectic maps are derived from them through nonlinearization procedures as an application they are used to calculate the finite genus solutions of some new integrable lattice models constructed from these dts

a finite genus solution of the h1 model the h1 model in the adler bobenko suris list i e the lattice potential kdv equation and the closely related lattice kdv equation with nijhoff s discretization are investigated a new lax pair of the h1 model is given by which integrable symplectic maps are constructed through a non linearization procedure resorting to these maps finite genus solutions of the h1 model as well as the lattice kdv equation are calculated

iterated greedy local search methods for unrelated parallel machine scheduling abstract this work deals with the parallel machine scheduling problem which consists in the assignment of n jobs on m parallel machines the most general variant of this problem is when the processing time depends on the machine to which each job is assigned to this case is known as the unrelated parallel machine problem similarly to most of the literature this paper deals with the minimization of the maximum completion time of the jobs commonly referred to as makespan c max many algorithms and methods have been proposed for this hard combinatorial problem including several highly sophisticated procedures by contrast in this paper we propose a set of simple iterated greedy local search based metaheuristics that produce solutions of very good quality in a very short amount of time extensive computational campaigns show that these solutions are most of the time better than the current state of the art methodologies by a statistically significant margin

a unified approach to scheduling on unrelated parallel machines we develop a single rounding algorithm for scheduling on unrelated parallel machines this algorithm works well with the known linear programming quadratic programming and convex programming relaxations for scheduling to minimize completion time makespan and other well studied objective functions this algorithm leads to the following applications for the general setting of unrelated parallel machines i a bicriteria algorithm for a schedule whose weighted completion time and makespan simultaneously exhibit the current best individual approximations for these criteria ii better than two approximation guarantees for scheduling to minimize the lp norm of the vector of machine loads for all 1

computing correlated equilibria in multi player games we develop polynomial time algorithms for finding correlated equilibria a well studied notion of rationality that generalizes the nash equilibrium in a broad class of succinctly representable multiplayer games encompassing graphical games anonymous games polymatrix games congestion games scheduling games local effect games as well as several generalizations our algorithm is based on a variant of the existence proof due to hart and schmeidler and employs linear programming duality the ellipsoid algorithm markov chain steady state computations as well as application specific methods for computing multivariate expectations over product distributions for anonymous games and graphical games of bounded tree width we provide a different polynomial time algorithm for optimizing an arbitrary linear function over the set of correlated equilibria of the game in contrast to our sweeping positive results for computing an arbitrary correlated equilibrium we prove that optimizing over correlated equilibria is np hard in all of the other classes of games that we consider

almost tight bounds and existence theorems for single commodity confluent flows a flow of a commodity is said to be confluent if at any node all the flow of the commodity leaves along a single edge in this article we study single commodity confluent flow problems where we need to route given node demands to a single destination using a confluent flow single and multi commodity confluent flows arise in a variety of application areas most notably in networking in fact most flows in the internet are multi commodity confluent flows since internet routing is destination based we present near tight approximation algorithms hardness results and existence theorems for minimizing congestion in single commodity confluent flows the maximum edge congestion of a single commodity confluent flow occurs at one of the incoming edges of the destination therefore finding a minimum congestion confluent flow is equivalent to the following problem given a directed graph g with k sinks and non negative demands on all the nodes of g determine a confluent flow that routes every node demand to some sink such that the maximum congestion at a sink is minimized the main result of this article is a polynomial time algorithm for determining a confluent flow with congestion at most 1 p ln k in g if g admits a splittable flow with congestion at most 1 we complement this result in two directions first we present a graph g that admits a splittable flow with congestion at most 1 yet no confluent flow with congestion smaller than hk the kth harmonic number thus establishing tight upper and lower bounds to within an additive constant less than 1 second we show that it is np hard to approximate the congestion of an optimal confluent flow to within a factor of log2k 2 thus resolving the polynomial time approximability to within a multiplicative constant we also consider a demand maximization version of the problem we show that if g admits a splittable flow of congestion at most 1 then a variant of the congestion minimization algorithm yields a confluent flow in g with congestion at most 1 that satisfies 1 3 fraction of total demand we show that the gap between confluent flows and splittable flows is much smaller if the underlying graph is k connected in particular we prove that k connected graphs with k sinks admit confluent flows of congestion less than c p dmax where c is the congestion of the best splittable flow and dmax is the maximum demand of any node in g the proof of this existence theorem is non constructive and relies on topological techniques introduced by lovasz