assouad type dimensions and homogeneity of fractals we investigate several aspects of the assouad dimension and the lower dimension which together form a natural dimension pair in particular we compute these dimensions for certain classes of self affine sets and quasiself similar sets and study their relationships with other notions of dimension such as the hausdorff dimension for example we also investigate some basic properties of these dimensions including their behaviour regarding unions and products and their set theoretic complexity

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

approximation techniques for average completion time scheduling we consider the problem of nonpreemptive scheduling to minimize average weighted completion time allowing for release dates parallel machines and precedence constraints recent work has led to constant factor approximations for this problem based on solving a preemptive or linear programming relaxation and then using the solution to get an ordering on the jobs we introduce several new techniques which generalize this basic paradigm we use these ideas to obtain improved approximation algorithms for one machine scheduling to minimize average completion time with release dates in the process we obtain an optimal randomized on line algorithm for the same problem that beats a lower bound for deterministic on line algorithms we consider extensions to the case of parallel machine scheduling and for this we introduce two new ideas first we show that a preemptive one machine relaxation is a powerful tool for designing parallel machine scheduling algorithms that simultaneously produce good approximations and have small running times second we show that a nongreedy rounding of the relaxation yields better approximations than a greedy one we also prove a general theorem relating the value of one machine relaxations to that of the schedules obtained for the original m machine problems this theorem applies even when there are precedence constraints on the jobs we apply this result to obtain improved approximation ratios for precedence graphs such as in trees out trees and series parallel graphs

on the relationship between combinatorial and lp based lower bounds for np hard scheduling problems enumerative approaches to solving optimization problems such as branch and bound require a subroutine that produces a lower bound on the value of the optimal solution in the domain of scheduling problems the requisite lower bound has typically been derived from either the solution to a linear programming lp relaxation of the problem or the solution to a combinatorial relaxation in this paper we investigate from a theoretical perspective the relationship between several lp based lower bounds and combinatorial lower bounds for three scheduling problems in which the goal is to minimize the average weighted completion time of the jobs scheduled we establish a number of facts about the relationship between these different sorts of lower bounds including the equivalence of certain lp based lower bounds for these problems to combinatorial lower bounds used in successful branch and bound algorithms as a result we obtain the first worst case analysis of the quality of the lower bounds delivered by these combinatorial relaxations

a time indexed formulation of non preemptive single machine scheduling problems we consider the formulation of non preemptive single machine scheduling problems using time indexed variables this approach leads to very large models but gives better lower bounds than other mixed integer programming formulations we derive a variety of valid inequalities and show the role of constraint aggregation and the knapsack problem with generalised upper bound constraints as a way of generating such inequalities a cutting plane branch and bound algorithm based on these inequalities has been implemented computational experience on small problems with 20 30 jobs and various constraints and objective functions is presented

on the equivalence of the max min transportation lower bound and the time indexed lower bound for single machine scheduling problems new observations are made about two lower bound schemes for single machine min sum scheduling problems we find that the strongest bound of those provided by transportation problem relaxations can be computed by solving a linear program we show the equivalence of this strongest bound and the bound provided by the lp relaxation of the time indexed integer programming formulation these observations lead to a new lower bound scheme that yields fast approximation of the time indexed bound several techniques are developed to facilitate the effective use of the new lower bound in branch and bound numerical experiments are conducted on 375 benchmark problems of the total weighted tardiness problem from or library results obtained with our new method are spectacular we are able to solve all 125 open problems to optimality

reference points and approximation algorithms in multicriteria discrete optimization abstract mathematical research on multicriteria optimization problems predominantly revolves around the set of pareto optimal solutions in practice on the other hand methods that output a single solution are more widespread reference point methods are a successful example of this approach and are widely used in real world multicriteria optimization a reference point solution is the solution closest to a given reference point in the objective space we study the connection between reference point methods and approximation algorithms for multicriteria optimization problems over discrete sets in particular we establish that in terms of computational complexity computing approximate reference point solutions is polynomially equivalent to approximating the pareto set complementing these results we show for a number of general algorithmic techniques in single criteria optimization how they can be lifted to reference point optimization in particular we lift the link between dynamic programming and fptas as well as certain lp rounding techniques the latter applies e g to set cover and several machine scheduling problems

scheduling to minimize average completion time off line and on line approximation algorithms in this paper we introduce two general techniques for the design and analysis of approximation algorithms for np hard scheduling problems in which the objective is to minimize the weighted sum of the job completion times for a variety of scheduling models these techniques yield the first algorithms that are guaranteed to find schedules that have objective function value within a constant factor of the optimum in the first approach we use an optimal solution to a linear programming relaxation in order to guide a simple list scheduling rule consequently we also obtain results about the strength of the relaxation our second approach yields on line algorithms for these problems in this setting we are scheduling jobs that continually arrive to be processed and for each time t we must construct the schedule until time t without any knowledge of the jobs that will arrive afterwards our on line technique yields constant performance guarantees for a variety of scheduling environments and in some cases essentially matches the performance of our off line lp based algorithms

on the approximability of single machine scheduling with precedence constraints we consider the single machine scheduling problem to minimize the weighted sum of completion times under precedence constraints in a series of recent papers it was established that this scheduling problem is a special case of minimum weighted vertex cover in this paper we show that the vertex cover graph associated with the scheduling problem is exactly the graph of incomparable pairs defined in the dimension theory of partial orders exploiting this relationship allows us to present a framework for obtaining 2 2 f approximation algorithms provided that the set of precedence constraints has fractional dimension of at most f our approach yields the best known approximation ratios for all previously considered special classes of precedence constraints and it provides the first results for bounded degree and orders of interval dimension 2 on the negative side we show that the addressed problem remains np hard even when restricted to the special case of interval orders furthermore we prove that the general problem if a fixed cost present in all feasible schedules is ignored becomes as hard to approximate as vertex cover we conclude by giving the first inapproximability result for this problem showing under a widely believed assumption that it does not admit a polynomial time approximation scheme

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

a heuristic for a scheduling problem with communication delays this paper addresses a scheduling problem with interprocessor communication delays the jobs and the communication delays are of unit length the number of processors is unbounded the aim is to minimize the makespan we develop a new list scheduling heuristic and we prove that its worst case relative performance is 4 3

worst case bound of an lrf schedule for the mean weighted flow time problem this paper studies the problem of scheduling a set of n independent tasks on m identical processors so as to minimize mean weighted flow time the problem is known to be np complete for m geqq 2 and to be np complete in the strong sense for m arbitrary the worst case behavior of a heuristic algorithm which requires time o n log n is investigated and it is shown that the mean weighted flow time obtained by the algorithm does not exceed sqrt 2 1 2 cong 1 0207 times that of an optimal schedule moreover the bound sqrt 2 1 2 is best possible

preemptive and non preemptive generalized min sum set cover in the non preemptive generalized min sum set cover problem we are given n ground elements and a collection of sets mathcal s s 1 s 2 ldots s m where each set s i in 2 n has a positive requirement kappa s i that has to be fulfilled we would like to order all elements to minimize the total weighted cover time of all sets the cover time of a set s i is defined as the first index j in the ordering such that the first j elements in the ordering contain kappa s i elements in s i this problem was introduced by azar et al 2009 with interesting motivations in web page ranking and broadcast scheduling for this problem constant approximations are known by bansal et al 2010 and skutella and williamson oper res lett 39 6 433 436 2011 we study the version where preemption is allowed the difference is that elements can be fractionally scheduled and a set s is covered in the moment when kappa s amount of elements in s are scheduled we give a 2 approximation for this preemptive problem our linear programming relaxation and analysis are completely different from the aforementioned previous works we also show that any preemptive solution can be transformed into a non preemptive one by losing a factor of 6 2 in the objective function as a byproduct we obtain an improved 12 4 approximation for the non preemptive problem

improved approximation algorithms for shop scheduling problems in the job shop scheduling problem there are m machines and n jobs a job consists of a sequence of operations each of which must be processed on a specified machine and the aim is to complete all jobs as quickly as possible this problem is strongly cal np hard even for very restrictive special cases the authors give the first randomized and deterministic polynomial time algorithms that yield polylogarithmic approximations to the optimal length schedule these algorithms also extend to the more general case where a job is given not by a linear ordering of the machines on which it must be processed but by an arbitrary partial order comparable bounds can also be obtained when there are m types of machines a specified number of machines of each type and each operation must be processed on one of the machines of a specified type as well as for the problem of scheduling unrelated parallel machines subject to chain precedence constraints

decentralized control of vehicle formations this paper investigates a method for decentralized stabilization of vehicle formations using techniques from algebraic graph theory the vehicles exchange information according to a pre specified communication digraph g a feedback control is designed using relative information between a vehicle and its in neighbors in g we prove that a necessary and sufficient condition for an appropriate decentralized linear stabilizing feedback to exist is that g has a rooted directed spanning tree we show the direct relationship between the rate of convergence to formation and the eigenvalues of the directed laplacian of g various special situations are discussed including symmetric communication graphs and formations with leaders several numerical simulations are used to illustrate the results

scheduling parallel machines with inclusive processing set restrictions and job release times we consider the problem of scheduling a set of jobs with different release times on parallel machines so as to minimize the makespan of the schedule the machines have the same processing speed but each job is compatible with only a subset of those machines the machines can be linearly ordered such that a higher indexed machine can process all those jobs that a lower indexed machine can process we present an efficient algorithm for this problem with a worst case performance ratio of 2 we also develop a polynomial time approximation scheme ptas for the problem as well as a fully polynomial time approximation scheme fptas for the case in which the number of machines is fixed

impulsive multiconsensus of second order multiagent networks using sampled position data a multiconsensus problem of multiagent networks is solved in this paper where multiconsensus refers to that the states of multiple agents in each subnetwork asymptotically converge to an individual consistent value when there exist information exchanges among subnetworks a distributed impulsive protocol is proposed to achieve multiconsensus of second order multiagent networks in terms of three categories 1 stationary multiconsensus 2 the first dynamic multiconsensus and 3 the second dynamic multiconsensus this impulsive protocol utilizes only sampled position data and is implemented at sampling instants for those three categories of multiconsensus the control parameters in the impulsive protocol are designed respectively moreover necessary and sufficient conditions are derived under which each multiconsensus can be reached asymptotically several simulations are finally provided to demonstrate the effectiveness of the obtained theoretical results

on the distance sets of ahlfors david regular sets abstract i prove that if k r 2 is a compact s ahlfors david regular set with s 1 then dim p d k 1 where d k x y x y k is the distance set of k and dim p stands for packing dimension the same proof strategy applies to other problems of similar nature for instance one can show that if k r 2 is a compact s ahlfors david regular set with s 1 then there exists a point x 0 k such that dim p k k x 0 1

average consensus in networks of dynamic agents with switching topologies and multiple time varying delays in this paper we discuss average consensus problem in undirected networks of dynamic agents with fixed and switching topologies as well as multiple time varying communication delays by employing a linear matrix inequality method we prove that all the nodes in the network achieve average consensus asymptotically for appropriate communication delays if the network topology is connected particularly several feasible linear matrix inequalities are established to determine the maximal allowable upper bound of time varying communication delays numerical examples are given to demonstrate the effectiveness and the sharpness of the theoretical results

convex quadratic and semidefinite programming relaxations in scheduling we consider the problem of scheduling unrelated parallel machines subject to release dates so as to minimize the total weighted completion time of jobs the main contribution of this paper is a provably good convex quadratic programming relaxation of strongly polynomial size for this problem the best previously known approximation algorithms are based on lp relaxations in time or interval indexed variables those lp relaxations however suffer from a huge number of variables as a result of the convex quadratic programming approach we can give a very simple and easy to analyze 2 approximation algorithm which can be further improved to performance guarantee 3 2 in the absence of release dates we also consider preemptive scheduling problems and derive approximation algorithms and results on the power of preemption which improve upon the best previously known results for these settings finally for the special case of two machines we introduce a more sophisticated semidefinite programming relaxation and apply the random hyperplane technique introduced by goemans and williamson for the maxcut problem this leads to an improved 1 2752 approximation

approximation bounds for a general class of precedence constrained parallel machine scheduling problems an important class of scheduling problems concerns parallel machines and precedence constraints we consider precedence delays which associate with each precedence constraint a certain amount of time that must elapse between the completion and start times of the corresponding jobs together with ordinary precedence constraints release dates and delivery times can be modeled in this manner we present a 4 approximation algorithm for the total weighted completion time objective for this general class of problems the algorithm is a rather simple form of list scheduling the list is in order of job midpoints derived from a linear programming relaxation our analysis unifies and simplifies that of a number of special cases heretofore separately studied while actually improving many of the former approximation results

a ptas for minimizing the total weighted completion time on identical parallel machines we consider the problem of scheduling a set of n jobs on m identical parallel machines so as to minimize the weighted sum of job completion times this problem is np hard in the strong sense the best approximation result known so far was a documentclass aastex usepackage amsbsy usepackage amsfonts usepackage amssymb usepackage bm usepackage mathrsfs usepackage pifont usepackage stmaryrd usepackage textcomp usepackage portland xspace usepackage amsmath amsxtra pagestyle empty declaremathsizes 10 9 7 6 begin document 1 over 2 1 sqrt 2 end document approximation algorithm that has been derived by kawaguchi and kyan back in 1986 the contribution of this paper is a polynomial time approximation scheme for this setting which settles a problem that was open for a long time moreover our result constitutes the first known approximation scheme for a strongly np hard scheduling problem with minsum objective

dimension growth for iterated sumsets we study dimensions of sumsets and iterated sumsets and provide natural conditions which guarantee that a set f subseteq mathbb r satisfies overline dim mathrm b f f overline dim mathrm b f or even dim mathrm h n f rightarrow 1 our results apply to for example all uniformly perfect sets which include ahlfors david regular sets our proofs rely on hochman s inverse theorem for entropy and the assouad and lower dimensions play a critical role we give several applications of our results including an erd s volkmann type theorem for semigroups and new lower bounds for the box dimensions of distance sets for sets with small dimension

single machine scheduling polyhedra with precedence constraints we consider nonpreemptive single machine scheduling subject to precedence constraints we define feasible schedules by the vector of the job completion times and study the structure of the convex hull of all feasible schedules called the scheduling polyhedron p we derive classes of valid inequalities for p and necessary and sufficient conditions under which they are facet inducing our main result is a complete description of the minimal linear system defining p when the precedence constraints are series parallel moreover this system consists of two classes of facet inducing inequalities associated with series and parallel compositions respectively if the precedence constraints are not series parallel we present another class of facet inducing inequalities for p associated with induced z subgraphs we also show that the convex hull of all preemptive feasible schedules is identical to p if and only if the precedence constraints form an out forest

structure of a simple scheduling polyhedron in a one machine nonpreemptive scheduling problem the feasible schedules may be defined by the vector of the corresponding job completion times for given positive processing times the associated simple scheduling polyhedronp is the convex hull of these feasible completion time vectors the main result of this paper is a complete description of the minimal linear system definingp we also give a complete combinatorial description of the face lattice ofp and a simple o n logn separation algorithm this algorithm has potential usefulness in cutting plane type algorithms for more difficult scheduling problems

parallel machine scheduling linear programming and parameter list scheduling heuristics in this paper we consider a class of parallel machine scheduling problems and their associated set partitioning formulations we show that the tightness of the linear programming relaxation of these formulations is directly related to the performance of a class of heuristics called parameter list scheduling heuristics this makes it possible to characterize the worst possible gap between optimal solutions for the scheduling problems and the corresponding linear programming relaxations in the case of the classical parallel machine weighted completion time model we also show that the solution to the linear programming relaxation of the set partitioning formulation is asymptotically optimal under mild assumptions on the distribution of job weights and processing times finally we extend most of the results to the time discretized formulation of machine scheduling problems

l 1 group consensus of multi agent systems with stochastic inputs under directed interaction topology we address l 1 group consensus problems of networked multi agent systems with fixed directed information exchange in this article a discrete time protocol is designed to solve the l 1 group consensus problems via general stochastic inputs under some mild assumptions we derive sufficient conditions to guarantee the l 1 group consensus based on algebraic graph theory and ergodicity theory for finite markov chains moreover the boundedness of the expected infinite sum of the inputs ensures the boundedness of the expected trajectory numerical simulations instantiating the inputs as varied independent and identically distributed random processes as well as autoregressive processes are presented to demonstrate the effectiveness of our theoretical results

minimizing average completion time in the presence of release dates a natural and basic problem in scheduling theory is to provide good average quality of service to a stream of jobs that arrive over time in this paper we consider the problem of schedulingn jobs that are released over time in order to minimize the average completion time of the set of jobs in contrast to the problem of minimizing average completion time when all jobs are available at time 0 all the problems that we consider are np hard and essentially nothing was known about constructing good approximations in polynomial time we give the first constant factor approximation algorithms for several variants of the single and parallel machine models many of the algorithms are based on interesting algorithmic and structural relationships between preemptive and nonpreemptive schedules and linear programming relaxations of both many of the algorithms generalize to the minimization of averageweighted completion time as well 1998 the mathematical programming society inc published by elsevier science b v

entropy of convolutions on the circle given ergodic p invariant measures f ig on the 1 torus w e give a sharp condition on their entropies guaranteeing that the entropy of the convolution 1 n converges to logp we also prove a variant of this result for joinings of full entropy on in conjunction with a method of host this yields the following denote aeq x qx mod 1 then for every p invariant ergodic with positive entropy 1 n p ni1 n 0 aecn converges weak to lebesgue measure as n i 1 under a certain mild combinatorial condition onfckg for instance the condition is satisfled if p 10 and ck 2 k 6 k or ck 2 2 k this extends a result of johnson and rudolph who considered the sequence ck q k when p and q are multiplicatively independent we also obtain the following corollary concerning hausdorfi dimension of sum sets for any sequence fsig of p invariant closed subsets of i f p dimh si j log dimh si j 1 then dimh s1 sn i 1

on self similar sets with overlaps and inverse theorems for entropy we study the dimension of self similar sets and measures on the line we show that if the dimension is less than the generic bound of minf1 sg where s is the similarity dimension then there are superexponentially close cylinders at all small enough scales this is a step towards the conjecture that such a dimension drop implies exact overlaps and conrms it when the generating similarities have algebraic coecients as applications we prove furstenberg s conjecture on projections of the one dimensional sierpinski gasket and achieve some progress on the bernoulli convolutions problem and more generally on problems about parametric families of self similar measures the key tool is an inverse theorem on the structure of pairs of probability measures whose mean entropy at scale 2 n has only a small amount of growth under convolution

quickest flows over time flows over time also called dynamic flows generalize standard network flows by introducing an element of time they naturally model problems where travel and transmission are not instantaneous traditionally flows over time are solved in time expanded networks that contain one copy of the original network for each discrete time step while this method makes available the whole algorithmic toolbox developed for static flows its main and often fatal drawback is the enormous size of the time expanded network we present several approaches for coping with this difficulty first inspired by the work of ford and fulkerson on maximal s t flows over time or maximal dynamic s t flows we show that static length bounded flows lead to provably good multicommodity flows over time second we investigate condensed time expanded networks which rely on a rougher discretization of time we prove that a solution of arbitrary precision can be computed in polynomial time through an appropriate discretization leading to a condensed time expanded network of polynomial size in particular our approach yields fully polynomial time approximation schemes for the np hard quickest min cost and multicommodity flow problems for single commodity problems we show that storage of flow at intermediate nodes is unnecessary and our approximation schemes do not use any

a new average case analysis for completion time scheduling we present a new average case analysis for the problem of scheduling n jobs on m machines so that the sum of job completion times is minimized our goal is to use the concept of competitive ratio which is a typical worst case notion also within an average case analysis we show that the classic sept scheduling strategy with n worst case competitive ratio achieves an average of o 1 under several natural distributions among them the exponential distribution our analysis technique allows to also roughly estimate the probability distribution of the competitive ratio thus our result bridges the gap between worst case and average case performance guarantee

consensus seeking in multiagent systems under dynamically changing interaction topologies this note considers the problem of information consensus among multiple agents in the presence of limited and unreliable information exchange with dynamically changing interaction topologies both discrete and continuous update schemes are proposed for information consensus this note shows that information consensus under dynamically changing interaction topologies can be achieved asymptotically if the union of the directed interaction graphs have a spanning tree frequently enough as the system evolves

the asymptotic performance ratio of an on line algorithm for uniform parallel machine scheduling with release dates jobs arriving over time must be non preemptively processed on one of m parallel machines each running at its own speed so as to minimize a weighted sum of the job completion times in this on line environment the processing requirement and weight of a job are not known before the job arrives the weighted shortest processing requirement wspr heuristic is a simple extension of the well known wspt heuristic which is optimal for the single machine problem without release dates according to wspr whenever a machine completes a job the next job assigned to it is the one with the least ratio of processing requirement to weight among all jobs available for processing at this point in time we analyze the performance of this heuristic and prove that its asymptotic competitive ratio is one for all instances with bounded job processing requirements and weights this implies that the wspr algorithm generates a solution whose relative error approaches zero as the number of jobs increases our proof does not require any probabilistic assumption on the job parameters and relies extensively on properties of optimal solutions to a single machine relaxation of the problem

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

scheduling multiple variable speed machines we examine scheduling problems where we control not only the assignment of jobs to machines but also the time used by the job on the machine for instance many tooling machines allow control of the speed at which a job is run increasing the speed incurs costs due to machine wear but also increases throughput we discuss some fundamental scheduling problems in this environment and give algorithms for some interesting cases some cases are inherently difficult so for these we give heuristics our approach illustrates the exploitation of underlying network structure in combinatorial optimization problems we create heuristics that optimally schedule a large portion of the jobs and then attempt to fit in the remainder this also gives a method for quickly finding valid inequalities violated by the linear relaxation solution for the problem of minimizing the sum of makespan and production costs a rounding heuristic is within a constant factor of optimal our heuristics are compared to other classical heu

vertex cover in graphs with locally few colors erdos et al defined the local chromatic number of a graph as the minimum number of colors that must appear within distance 1 of a vertex for any d 2 there are graphs with arbitrarily large chromatic number that can be colored so that i no vertex neighborhood contains more than d different colors bounded local colorability and ii adjacent vertices from two color classes induce a complete bipartite graph biclique coloring we investigate the weighted vertex cover problem in graphs when a locally bounded coloring is given this generalizes the vertex cover problem in bounded degree graphs to a class of graphs with arbitrarily large chromatic number assuming the unique game conjecture ugc we provide a tight characterization we prove that it is ugc hard to improve the approximation ratio of 2 2 d 1 on d 1 locally but not necessarily biclique colorable graphs a matching upper bound is also provided vice versa when properties i and ii hold we present a randomized algorithm with approximation ratio of 2 w 1 lnln dln d this matches known inapproximability results for the special case of bounded degree graphs moreover we show that when both the above two properties i and ii hold the obtained result finds a natural application in a classical scheduling problem namely the precedence constrained single machine scheduling problem to minimize the total weighted completion time denoted as 1 prec w jc j in standard scheduling notation in a series of recent papers it was established that this scheduling problem is a special case of the minimum weighted vertex cover in graphs g p of incomparable pairs defined in the dimension theory of partial orders we show that g p satisfies properties i and ii where d 1 is the maximum number of predecessors or successors of each job

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

a polyhedral approach to single machine scheduling problems we report new results for a time indexed formulation of nonpreemptive single machine scheduling problems we give complete characterizations of all facet inducing inequalities with integral coefficients and right hand side 1 or 2 for the convex hull of the set of feasible partial schedules i e schedules in which not all jobs have to be started furthermore we identify conditions under which these facet inducing inequalities are also facet inducing for the original polytope which is the convex hull of the set of feasible complete schedules i e schedules in which all jobs have to be started to obtain insight in the effectiveness of these classes of facet inducing inequalities we develop a branch and cut algorithm based on them we evaluate its performance on the strongly np hard single machine scheduling problem of minimizing the weighted sum of the job completion times subject to release dates

decompositions network flows and a precedence constrained single machine scheduling problem we present an in depth theoretical algorithmic and computational study of a linear programming lp relaxation to the precedence constrained single machine scheduling problem 1 prec s jw jc jto minimize a weighted sum of job completion times on the theoretical side we study the structure of tight parallel inequalities in thelp relaxation and show thatevery permutation schedule that is consistent with sidney s decomposition has total cost no more than twice the optimum on the algorithmic side we provide a parametric extension to sidney s decomposition and show that a finest decomposition can be obtained by essentially solving a parametric minimum cut problem finally we report results obtained by an algorithm based on these developments on randomly generated instances with up to 2 000 jobs

asymptotic performance ratio of an online algorithm for the single machine scheduling with release dates we analyze the single machine weighted completion time problem with release dates and prove that the asymptotic performance ratio of a simple online algorithm is one this implies that this algorithm generates a solution whose relative error decreases to zero as the number of jobs increases our proof does not require any probabilistic assumption on the job parameters and relies heavily on properties of optimal solutions to a linear programming relaxation of the problem

minimizing mean flow time with release time constraint abstract we consider the problem of preemptively scheduling a set of n independent tasks with release times on m identical processors with the objective of minimizing the mean flow time for one processor baker gives an o n log n time algorithm to find an optimal schedule lawler asks the question whether the problem can be shown to be solvable in polynomial time or shown to be np hard for m 2 in this paper we answer this question by showing that it is np hard for each fixed m 2

stability of multiagent systems with time dependent communication links we study a simple but compelling model of network of agents interacting via time dependent communication links the model finds application in a variety of fields including synchronization swarming and distributed decision making in the model each agent updates his current state based upon the current information received from neighboring agents necessary and or sufficient conditions for the convergence of the individual agents states to a common value are presented thereby extending recent results reported in the literature the stability analysis is based upon a blend of graph theoretic and system theoretic tools with the notion of convexity playing a central role the analysis is integrated within a formal framework of set valued lyapunov theory which may be of independent interest among others it is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence even for the simple models discussed in the present paper

the achievable region approach to the optimal control of stochastic systems summary the achievable region approach seeks solutions to stochastic optimization problems by characterizing the space of all possible performances the achievable region of the system of interest and optimizing the overall system wide performance objective over this space this is radically different from conventional formulations based on dynamic programming the approach is explained with reference to a simple two class queuing system powerful new methodologies due to the authors and co workers are deployed to analyse a general multiclass queuing system with parallel servers and then to develop an approach to optimal load distribution across a network of interconnected stations finally the approach is used for the first time to analyse a class of intensity control problems