Time-indexed linear programming formulations have recently received a great deal of attention for their practical effectiveness in solving a number of single-machine scheduling problems. We show that these formulations are also an important tool in the design of approximation algorithms with good worst-case performance guarantees. We give simple new rounding techniques to convert an optimal fractional solution into a feasible schedule for which we can prove a constant-factor performance guarantee, thereby giving the first theoretical evidence of the strength of these relaxations. Specifically, we consider the problem of minimizing the total weighted job completion time on a single machine subject to precedence constraints, and give a polynomialtime (4 + ffl)-approximation algorithm, for any ffl ? 0; the best previously known guarantee for this problem was superlogarithmic. With somewhat larger constants, we also show how to extend this result to the case with release date constraints, ...

. This paper describes an algorithm designed to minimally recongure schedules in response to a changing environment. External factors have caused an existing schedule to become invalid, perhaps due to the withdrawal of resources, or because of changes to the set of scheduled activities. The total shift in the start and end times of already scheduled activities should be kept to a minimum. This optimization requirement may be captured using a linear optimization function over linear constraints. However, the disjunctive nature of the resource constraints impairs traditional mathematical programming approaches. The unimodular probing algorithm interleaves constraint programming and linear programming. The linear programming solver handles only a controlled subset of the problem constraints, to guarantee that the values returned are discrete. Using probe backtracking, a complete, repair-based method for search, these values are simply integrated into constraint programming. Unimodular p...

We provide a review and synthesis of polyhedral approaches to machine scheduling problems. The choice of decision variables is the prime determinant of various formulations for such problems. Constraints, such as facet inducing inequalities for corresponding polyhedra, are often needed, in addition to those just required for the validity of the initial formulation, in order to obtain useful lower bounds and structural insights. We review formulations based on time–indexed variables; on linear ordering, start time and completion time variables; on assignment and positional date variables; and on traveling salesman variables. We point out relationship between various models, and provide a number of new results, as well as simplified new proofs of known results. In particular, we emphasize the important role that supermodular polyhedra and greedy algorithms play in many formulations and we analyze the strength of the lower and upper bounds obtained from different formulations and relaxations. We discuss separation algorithms for several classes of inequalities, and their potential applicability in generating cutting planes for the practical solution of such scheduling problems. We also review some recent results on approximation algorithms based on some of these formulations.

A well studied and difficult class of scheduling problems concerns parallel machines and precedence constraints. In order to model more realistic situations, we consider precedence delays, associating with each precedence constraint a certain amount of time which must elapse between the completion and start times of the corresponding jobs. Release dates, among others, may be modeled in this fashion. We provide the first constant-factor approximation algorithms for the makespan and the total weighted completion time objectives in this general class of problems. These algorithms are rather simple and practical forms of list scheduling. Our analysis also unifies and simplifies that of a number of special cases heretofore separately studied, while actually improving some of the former approximation results.

We consider the problem of scheduling a set of jobs on a single machine with the objective of minimizing sum of weighted completion times. The problem is NP-hard when there are precedence constraints between jobs [15]. We provide an efficient combinatorial 2-approximation algorithm for this problem. In contrast to our work, earlier approximation algorithms [12] achieving constant factor approximations are based on solving a linear programming relaxation of the problem. We also show that the linear ordering relaxation of Potts [20] has an integrality gap of 2. 1 Introduction We consider the following scheduling problem. We are given a set of jobs J 1 ; J 2 ; : : : ; J n where each job J i has a processing time p i and a weight w i . Jobs have precedence constraints between them that are specified in the form of a directed acyclic graph. If i OE j, J j cannot be scheduled before J i completes. The objective is to find a non-preemptive schedule of the jobs on a single machine (or equiva...

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 | sum(w_jC_j) to minimize a weighted sum of job completion times. On the theoretical side, we study the structure of tight parallel inequalities in the LP relaxation and show that every permutation schedule which 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.

In this note we give an alternate proof that a scheduling algorithm of Lawler [E.L.

We consider the problem of nonpreemptively scheduling a set of jobs with identical processing requirements (unit jobs) on parallel machines with nonstationary speeds. In addition to the case of uniform machines, this allows for such predictable effects as operator learning and tool wear and tear, as well as such planned activities as machine upgrades, maintenance and the preassignment of other operations, all of which may affect the available processing speed of the machine at different points in time. We also allow release dates that satisfy a certain compatibility property. We show that the convex hull of feasible completion time vectors is a supermodular polyhedron. For nonidentical but compatible release dates, the supermodular function defining this polyhedron is the Dilworth truncation of a (non supermodular) function defined in a natural way from the release dates. This supermodularity result implies that the total weighted flow time can be minimized by a greedy algor...

Given a poset P as a precedence relation on a set of jobs with processing time vector p, the generalized permutahedron perm(P; p) of P is defined as the convex hull of all job completion time vectors corresponding to a linear extension of P . Thus, the generalized permutahedron allows for the single machine weighted flowtime scheduling problem to be formulated as a linear programming problem over perm(P; p). Queyranne and Wang [8] as well as von Arnim and Schrader [2] gave a collection of valid inequalities for this polytope. Here we present a description of its geometric structure that depends on the series decomposition of the poset P , prove a dimension formula for perm(P; p), and characterize the facet inducing inequalities under the known classes of valid inequalities.

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