Batch processing with interval graph compatibilities between tasks

We analyze batch-scheduling problems that arise in connection with certain industrial applications. The models concern processing on a single max-batch machine with the additional feature that the tasks of the same batch have to be compatible. Compatibility is a symmetric binary relation—the compatible pairs are described with an undirected “compatibility graph”, which is often an interval graph according to some natural practical conditions that we present. We consider several models with varying batch capacities, processing times or compatibility graphs. We summarize known results, and present a min-max formula and polynomial time algorithms.     

Minimizing the number of machines for minimum length schedules

In this paper, we present a new objective function for scheduling on parallel machines: minimizing the number of machines for schedules of minimum length. We study its complexity and we prove the NP-completeness of this problem, even if there is no precedences or for unitary execution times. We propose several polynomial algorithms for various particular cases.     

Tight MIP formulations for bounded up/down times and interval-dependent start-ups

Mathematical Programming - Switching machines on and off is an important aspect of unit commitment problems and production planning problems, among others. Here we study tight mixed integer...     

On finding the K best cuts in a network

We show that the O(K · n⁴) algorithm of Hamacher (1982) for finding the K best cut-sets fails because it may produce cuts rather than cut-sets. With the convention that two cuts $\text{(X,}\text{X}\text{)}$ and $\text{(Y,}\text{Y}\text{)}$ are different whenever X ≠ Y the K best cut problem can be solved in O(K · n⁴).     

Structural and algorithmic properties for parametric minimum cuts

We consider the minimum s, t-cut problem in a network with parametrized arc capacities. Following the seminal work of Gallo et?al. (SIAM J. Comput. 18(1):30–55, 1989), classes of this parametric problem have been shown to enjoy the nice Structural Property that minimum cuts are nested, and the nice Algorithmic Property that all minimum cuts can be computed in the same asymptotic time as a single minimum cut by using a clever Flow Update step to move from one value of the parameter to the next. We present a general framework for parametric minimum cuts that extends and unifies such results. We define two conditions on parametrized arc capacities that are necessary and sufficient for (strictly) decreasing differences of the parametric cut function. Known results in parametric submodular optimization then imply the Structural Property. We show how to construct appropriate Flow Updates in linear time under the above conditions, implying that the Algorithmic Property also holds under these conditions. We then consider other classes of parametric minimum cut problems, without decreasing differences, for which we establish the Structural and/or the Algorithmic Property, as well as other cases where nested minimum cuts arise.     

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. Research supported in part by ONR Contracts N00014-90-J-1649 and N00014-95-1-0232, NSF Contracts DDM-9322828, DMI-9732795, DMI-0085683 and DMI-0245352, NUS Academic Research Grant R314-000-046-112, and a research grant from the Natural Sciences and Research Council of Canada (NSERC).     

Sublattices of product spaces: Hulls, representations and counting

The Cartesian product of lattices is a lattice, called a product space, with componentwise meet and join operations. A sublattice of a lattice L is a subset closed for the join and meet operations of L. The sublattice hullLQ of a subset Q of a lattice is the smallest sublattice containing Q. We consider two types of representations of sublattices and sublattice hulls in product spaces: representation by projections and representation with proper boundary epigraphs. We give sufficient conditions, on the dimension of the product space and/or on the sublattice hull of a subset Q, for LQ to be entirely defined by the sublattice hulls of the two-dimensional projections of Q. This extends results of Topkis (1978) and of Veinott [Representation of general and polyhedral subsemilattices and sublattices of product spaces, Linear Algebra Appl. 114/115 (1989) 681-704]. We give similar sufficient conditions for the sublattice hull LQ to be representable using the epigraphs of certain isotone (i.e., nondecreasing) functions defined on the one-dimensional projections of Q. This also extends results of Topkis and Veinott. Using this representation we show that LQ is convex when Q is a convex subset in a vector lattice (Riesz space), and is a polyhedron when Q is a polyhedron in Rⁿ.We consider in greater detail the case of a finite product of finite chains (i.e., totally ordered sets). We use the representation with proper boundary epigraphs and provide upper and lower bounds on the number of sublattices, giving a partial answer to a problem posed by Birkhoff in 1937. These bounds are close to each other in a logarithmic sense. We define a corner representation of isotone functions and use it in conjunction with the representation with proper boundary epigraphs to define an encoding of sublattices. We show that this encoding is optimal (up to a constant factor) in terms of memory space. We also consider the sublattice hull membership problem of deciding whether a given point is in the sublattice hull LQ of a given subset Q. We present a good characterization and a polynomial time algorithm for this sublattice hull membership problem. We construct in polynomial time a data structure for the representation with proper boundary epigraphs, such that sublattice hull membership queries may be answered in time logarithmic in the size |Q| of the given subset.     

On the integer-valued variables in the linear vertex packing problem

Given a graph with weights on vertices, the vertex packing problem consists of finding a vertex packing (i.e. a set of vertices, no two of them being adjacent) of maximum weight. A linear relaxation of one binary programming formulation of this problem has these two well-known properties: (i) every basic solution is (0, 1/2, 1)-valued, (ii) in an optimum linear solution, an integer-valued variable keeps the same value in an optimum binary solution.As an answer to an open problem from Nemhauser and Trotter, it is shown that there is a unique maximal set of variables which are integral in optimal (VLP) solutions.This research was supported by National Research Council of Canada GRANT A8528 and RD 804.