Analysis of a linear programming heuristic for scheduling unrelated parallel machines

AbstractEach of n jobs is to be processed without interruption on one of m unrelated parallel machines. The objectives is to minimize the maximum completion time. A heuristic method is presented, the first stage of which uses linear programming to form a partial schedule leaving at most m−1 jobs unscheduled: the second stage schedules these m−1 jobs using an enumerative method. For m≥3, it is shown that the heuristic has a (best possible) worst-case performance ratio of 2 and has a computational requirement which is polynomial in n although it is exponential in m. For m = 2, it is shown that the heuristic has a (best possible) worst-case performance ratio of 1 +5)2 and requires linear time. A modified version of the heuristic is presented for m = 2 which is shown to have a (best possible) worst-case performance ratio of 32 while still requiring linear time

Scheduling identical parallel machines to minimize total weighted completion time

AbstractA branch and bound algorithm is proposed for the problem of scheduling jobs on identical parallel machines to minimize the total weighted completion time. Based upon a formulation which partitions the period of processing into unit time intervals, the lower bounding scheme is derived by performing a Lagrangean relaxation of the machine capacity constraints. A special feature is that the multipliers are obtained by a simple heuristic method which allows each lower bound to be computed in polynomial time. This bounding scheme, along with a new dominance rule, is incorporated into a branch and bound algorithm. Computational experience indicates that it is superior to known algorithms

A survey of algorithms for the single machine total weighted tardiness scheduling problem

AbstractThis paper surveys algorithms for the problem of scheduling jobs on a single machine to minimize total weighted tardiness. Special attention is given to two dynamic programming and four branch and bound algorithms. The dynamic programming algorithms both use the same recursion defined on sets of jobs, but they generate the sets in lexicographic order and cardinality order respectively. Two of the branch and bound algorithms use the quickly computed but possibly rather weak lower bounds obtained from linear and exponential functions of completion times problems. These algorithms rely heavily on dominance rules to restrict the search. The other two branch and bound algorithms use lower bounds obtained from the Lagrangean relaxation of machine capacity constraints and from dynamic programming state-space relaxation. They invest a substantial amount of computation time at each node of the search tree in an attempt to generate tight lower bounds and thereby generate only small search trees. A computational comparison of all these algorithms on problems with up to 50 jobs is given

Potts model on recursive lattices: some new exact results

We compute the partition function of the Potts model with arbitrary values of $q$ and temperature on some strip lattices. We consider strips of width $L_y=2$, for three different lattices: square, diced and `shortest-path' (to be defined in the text). We also get the exact solution for strips of the Kagome lattice for widths $L_y=2,3,4,5$. As further examples we consider two lattices with different type of regular symmetry: a strip with alternating layers of width $L_y=3$ and $L_y=m+2$, and a strip with variable width. Finally we make some remarks on the Fisher zeros for the Kagome lattice and their large q-limit.Comment: 17 pages, 19 figures. v2 typos corrected, title changed and references, acknowledgements and two further original examples added. v3 one further example added. v4 final versio

Strong LP formulations for scheduling splittable jobs on unrelated machines

International audienceA 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+ϕ1+ϕ , where ϕϕ 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 ee−1≈1.582(unless =)ee−1≈1.582(unless P=NP) , which is larger than the inapproximability bound of 1.5 for the original problem

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial

We derive some new structural results for the transfer matrix of square-lattice Potts models with free and cylindrical boundary conditions. In particular, we obtain explicit closed-form expressions for the dominant (at large |q|) diagonal entry in the transfer matrix, for arbitrary widths m, as the solution of a special one-dimensional polymer model. We also obtain the large-q expansion of the bulk and surface (resp. corner) free energies for the zero-temperature antiferromagnet (= chromatic polynomial) through order q^{-47} (resp. q^{-46}). Finally, we compute chromatic roots for strips of widths 9 <= m <= 12 with free boundary conditions and locate roughly the limiting curves.Comment: 111 pages (LaTeX2e). Includes tex file, three sty files, and 19 Postscript figures. Also included are Mathematica files data_CYL.m and data_FREE.m. Many changes from version 1: new material on series expansions and their analysis, and several proofs of previously conjectured results. Final version to be published in J. Stat. Phy

Transfer matrices and partition-function zeros for antiferromagnetic Potts models. VI. Square lattice with special boundary conditions

We study, using transfer-matrix methods, the partition-function zeros of the square-lattice q-state Potts antiferromagnet at zero temperature (= square-lattice chromatic polynomial) for the special boundary conditions that are obtained from an m x n grid with free boundary conditions by adjoining one new vertex adjacent to all the sites in the leftmost column and a second new vertex adjacent to all the sites in the rightmost column. We provide numerical evidence that the partition-function zeros are becoming dense everywhere in the complex q-plane outside the limiting curve B_\infty(sq) for this model with ordinary (e.g. free or cylindrical) boundary conditions. Despite this, the infinite-volume free energy is perfectly analytic in this region.Comment: 114 pages (LaTeX2e). Includes tex file, three sty files, and 23 Postscript figures. Also included are Mathematica files data_Eq.m, data_Neq.m,and data_Diff.m. Many changes from version 1, including several proofs of previously conjectured results. Final version to be published in J. Stat. Phy