A note on the binomial model with simplex constraints

Liu (2000) considered maximum likelihood estimation and Bayesian estimation in a binomial model with simplex constraints using the expectation-maximization (EM) and data augmentation (DA) algorithms. By introducing latent variables {Z_ij} and {Y_ij} (to be defined later), he formulated the constrained parameter problem into a missing data problem. However, the derived DA algorithm does not work because he actually assumed that the {Y_ij} are known. Furthermore, although the final results from the derived EM algorithm are correct, his findings are based on the assumption that the {Y_ij} are observable. This note provides a correct DA algorithm. In addition, we obtained the same E-step and M-step under the assumption that the {Y_ij} are unobservable. A real example is used for illustration.     

Heuristic methods for the identical parallel machine flowtime problem with set-up times

We consider the scheduling of N jobs divided into G families for processing on M identical parallel machines. No set-up is necessary between jobs belonging to the same family. A set-up must be scheduled when switching from the processing of family i jobs to those of another family j, i≠j, the duration of this set-up being the sequence-independent set-up time s_j for family j. We propose heuristics for this problem and computationally evaluate the performance of the heuristics relative to lower bounds and solutions obtained using an exact algorithm.Scope and purposeWe study a machine-scheduling problem within which we have identical parallel machines, jobs arranged into families, and sequence-independent set-up times between jobs of different families on these machines. Our purpose is to develop simple, effective and efficient heuristics for this problem, and we seek to maximise the use of ideas and algorithms that have appeared previously in the literature for related problems. In our computational experiments, we seek to study the behaviour of these heuristics and uncover relevant properties of the scheduling problem. Within this experiment, we compare the observed performance of the heuristics relative to lower bounds and optimal solutions.     

Fitting a least squares piecewise linear continuous curve in two dimensions

An optimal piecewise linear continuous fit to a given set of n data points D = {(x_i, y_i) : 1 ≤ i ≤ n} in two dimensions consists of a continuous curve defined by k linear segments {L₁, L₂,…,L_k} which minimizes a weighted least squares error function with weight w_i at (x_i, y_i), where k ≥ 1 is a given integer. A key difficulty here is the fact that the linear segment L_j, which approximates a subset of consecutive data points D_j ⊂ D in an optimal solution, is not necessarily an optimal fit in itself for the points D_j. We solve the problem for the special case k = 2 by showing that an optimal solution essentially consists of two least squares linear regression lines in which the weight w_j of some data point (x_j, y_j) is split into the weights λw_j and (1 − λ)w_j, 0 ≤ λ ≤ 1, for computations of these lines. This gives an algorithm of worst-case complexity O(n) for finding an optimal solution for the case k = 2.     

A Time-Optimal Multiple Search Algorithm on Enhanced Meshes,with Applications

Given a sorted sequence A = a₁, a₂, ..., a_n of items from a totally ordered universe, along with an arbitrary sequence Q = q₁, q₂, ..., q_m (1 ≤ m ≤ n) of queries, the multiple search problem involves computing for every q_j (1 ≤ j ≤ m) the unique a_i for which a_i−1 ≤ q_j < a_i. In this paper we propose a time-optimal algorithm to solve the multiple search problem on meshes with multiple broadcasting. More specifically, on a [formula] × [formula] mesh with multiple broadcasting, our algorithm runs in [formula] time which is shown to be time-optimal. We also present some surprising applications of the multiple search algorithm to computer graphics, image processing, robotics, and computational geometry.     

Thek-track assignment problem

Thek-track assignment problem is a scheduling problem withn jobs andk machines. Each machinej has a certain operational period (track) which starts at timea _j and ends at timeb _j . Each jobi has a specific start times _i and a specific finish timet _i . A schedule is an assignment of certain jobs to machines such that the intervals [s _i ,t _i [assigned to the same machinej do not overlap and fit into track [a _j ,b _j [. We are interested in a schedule which maximizes the number of assigned jobs. AO(n ^k?1 k!k ^k+1 )-algorithm which solves this problem is presented. Furthermore it is shown that the more general problem, in which for each track only a given set of jobs can be scheduled on that track, can be solved inO(n ^k k!k ^k )-time.     

Performance bound analysis of a heuristic for the total weighted flowtime problem with fixed delivery dates

We consider single-machine scheduling with fixed delivery dates, which are given or determined before the jobs are processed. A job is delivered on the earliest fixed delivery date that is no earlier than its completion time. The flowtime of a job is defined as its delivery date. The objective is to minimize the total weighted flowtime of the jobs. The largest ratio first (LRF) rule is a heuristic that sequences jobs in nonincreasing order of w_j/p_j, where p_j and w_j are the processing time and weight of job J_j, respectively. We investigate the performance bounds of the LRF heuristic under different scenarios of the problem. We conducted computational experiments to test the performance of the heuristic. The results show that the LRF heuristic is able to produce near-optimal and optimal solutions.     

A new Lagrangian relaxation algorithm for hybrid flowshop scheduling to minimize total weighted completion time

We investigate the problem of scheduling n jobs in s-stage hybrid flowshops with parallel identical machines at each stage. The objective is to find a schedule that minimizes the sum of weighted completion times of the jobs. This problem has been proven to be NP-hard. In this paper, an integer programming formulation is constructed for the problem. A new Lagrangian relaxation algorithm is presented in which precedence constraints are relaxed to the objective function by introducing Lagrangian multipliers, unlike the commonly used method of relaxing capacity constraints. In this way the relaxed problem can be decomposed into machine type subproblems, each of which corresponds to a specific stage. A dynamic programming algorithm is designed for solving parallel identical machine subproblems where jobs may have negative weights. The multipliers are then iteratively updated along a subgradient direction. The new algorithm is computationally compared with the commonly used Lagrangian relaxation algorithms which, after capacity constraints are relaxed, decompose the relaxed problem into job level subproblems and solve the subproblems by using the regular and speed-up dynamic programming algorithms, respectively. Numerical results show that the new Lagrangian relaxation method produces better schedules in much shorter computation time, especially for large-scale problems.     

Minimizing the number of tardy jobs under piecewise-linear deterioration

In this paper, we consider a single machine scheduling problem with piecewise-linear deterioration where its objective is to minimize the number of tardy jobs, in which the processing time of each job depends on its starting time where all the jobs have a specific deterioration rate. The problem is known to be NP-hard; therefore a Branch and Bound algorithm and a heuristic algorithm with O(n²) are proposed. The proposed heuristic algorithm has been utilized for solving large scale problems and upper bound of the B&B algorithm. Computational experiments on 1840 problems demonstrate that the Branch and Bound procedure can solve problems with 28 jobs and 85.4% of all the sample problems optimally showing the high capability of the proposed procedure. Also it is shown that the average value of the ratio of optimal answer to the heuristic algorithm result with the objective ∑(1-_Ui)$\sum (1-U_i)$ is at last 1.08 which is more efficient in contrast to other proposed algorithms in related studies in the literature. According to high efficacy of the heuristic algorithm, large scale samples are also being solved and the results are presented. A specific form of this problem is also being considered and it is proven that the B&B procedure can handle problems with more jobs even up to 44 jobs.     

Coupling Genetic Local Search and Recovering Beam Search algorithms for minimizing the total completion time in the single machine scheduling problem subject to release dates

In this paper we consider the well-known single machine scheduling problem with release dates and minimization of the total job completion time. For solving this problem, denoted by 1|r_j|∑C_j, we provide a new metaheuristic which is an extension of the so-called filtered beam search proposed by Ow and Morton [30]. This metaheuristic, referred to as a Genetic Recovering Beam Search (GRBS), takes advantages of a Genetic Local Search (GLS) algorithm and a Recovering Beam Search (RBS) in order to efficiently explore the solution space. In this paper we present the GRBS framework and its application to the 1|r_j|∑C_j problem. Computational results show that it consistently yields optimal or near-optimal solutions and that it provides interesting results by comparison to GLS and RBS algorithms. Moreover, these results highlight that the proposed algorithm outperforms the state-of-the-art heuristics.     

Dominating set is fixed parameter tractable in claw-free graphs

We show that the Dominating Set problem parameterized by solution size is fixed-parameter tractable (FPT) in graphs that do not contain the claw (K_1,3, the complete bipartite graph on four vertices where the two parts have one and three vertices, respectively) as an induced subgraph. We present an algorithm that uses 2^O(k2)n^O(1) time and polynomial space to decide whether a claw-free graph on n vertices has a dominating set of size at most k. Note that this parameterization of Dominating Set is W[2]-hard on the set of all graphs, and thus is unlikely to have an FPT algorithm for graphs in general.The most general class of graphs for which an FPT algorithm was previously known for this parameterization of Dominating Set is the class of K_i,j-free graphs, which exclude, for some fixed i,j∈N, the complete bipartite graph K_i,j as a subgraph. For i,j≥2, the class of claw-free graphs and any class of K_i,j-free graphs are not comparable with respect to set inclusion. We thus extend the range of graphs over which this parameterization of Dominating Set is known to be fixed-parameter tractable.We also show that, in some sense, it is the presence of the claw that makes this parameterization of the Dominating Set problem hard. More precisely, we show that for any t≥4, the Dominating Set problem parameterized by the solution size is W[2]-hard in graphs that exclude the t-claw K_1,t as an induced subgraph. Our arguments also imply that the related Connected Dominating Set and Dominating Clique problems are W[2]-hard in these graph classes.Finally, we show that for any t∈N, the Clique problem parameterized by solution size, which is W[1]-hard on general graphs, is FPT in t-claw-free graphs. Our results add to the small and growing collection of FPT results for graph classes defined by excluded subgraphs, rather than by excluded minors.     

Solving the bi-objective personnel assignment problem using particle swarm optimization

A particle swarm optimization (PSO) algorithm combined with the random-key (RK) encoding scheme (named as PSORK) for solving a bi-objective personnel assignment problem (BOPAP) is presented. The main contribution of this work is to improve the f₁_f₂ heuristic algorithm which was proposed by Huang et al. [3]. The objective of the f₁_f₂ heuristic algorithm is to get a satisfaction level (SL) value which is satisfied to the bi-objective values f₁, and f₂ for the personnel assignment problem. In this paper, PSORK algorithm searches the solution of BOPAP space thoroughly. The experimental results show that the solution quality of BOPAP based on the proposed method is far better than that of the f₁_f₂ heuristic algorithm.     

Hybrid column generation for large-size Covering Integer Programs: Application to transportation planning

The well-known column generation scheme is often an efficient approach for solving the linear relaxation of large-size Covering Integer Programs (CIP). In this paper, this technique is hybridized with an extension of the best-known CIP approximation heuristic, taking advantage of distinct criteria of columns selection. This extension uses fractional optimization for solving pricing subproblems. Numerical results on a real-case transportation planning problem show that the hybrid scheme accelerates the convergence of column generation both in terms of number of iterations and computational time. The integer solutions generated at the end of the process can also be improved for a significant proportion of instances, highlighting the potential of diversification of the approximation heuristic.     

Grade of Service Steiner Minimum Trees in the Euclidean Plane

Abstract. Let P = {p ₁ , p ₂ , \ldots, p _n } be a set of n {\lilsf terminal points} in the Euclidean plane, where point p _i has a {\lilsf service request of grade} g(p _i ) ∈ {1, 2, \ldots, n} . Let 0 < c(1) < c(2) < ?s < c(n) be n real numbers. The {\lilsf Grade of Service Steiner Minimum Tree (GOSST)} problem asks for a minimum cost network interconnecting point set P and some {\lilsf Steiner points} with a service request of grade 0 such that (1) between each pair of terminal points p _i and p _j there is a path whose minimum grade of service is at least as large as \min(g(p _i ), g(p _j )) ; and (2) the cost of the network is minimum among all interconnecting networks satisfying (1), where the cost of an edge with service of grade g is the product of the Euclidean length of the edge with c(g) . The GOSST problem is a generalization of the Euclidean Steiner minimum tree problem where all terminal points have the same grade of service request. When there are only two (three, respectively) different grades of service request by the terminal points, we present a polynomial time approximation algorithm with performance ratio \frac 4 3 ρ (((5+4\sqrt 2 )/7)ρ , respectively), where ρ is the performance ratio achieved by an approximation algorithm for the Euclidean Steiner minimum tree problem. For the general case, we prove that there exists a GOSST that is the minimum cost network under a full Steiner topology or its degeneracies. A powerful interior-point algorithm is used to find a (1+ε) -approximation to the minimum cost network under a given topology or its degeneracies in O(n ^1.5 (log n + log (1/ε))) time. We also prove a lower bound theorem which enables effective pruning in a branch-and-bound method that partially enumerates the full Steiner topologies in search for a GOSST. We then present a k -optimal heuristic algorithm to compute good solutions when the problem size is too large for the branch-and-bound algorithm. Preliminary computational results are presented.     

Optimal algorithms for the average-constrained maximum-sum segment problem

Given a real number sequence A=(a₁,a₂,…,an), an average lower bound L, and an average upper bound U, the Average-Constrained Maximum-Sum Segment problem is to locate a segment A(i,j)=(ai,ai+1,…,aj) that maximizes _∑i?k?jak subject to . In this paper, we give an O(n)-time algorithm for the case where the average upper bound is ineffective, i.e., U=∞. On the other hand, we prove that the time complexity of the problem with an effective average upper bound is Ω(nlogn) even if the average lower bound is ineffective, i.e., L=−∞.     

A fast algorithm for computing a longest common increasing subsequence

Let A=〈a₁,a₂,…,am〉 and B=〈b₁,b₂,…,bn〉 be two sequences, where each pair of elements in the sequences is comparable. A common increasing subsequence of A and B is a subsequence 〈ai₁=bj₁,ai₂=bj₂,…,ail=bjl〉, where i₁<i₂<?<il and j₁<j₂<?<jl, such that for all 1?k<l, we have aik<aik+1. A longest common increasing subsequence of A and B is a common increasing subsequence of the maximum length. This paper presents an algorithm for delivering a longest common increasing subsequence in O(mn) time and O(mn) space.     

Fully symmetric swapped networks based on bipartite cluster connectivity

The class of swapped or OTIS (optical transpose interconnect system) networks, built of n copies of an n-node cluster by connecting node i in cluster j to node j in cluster i for i≠j, has been studied extensively. One problem with such networks is that node i of cluster i has no intercluster link. This slight asymmetry complicates a number of algorithms and hinders both theoretical investigations and practical pursuits, such as building parallel node-disjoint paths for fault tolerance. We introduce biswapped networks that are fully symmetric and have cluster connectivity very similar to swapped/OTIS networks. We derive basic topological parameters, present a simple distributed shortest-path routing algorithm, and point to a number of other interesting properties under investigation for biswapped networks.     

Optimal algorithm for a special point-labeling problem

A special class of map labeling problem is studied. Let P={p₁,p₂,…,p_n} be a set of point sites distributed on a 2D map. A label associated with each point p_i is an axis-parallel rectangle r_i of specified width. The height of all are same. The placement of r_i must contain p_i at its top-left or bottom-left corner, and it does not obscure any other point in P. The objective is to label the maximum number of points on the map so that the placed labels are mutually non-overlapping. We first consider a simple model for this problem. Here, for each point p_i, the corner specification (i.e., whether the point p_i would appear at the top-left or bottom-left corner of the label) is known a priori. We show that the time complexity of this problem is , and then propose an algorithm for this problem which runs in O(nlogn) time. If the corner specifications of the points in P are not known, our algorithm is a 2-approximation algorithm. Here we propose an efficient heuristic algorithm that is easy to implement. Experimental evidences show that it produces optimal solutions for most of the randomly generated instances and for all the standard benchmarks available in http://www.math-inf.uni-greifswald.de/map-labeling/.     

Numerical solution for the one-phase Stefan problem by Piecewise constant approximation of the interface

The classical one-phase one-dimensional Stefan problem is numerically solved on rectangles,R _j , of increasing size controlled by the Stefan condition. This approach is based on a scheme introduced by E. Di Benedetto and R. Spigler in 1983. The practical implementation rests on the representation viathermal potentials of the solutionu ^j (x, t) to the heat equation inR _j . The quantityu _x ^j (x _j ,jΔt) which determines the (j+1)-th rectangle is evaluatedanalytically by solving explicitly an integral equation. The solution inR _j+1 is then obtained bynumerically evaluating a further integral expression. The algorithm is tested by solving two problems whose solution is explicitly known. Convergence, stability and convergence rate as Δx→0, Δt→0 have been tested and plots are shown.     

Embedded local search approaches for routing optimization

This paper presents a new class of heuristics which embed an exact algorithm within the framework of a local search heuristic. This approach was inspired by related heuristics which we developed for a practical problem arising in electronics manufacture. The basic idea of this heuristic is to break the original problem into small subproblems having similar properties to the original problem. These subproblems are then solved using time intensive heuristic approaches or exact algorithms and the solution is re-embedded into the original problem. The electronics manufacturing problem where we originally used the embedded local search approach, contains the Travelling Salesman Problem (TSP) as a major subproblem. In this paper we further develop our embedded search heuristic, HyperOpt, and investigate its performance for the TSP in comparison to other local search based approaches. We introduce an interesting hybrid of HyperOpt and 3-opt for asymmetric TSPs which proves more efficient than HyperOpt or 3-opt alone. Since pure local search seldom yields solutions of high quality we also investigate the performance of the approaches in an iterated local search framework. We examine iterated approaches of Large-Step Markov Chain and Variable Neighbourhood Search type and investigate their performance when used in combination with HyperOpt. We report extensive computational results to investigate the performance of our heuristic approaches for asymmetric and Euclidean Travelling Salesman Problems. While for the symmetric TSP our approaches yield solutions of comparable quality to 2-opt heuristic, the hybrid methods proposed for asymmetric problems seem capable of compensating for the time intensive embedded heuristic by finding tours of better average quality than iterated 3-opt in many less iterations and providing the best heuristic solutions known, for some instance classes.