The bottleneck k-MST

The k-MST is a well known NP-hard problem and several approximation algorithms exist to solve this problem with a guaranteed performance bound. A closely related problem, called the bottleneck k-MST (BMST(k)) can however be solved in O(mlogn) time on graph with n nodes and m edges. We propose two algorithms to solve BMST(k), one of complexity O(m+nlogn) and the other of O(m) time. We also consider a generalization of BMST(k) which subsumes many bottleneck problems studied in the literature and show that this generalized problem can also be solved in O(m) time.     

The optimum assignments and a new heuristic approach for the traveling salesman problem

In this paper the equivalent problem approach to the n × n optimum assignment problem is exploited for providing a heuristic solution to the traveling salesman problem. It is shown that the simplified problem of finding an optimum tour using the n ? k arcs of the cycles of an n × n optimum assignment and k other arcs is NP-hard. Next by using the “nearest neighbor rule” for zero cost cycles, an O(n²) algorithm is presented for obtaining a suboptimal solution to the simplified problem. Using the notion of a path switching operation that always results in a new tour having a lower cost than the original tour, an algorithm for a systematic refinement of the suboptimal tour is given. The algorithm presented in this paper efficiently solves the problem for k = 2,3 for any n. Examples illustrating the algorithm are given, and the time complexities as well as error bounds have been studied. Further work needed in the area is indicated.     

A Parallel Algorithm for Traffic Control Problems in Three-Stage Connecting Networks

A parallel heuristic algorithm for traffic control problems in three-stage connecting networks is presented in this paper. A three-stage connecting network consists of an input crossbar switching stage, an intermediate crossbar switching stage, and an output crossbar switching stage. The goal of our algorithm is to quickly and efficiently find a conflict-free switching assignment for communication demands through the network. The algorithm requires n² × m processing elements for the network composed of n input/output switches and m intermediate switches, where it runs not only on a sequential machine, but also on a parallel machine with maximally n² × m processors. The algorithm was verified by 1100 simulation runs with the network size from 10² × 7 to 50² × 27. The simulation results show that the algorithm can find a solution in nearly constant time with n² × m processors.     

The two-stage assembly scheduling problem to minimize total completion time with setup times

We address the two-stage assembly scheduling problem where there are m machines at the first stage and an assembly machine at the second stage. The objective is to schedule the available n jobs so that total completion time of all n jobs is minimized. Setup times are treated as separate from processing times. This problem is NP-hard, and therefore we present a dominance relation and propose three heuristics. The heuristics are evaluated based on randomly generated data. One of the proposed heuristics is known to be the best heuristic for the case of zero setup times while another heuristic is known to perform well for such problems. A new version of the latter heuristic, which utilizes the dominance relation, is proposed and shown to perform much better than the other two heuristics.     

A comparative runtime analysis of heuristic algorithms for satisfiability problems

The satisfiability problem is a basic core NP-complete problem. In recent years, a lot of heuristic algorithms have been developed to solve this problem, and many experiments have evaluated and compared the performance of different heuristic algorithms. However, rigorous theoretical analysis and comparison are rare. This paper analyzes and compares the expected runtime of three basic heuristic algorithms: RandomWalk, (1+1) EA, and hybrid algorithm. The runtime analysis of these heuristic algorithms on two 2-SAT instances shows that the expected runtime of these heuristic algorithms can be exponential time or polynomial time. Furthermore, these heuristic algorithms have their own advantages and disadvantages in solving different SAT instances. It also demonstrates that the expected runtime upper bound of RandomWalk on arbitrary k-SAT (k?3) is O(n(k−1)), and presents a k-SAT instance that has Θ(n(k−1)) expected runtime bound.     

A variable neighborhood search approach for planning and scheduling of jobs on unrelated parallel machines

In this paper, we study a planning and scheduling problem for unrelated parallel machines. There are n jobs that have to be assigned and sequenced on m unrelated parallel machines. Each job has a weight that represents the priority of the corresponding customer order, a given due date, and a release date. An Automated Guided Vehicle is used to transport at maximum Load _max jobs into a storage space in front of the machines in a given period of time. We consider t _max consecutive periods. We are interested in minimizing the total weighted tardiness of the jobs across the periods. This measure is important when we are interested in a good on-time delivery performance. We present an appropriate mixed integer program. To solve this NP-hard problem, we develop a heuristic methodology based on decomposition and variable neighborhood search (VNS). The proposed approaches are assessed using randomly generated problem instances. We compare them with a simple heuristic based on decomposition and list scheduling using the Apparent Tardiness Cost dispatching rule. The results demonstrate that the heuristic approach based on VNS performs comparably to the mixed integer program while having reasonable solution times and outperforms the simple heuristic and a genetic algorithm (GA) from previous research.     

An analog computer method to solve Fredholm integral equations of the first kind

Many problems of contemporary interest are characterized by Fredholm type integral equations of the first kind. These equations are inherently ill-posed and difficult to solve. It is customary to convert the equation into a set of m algebraic equations Af = g in n unknowns with m not necessarily equal to n. Then one can solve these m equations in a least square sense. Among the class of vectors f that minimize the Euclidean norm of the error, there exists a unique vector A⁺g which is of least norm where A⁺ is the generalized inverse of A. One method of finding the generalized inverse of A is to reformulate the problem into an equivalent system of first order ordinary differential equations with specified initial conditions. The steady state solution of this system is A⁺g, the required value of f. This procedure was implemented on an analog computer and the results presented.     

Scheduling nonpreemptive jobs on parallel machines subject to exponential unrecoverable interruptions

In this paper we consider the problem of scheduling n independent jobs on m parallel machines. If, while a machine is processing a job, a failure (unrecoverable interruption) occurs, the current job as well as subsequently scheduled jobs on that machine cannot be performed, and hence do not contribute to the overall revenue or throughput. The objective is to maximize the expected amount of work done before an interruption occurs. In this paper, we investigate the problem when failures are exponentially distributed. We show that the problem is NP-hard, and characterize a polynomially solvable special case. We then propose both an exact algorithm having pseudopolynomial complexity and a heuristic algorithm. A combinatorial upper bound is also proposed for the problem. Experimental results show the effectiveness of the heuristic approach.     

Unrelated parallel-machine scheduling with rate-modifying activities to minimize the total completion time

Zhao et al. (2009) [24] study the m identical parallel-machine scheduling problem with rate-modifying activities to minimize the total completion time. They show that the problem can be solved in O(n^2m+3) time. In this study we extend the scheduling environment to the unrelated parallel-machine setting and present a more efficient algorithm to solve the extended problem. For the cases where the rate-modifying rate is (i) larger than 0 and not larger than 1, and (ii) larger than 0, we show that the problem can be solved in O(n^m+3) and O(n^2m+2) time, respectively.     

Completion time variance minimization in single machine and multi-machine systems

In this paper we consider the problem of minimizing the completion time variance of n jobs on a single machine with deterministic processing times. We propose a new heuristic and compare the results with existing popular heuristics for the problem. We also propose a method based on genetic algorithms to solve the problem. We present the worst case performance analysis of the proposed heuristic. We also consider the problem of minimizing the completion time variance of n jobs on m identical parallel machines in both restricted and unrestricted versions. A heuristic method and a method based on genetic algorithms are presented for both the cases and results of computational testing are provided. It is concluded that the proposed methods provide better results compared to existing methods for the single machine case as well as for the multi-machine case.     

Time-slot assignment for TDMA-systems

In satellite communication as in other technical systems using the TDMA-technique (time division multiple access) the problem arises to decompose a given (n×n)-matrix in a weighted sum of permutation matrices such that the sum of the weights becomes minimal. We show at first that an optimal solution of this problem can be obtained inO(n ⁴)-time using at mostO(n ²) different permutation matrices. Thereafter we discuss shortly the decomposition inO(n) different matrices which turns out to be NP-hard. Moreover it is shown that an optimal decomposition using a class of 2n permutation matrices which are fixed in advance can be obtained by solving a classical assignment problem. This latter problem can be generalized by taking arbitrary Boolean matrices. The corresponding decomposition problem can be transformed to a special max flow min cost network flow problem, and is thus soluble by a genuinely polynomial algorithm.     

Clustering Large Graphs via the Singular Value Decomposition

We consider the problem of partitioning a set of m points in the n-dimensional Euclidean space into k clusters (usually m and n are variable, while k is fixed), so as to minimize the sum of squared distances between each point and its cluster center. This formulation is usually the objective of the k-means clustering algorithm (Kanungo et al. (2000)). We prove that this problem in NP-hard even for k = 2, and we consider a continuous relaxation of this discrete problem: find the k-dimensional subspace V that minimizes the sum of squared distances to V of the m points. This relaxation can be solved by computing the Singular Value Decomposition (SVD) of the m × n matrix A that represents the m points; this solution can be used to get a 2-approximation algorithm for the original problem. We then argue that in fact the relaxation provides a generalized clustering which is useful in its own right. Finally, we show that the SVD of a random submatrix—chosen according to a suitable probability distribution—of a given matrix provides an approximation to the SVD of the whole matrix, thus yielding a very fast randomized algorithm. We expect this algorithm to be the main contribution of this paper, since it can be applied to problems of very large size which typically arise in modern applications.     

An improved NEH heuristic to minimize makespan in permutation flow shops

For over 20 years the NEH heuristic of Nawaz, Enscore, and Ham [A heuristic algorithm for the m-machine, n-job flow-shop sequencing problem. Omega, The International Journal of Management Science 1983;11:91–5] has been commonly regarded as the best heuristic for solving the NP-hard problem of minimizing the makespan in permutation flow shops. The strength of NEH lies mainly in its priority order according to which jobs are selected to be scheduled during the insertion phase. Framinan et al. [Different initial sequences for the heuristic of Nawaz, Enscore and Ham to minimize makespan, idle time or flowtime in the static permutation flowshop problem. International Journal of Production Research 2003;41:121–48] presented the results of an extensive study to conclude that the NEH priority order is superior to 136 different orders examined. Based upon the concept of Johnson's algorithm, we propose a new priority order combined with a simple tie-breaking method that leads to a heuristic that outperforms NEH for all problem sizes.     

On Some Proximity Problems of Colored Sets

The maximum diameter color-spanning set problem（MaxDCS） is defined as follows： given n points with m colors, select m points with m distinct colors such that the diameter of the set of chosen points is maximized. In this paper, we design an optimal O（n log n） time algorithm using rotating calipers for MaxDCS in the plane. Our algorithm can also be used to solve the maximum diameter problem of imprecise points modeled as polygons. We also give an optimal algorithm for the all farthest foreign neighbor problem（AFFN） in the plane, and propose algorithms to answer the farthest foreign neighbor query（FFNQ） of colored sets in two- and three-dimensional space. Furthermore, we study the problem of computing the closest pair of color-spanning set（CPCS） in d-dimensional space, and remove the log m factor in the best known time bound if d is a constant.     

A new set of high-performing heuristics to minimise flowtime in permutation flowshops

This paper addresses the problem of scheduling jobs in a permutation flowshop with the objective of total completion time minimisation. Since this problem is known to be NP-hard, most research has focussed on obtaining procedures – heuristics – able to provide good, but not necessarily optimal, solutions with a reasonable computational effort. Therefore, a full set of heuristics efficiently balancing both aspects (quality of solutions and computational effort) has been developed. 12 out of these 14 efficient procedures are composite heuristics based on the LR heuristic by Liu and Reeves (2001), which is of complexity $n^{3}m$. In our paper, we propose a new heuristic of complexity $n^{2}m$ for the problem, which turns out to produce better results than LR. Furthermore, by replacing the heuristic LR by our proposal in the aforementioned composite heuristics, we obtain a new set of 17 efficient heuristics for the problem, with 15 of them incorporating our proposal. Additionally, we also discuss some issues related to the evaluation of efficient heuristics for the problem and propose an alternative indicator.     

On the K best integer network flows

We address the problem of finding the K best integer solutions of a linear integer network flow problem. We design an O(f(n,m,L,U)+KmS(n,m,L)) time and O(K+m) memory space algorithm to determine the K best integer solutions, in a directed network with n nodes, m arcs, maximum absolute value cost L, and an upper bound U on arc capacities and node supplies. f(n,m,L,U) is the best time needed to solve the minimum cost flow problem in a directed network and S(n,m,L) is the best time to solve the single-source shortest path problem in a network with non-negative lengths. The introduced algorithm efficiently determines a “proper minimal cycle” by taking advantage of the relationship between the best solutions. This way, we improve the theoretical as well as practical memory space bounds of the well-known method due to Hamacher. Our computational experiments confirm this result.     

On heuristics for minimum length rectilinear partitions

The problem of partitioning a rectilinear figure into rectangles with minimum length is NP-hard and has bounded heuristics. In this paper we study a related problem,Elimination Problem (EP), in which a rectilinear figure is partitioned into a set of rectilinear figures containing no concave vertices of a fixed direction with minimum length. We show that a heuristic for EP within a factor of 4 from optimal can be computed in timeO(n ²), wheren is the number of vertices of the input figure, and a variant of this heuristic, within a factor of 6 from optimal, can be computed in timeO(n logn). As an application, we give a bounded heuristic for the problem of partitioning a rectilinear figure into histograms of a fixed direction with minimum length. An auxiliary result is that an optimal rectangular partition of a monotonic histogram can be computed in timeO(n ²), using a known speed-up technique in dynamic programming.     

Branch-and-bound method for minimizing the weighted completion time scheduling problem on a single machine with release dates

In this paper, we consider a single-machine scheduling problem with release dates. The aim is to minimize the total weighted completion time. This problem is known to be strongly NP-hard. We propose two new lower bounds that can be, respectively, computed in O(n²) and in O(nlogn) time where n is the number of jobs. We prove a sufficient and necessary condition for local optimality, which can also be considered as a priority rule. We present an efficient heuristic using such a condition. We also propose some dominance properties. A branch-and-bound algorithm incorporating the heuristic, the lower bounds and the dominance properties is proposed and tested on a large set of instances.     

Faster Algorithms for Stable Allocation Problems

We consider a high-multiplicity generalization of the classical stable matching problem known as the stable allocation problem, introduced by Baïou and Balinski in 2002. By leveraging new structural properties and sophisticated data structures, we show how to solve this problem in O(mlog?n) time on a bipartite instance with n vertices and m edges, improving the best known running time of O(mn). Building on this algorithm, we provide an algorithm for the non-bipartite stable allocation problem running in O(mlog?n) time with high probability. Finally, we give a polynomial-time algorithm for solving the “optimal” variant of the bipartite stable allocation problem, as well as a 2-approximation algorithm for the NP-hard “optimal” variant of the non-bipartite stable allocation problem.     

Minimizing the number of tardy jobs on a proportionate flowshop with general position-dependent processing times

In various real life scheduling systems job processing times vary according to the number of jobs previously processed. The vast majority of studies assume a restrictive functional form to describe job processing times. In this note, we address a scheduling problem with the most general job processing time functions. The machine setting assumed is an m-machine proportionate flowshop, and the objective function is minimum number of tardy jobs. We show that the problem can be formulated as a bottleneck assignment problem with a maximum cardinality constraint. An efficient polynomial time (O(n⁴ log n)) solution is introduced.