An approximation algorithm for the m-machine permutation flow shop scheduling problem with controllable processing times

The paper deals with the m-machine permutation flow shop scheduling problem in which job processing times, along with a processing order, are decision variables. It is assumed that the cost of processing a job on each machine is a linear function of its processing time and the overall schedule cost to be minimized is the total processing cost plus maximum completion time cost. A algorithm for the problem with m = 2 is provided; the best approximation algorithm until now has a worst-case performance ratio equal to . An extension to the m-machine (m ≥2) permutation flow shop problem yields an approximation algorithm with a worst-case bound equal to

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, where is the worst-case performance ratio of a procedure used, in the proposed algorithm, for solving the (pure) sequencing problem. Moreover, examples which achieve this bound for = 1 are also presented.     

A polynomial algorithm with approximation ratio 7/9 for the maximum two peripatetic salesmen problem

Under study is the maximum 2 peripatetic salesmen problem which consists in finding two edge-disjoint Hamiltonian cycles with maximal total weight in a complete undirected graph. We present a cubic time approximation algorithm for this problem with the performance guarantee 7/9 which is the best known estimation by now. We also present a modification of this algorithm for the case when the weights of graph edges are values in a given range [1, q] with the performance guarantee (7q + 3)/(9q + 1) which is also the best known estimation by now and has cubic time complexity too.     

A Globally Convergent Algorithm for a PDE-Constrained Optimization Problem Arising in Electrical Impedance Tomography

We study the convergence properties of an algorithm for the inverse problem of electrical impedance tomography, which can be reduced to a partial differential equation (PDE) constrained optimization problem. The direct problem consists of the potential equation div(??u) = 0 in a circle, with Neumann condition describing the behavior of the electrostatic potential in a medium with conductivity given by the function ?(x, y). We suppose that at each time a current ψ _i is applied to the boundary of the circle (Neumann's data), and that it is possible to measure the corresponding potential ? _i (Dirichlet data). The inverse problem is to find ?(x, y) given a finite number of Cauchy pairs measurements (? _i , ψ _i ), i = 1,…, N. The problem is formulated as a least squares problem, and the developed algorithm solves the continuous problem using descent iterations in its corresponding finite element approximations. Wolfe's conditions are used to ensure the global convergence of the optimization algorithm for the continuous problem. Although exact data are assumed, measurement errors in data and regularization methods shall be considered in a future work.     

Efficient and accurate spectral method for the time-fractional dual-phase-lag heat transfer model and its parameter estimation

In the current paper, a heat transfer model is suggested based on a time-fractional dual-phase-lag (DPL) model. We discuss the model in two parts, the direct problem and the inverse problem. Firstly, for solving it, the finite difference/Legendre spectral method is constructed. In the temporal direction, we employ the weighted and shifted Grünwald approximation, which can achieve second order convergence. In the spatial direction, the Legendre spectral method is used, it can obtain spectral accuracy. The stability and convergence are theoretically analyzed. For the inverse problem, the Bayesian method is used to construct an algorithm to estimate the four parameters for the model, namely, the time-fractional order α, the time-fractional order β, the delay time τ_T, and the relaxation time τ_q. Next, numerical experiments are provided to demonstrate the effectiveness of our scheme, with the values of τ_q and τ_T for processed meat employed. We also make a comparison with another method. The data obtained for the direct problem are used in the parameter estimation. The paper provides an accurate and efficient numerical method for the time-fractional DPL model.     

NOTE ON INVERSE PROBLEM WITH l∞ OBJECTIVE FUNCTION

In this note, the author proves that the inverse problem of submodular function on digraphs with l∞ objective function can be solved by strongly polynomial algorithm. The result shows that most inverse network optimization problems with l∞ objective function can be solved in the polynomial time.     

Solving the Inverse Generalized Problem of Vertical Seismic Profiling

The dynamic inverse seismics problem is considered in a generalized setting. We investigate whether the wave propagation problem in a vertically nonhomogeneous medium is well-posed. We show that the regular part of the solution is an L ₂ function and the inverse problem, i.e., the determination of the reflection coefficient, is thus reducible to minimizing the error functional. The gradient of the functional is obtained in explicit form from the conjugate problem, and approximate formulas for its evaluation are derived. A regularization algorithm for the solution of the inverse problem is considered; simulation results using various excitation sources are reported.     

Factorisation of Non-negative Fredholm Operators and Inverse Spectral Problems for Bessel Operators

We study the problem of factorisation of non-negative Fredholm operators acting in the Hilbert space L₂(0, 1) and its relation to the inverse spectral problem for Bessel operators. In particular, we derive an algorithm of reconstructing the singular potential of the Bessel operator from its spectrum and the sequence of norming constants.     

Inverse median location problems with variable coordinates

Given n points in \mathbbR^d{\mathbb{R}^d} with nonnegative weights, the inverse 1-median problem with variable coordinates consists in changing the coordinates of the given points at minimum cost such that a prespecified point in \mathbbR^d{\mathbb{R}^d} becomes the 1-median. The cost is proportional to the increase or decrease of the corresponding point coordinate. If the distances between points are measured by the rectilinear norm, the inverse 1-median problem is NP{\mathcal{NP}}-hard, but it can be solved in pseudo-polynomial time. Moreover, a fully polynomial time approximation scheme exists in this case. If the point weights are assumed to be equal, the corresponding inverse problem can be reduced to d continuous knapsack problems and is therefore solvable in O(nd) time. In case that the squared Euclidean norm is used, we derive another efficient combinatorial algorithm which solves the problem in O(nd) time. It is also shown that the inverse 1-median problem endowed with the Chebyshev norm in the plane is NP{\mathcal{NP}}-hard. Another pseudo-polynomial algorithm is developed for this case, but it is shown that no fully polynomial time approximation scheme does exist.     

Markov chain Monte Carlo Using an Approximation

This article presents a method for generating samples from an unnormalized posterior distribution f(·) using Markov chain Monte Carlo (MCMC) in which the evaluation of f(·) is very difficult or computationally demanding. Commonly, a less computationally demanding, perhaps local, approximation to f(·) is available, say f^**_x(·). An algorithm is proposed to generate an MCMC that uses such an approximation to calculate acceptance probabilities at each step of a modified Metropolis–Hastings algorithm. Once a proposal is accepted using the approximation, f(·) is calculated with full precision ensuring convergence to the desired distribution. We give sufficient conditions for the algorithm to converge to f(·) and give both theoretical and practical justifications for its usage. Typical applications are in inverse problems using physical data models where computing time is dominated by complex model simulation. We outline Bayesian inference and computing for inverse problems. A stylized example is given of recovering resistor values in a network from electrical measurements made at the boundary. Although this inverse problem has appeared in studies of underground reservoirs, it has primarily been chosen for pedagogical value because model simulation has precisely the same computational structure as a finite element method solution of the complete electrode model used in conductivity imaging, or “electrical impedance tomography.” This example shows a dramatic decrease in CPU time, compared to a standard Metropolis–Hastings algorithm.     

On-line scheduling on a batch processing machine with unbounded batch size to minimize the makespan

We present on-line algorithms to minimize the makespan on a single batch processing machine. We consider a parallel batching machine that can process up to b jobs simultaneously. Jobs in the same batch complete at the same time. Such a model of a batch processing machine has been motivated by burn-in ovens in final testing stage of semiconductor manufacturing. We deal with the on-line scheduling problem when jobs arrive over time. We consider a set of independent jobs. Their number is not known in advance. Each job is available at its release date and its processing requirement is not known in advance. This general problem with infinite machine capacity is noted 1∣p − batch, r_j, b = ∞∣C_max. Deterministic algorithms that do not insert idle-times in the schedule cannot be better than 2-competitive and a simple rule based on LPT achieved this bound [Z. Liu, W. Yu, Scheduling one batch processor subject to job release dates, Discrete Applied Mathematics 105 (2000) 129–136]. If we are allowed to postpone start of jobs, the performance guarantee can be improved to 1.618. We provide a simpler proof of this best known lower bound for bounded and unbounded batch sizes. We then present deterministic algorithms that are best possible for the problem with unbounded batch size (i.e., b = ∞) and agreeable processing times (i.e., there cannot exist an on-line algorithm with a better performance guarantee). We then propose another algorithm that leads to a best possible algorithm for the general problem with unbounded batch size. This algorithm improves the best known on-line algorithm (i.e. [G. Zhang, X. Cai, C.K. Wong, On-line algorithms for minimizing makespan on batch processing machines, Naval Research Logistics 48 (2001) 241–258]) in the sense that it produces a shortest makespan while ensuring the same worst-case performance guarantee.     

Inverse sorting problem by minimizing the total weighted number of changes and partial inverse sorting problems

In this paper, we consider two types of inverse sorting problems. The first type is an inverse sorting problem by minimizing the total weighted number of changes with bound constraints. We present an O(n ²) time algorithm to solve the problem. The second type is a partial inverse sorting problem and a variant of the partial inverse sorting problem. We show that both the partial inverse sorting problem and the variant can be solved by a combination of a sorting problem and an inverse sorting problem. Supported by the Hong Kong Universities Grant Council (CERG CITYU 103105) and the National Key Research and Development Program of China (2002CB312004) and the National Natural Science Foundation of China (700221001, 70425004).     

The Parallel Simplicity of Compaction and Chaining

Given a set of values x₁, x₂,..., x_n, of which k are nonzero, the compaction problem is the problem of moving the nonzero elements into the first k consecutive memory locations. The chaining problem asks that the nonzero elements be put into a linked list. One can in addition require that the elements remain in the same order, leading to the problems of ordered compaction and ordered chaining, respectively. This paper introduces a technique involving perfect hash functions that leads to a deterministic algorithm for ordered compaction running on a CRCW PRAM in time O(log k/log log n) using n processors. A matching lower bound for unordered compaction is given. The ordered chaining problem is shown to be solvable in time O(α(k)) with n processors (where α is a functional inverse of Ackermann′s function) and unordered chaining is shown to he solvable in constant time with n processors when k < n^{1/4− ε}.     

Distributed subgradient method for multi‐agent optimization with quantized communication

This paper focuses on a distributed optimization problem associated with a time‐varying multi‐agent network with quantized communication, where each agent has local access to its convex objective function, and cooperatively minimizes a sum of convex objective functions of the agents over the network. Based on subgradient methods, we propose a distributed algorithm to solve this problem under the additional constraint that agents can only communicate quantized information through the network. We consider two kinds of quantizers and analyze the quantization effects on the convergence of the algorithm. Furthermore, we provide explicit error bounds on the convergence rates that highlight the dependence on the quantization levels. Finally, some simulation results on a l₁‐regression problem are presented to demonstrate the performance of the algorithm. Copyright © 2016 John Wiley & Sons, Ltd.     

On the K shortest path trees problem

We address the problem of finding the K best path trees connecting a source node with any other non-source node in a directed network with arbitrary lengths. The main result in this paper is the proof that the kth shortest path tree is adjacent to at least one of the previous (k-1) shortest path trees. Consequently, we design an O(f(n,m,C_max)+Km) time and O(K+m) space algorithm to determine the K shortest path trees, in a directed network with n nodes, m arcs and maximum absolute length C_max, where O(f(n,m,C_max)) is the best time needed to solve the shortest simple paths connecting a source node with any other non-source node.     

A conformal mapping algorithm for the Bernoulli free boundary value problem

We propose a new numerical method for the solution of the Bernoulli free boundary value problem for harmonic functions in a doubly connected domain D in where an unknown free boundary Γ₀ is determined by prescribed Cauchy data on Γ₀ in addition to a Dirichlet condition on the known boundary Γ₁. Our main idea is to involve the conformal mapping method as proposed and analyzed by Akduman, Haddar, and Kress for the solution of a related inverse boundary value problem. For this, we interpret the free boundary Γ₀ as the unknown boundary in the inverse problem to construct Γ₀ from the Dirichlet condition on Γ₀ and Cauchy data on the known boundary Γ₁. Our method for the Bernoulli problem iterates on the missing normal derivative on Γ₁ by alternating between the application of the conformal mapping method for the inverse problem and solving a mixed Dirichlet–Neumann boundary value problem in D. We present the mathematical foundations of our algorithm and prove a convergence result. Some numerical examples will serve as proof of concept of our approach. Copyright © 2015 John Wiley & Sons, Ltd.     

Improved 0/1-interchange scheduling

Consideration is given to the problem of nonpreemptively scheduling a set ofN independent tasks to a system ofM identical processors, with the objective to minimize the overall finish time. It is proved that the 0/1-INTERCHANGE scheduling heuristic can be modified, without increasing its time complexity fromO(N logM), so that its worst-case performance bound is reduced from 2 to 4/3 times optimal.     

Minimizing Completion Time Variance with Compressible Processing Times

We introduce a new formulation of the standard completion time variance (CTV) problem with n jobs and one machine, in which the job sequence and the processing times of the jobs are all decision variables. The processing time of job i (i=1, ,n) can be compressed by an amount within [l_i, u_i], which however will incur a compression cost. The compression cost is a general convex non-decreasing function of the amount of the job processing time compressed. The objective is to minimize a weighted combination of the completion time variance and the total compression cost. We show that, under an agreeable condition on the bounds of the processing time compressions, a pseudo-polynomial algorithm can be derived to find an optimal solution for the problem. Based on the pseudo-polynomial time algorithm, two heuristic algorithms H1 and H2 are proposed for the general problem. The relative errors of both heuristic algorithms are guaranteed to be no more than , where is a measure of deviation from the agreeable condition. While H1 can find an optimal solution for the agreeable problem, H2 is dominant for solving the general problem. We also derive a tight lower bound for the optimal solution of the general problem. The performance of H2 is evaluated by complete enumeration for small n, and by comparison with this tight lower bound for large n. Computational results (with n up to 80) are reported, which show that the heuristic algorithm H2 in general can efficiently yield near optimal solutions, when n is large.     

Directed Graph Pattern Matching and Topological Embedding

Pattern matching in directed graphs is a natural extension of pattern matching in trees and has many applications to different areas. In this paper, we study several pattern matching problems in ordered labeled directed graphs. For the rooted directed graph pattern matching problem, we present an efficient algorithm which, given a pattern graphPand a target graphT, runs in time and spaceO(|E_P| |V_T| + |E_T|). It is faster than the best known method by a factor ofmin{|E_T|, |E_P| |V_T|}. This algorithm can also solve the directed graph pattern matching problem without increasing time or space complexity. Our solution to this problem outperforms the best existing method by Katzenelson, Pinter and Schenfeld by a factor ofmin{|V_P| |E_T|, |V_P| |E_P| |V_T|}. We also present an algorithm for the directed graph topological embedding problem which runs in timeO(|V_P| |E_T| + |E_P|) and spaceO(|V_P| |V_T| + |E_P| + |E_T|). To our knowledge, this algorithm is the first one for this problem.     

On dual minimum cost flow algorithms

We describe a new dual algorithm for the minimum cost flow problem. It can be regarded as a variation of the best known strongly polynomial minimum cost flow algorithm, due to Orlin. Indeed we obtain the same running time of O(m log m(m+n log n)), where n and m denote the number of vertices and the number of edges. However, in contrast to Orlin's algorithm we work directly with the capacitated network (rather than transforming it to a transshipment problem). Thus our algorithm is applicable to more general problems (like submodular flow) and is likely to be more efficient in practice.  Our algorithm can be interpreted as a cut cancelling algorithm, improving the best known strongly polynomial bound for this important class of algorithms by a factor of m. On the other hand, our algorithm can be considered as a variant of the dual network simplex algorithm. Although dual network simplex algorithms are reportedly quite efficient in practice, the best worst-case running time known so far exceeds the running time of our algorithm by a factor of n.     

AnO(m + n log n) Algorithm for the Maximum-Clique Problem in Circular-Arc Graphs

We present an algorithm to compute, inO(m + n log n) time, a maximum clique in circular-arc graphs (withnvertices andmedges) provided a circular-arc model of the graph is given. If the circular-arc endpoints are given in sorted order, the time complexity isO(m). The algorithm operates on the geometric structure of the circular arcs, radially sweeping their endpoints; it uses a very simple data structure consisting of doubly linked lists. Previously, the best time bound for this problem wasO(m log log n + n log n), using an algorithm that solved an independent subproblem for each of thencircular arcs. By using the radial-sweep technique, we need not solve each of these subproblems independently; thus we eliminate the log log nfactor from the running time of earlier algorithms. For vertex-weighted circular-arc graphs, it is possible to use our approach to obtain anO(m log log n + n log n) algorithm for finding a maximum-weight clique—which matches the best known algorithm.