Worst-case Performance Ratio Research Articles

Unbounded parallel batch scheduling with job delivery to minimize makespan

We consider unbounded parallel batch scheduling with job delivery to minimize makespan. When the jobs have identical size, we provide a polynomial-time algorithm. When the jobs have non-identical sizes, we provide a heuristic with a worst-case performance ratio 7/4.

SINGLE MACHINE SCHEDULING WITH JOB DELIVERY TO MINIMIZE MAKESPAN

In the single machine scheduling problem with job delivery to minimize makespan, jobs are processed on a single machine and delivered by a capacitated vehicle to their respective customers. We first consider the special case with a single customer, that is, all jobs have the same transportation time. Chang and Lee (2004) proved that this case is strongly NP-hard. They also provided a heuristic with the worst-case performance ratio [Formula: see text], and pointed out that no heuristic can have a worst-case performance ratio less than [Formula: see text] unless P = NP. In this paper, we provide a new heuristic which has the best possible worst-case performance ratio [Formula: see text]. We also consider an extended version in which the jobs have non-identical transportation times and the transportation time of a delivery batch is defined as the maximum transportation time of the jobs contained in it. We provide a heuristic with the worst-case performance ratio 2 for the extended version, and show that this bound is tight.

Coordinating scheduling with batch deliveries in a two-machine flow shop

In the demanding world of supply chain management, traditional scheduling models which only address the optimization of production sequence at certain stage are often not globally optimized. Rather, the extension, including the distribution stage following the production, can bring a more holistic view to the decision makers. This research focuses mainly on a class of two-machine flow shop problem in which jobs need to be delivered to customers by vehicles after their production stages. Two performance measures—the sum of job completion times (∑C j ) and the makespan (C max)—are investigated separately. For the objective of ∑C j , this study shows that it is strongly NP-hard whether the job sizes are assumed to be equal or not. On the other hand, with regard to the C max objective, this paper concentrates only on the problem of different job sizes and provides a proof of its NP-hardness. A heuristic method that guarantees to contain a worst-case performance ratio of 2 is developed for the C max problem.

Performance of service policies in a specialized service system with parallel servers

This paper considers a scheduling problem occurring in a specialized service system with parallel servers. In the system, customers are divided into the “ordinary” and “special” categories according to their service needs. Ordinary customers can be served by any server, while special customers can be served only by the flexible servers. We assume that the service time for any ordinary customer is the same and all special customers have another common service time. We analyze three classes of service policies used in practice, namely, policies with priority, policies without priority and mixed policies. The worst-case performance ratios are obtained for all of these service policies.

Designing Routes for Source Coding With Explicit Side Information in Sensor Networks

In this paper, we study the problem of designing routes for source coding with explicit side information (i.e., with side information at both the encoder and the decoder) in sensor networks. Two difficulties in constructing such data-centric routes are the lack of reasonably practical data aggregation models and the high computational complexity resulting from the coupling of routing and in-network data fusion. Our data aggregation model is built upon the observation that in many physical situations the side information providing the most coding gain comes from a small number of nearby sensors. Based on this model, we formulate an optimization problem to minimize the communication cost, and show that finding the exact solution of this problem is NP-hard. Subsequently, two suboptimal algorithms are proposed. One is inspired by the balanced trees that have small total weights and reasonable distance from each sensor to the fusion center [6]. The other separately routes the explicit side information to achieve cost minimization. Bounds on the worst-case performance ratios of two methods to the optimal solution are derived for a special class of rate models, and simulations are conducted to shed light on their average behaviors.

Approximation algorithms for the makespan minimization with positive tails on a single machine with a fixed non-availability interval

In this article, we consider the non-resumable case of the single machine scheduling problem with a fixed non-availability interval. We aim to minimize the makespan when every job has a positive tail. We propose a polynomial approximation algorithm with a worst-case performance ratio of 3/2 for this problem. We show that this bound is tight. We present a dynamic programming algorithm and we show that the problem has an FPTAS (Fully Polynomial Time Approximation Algorithm) by exploiting the well-known approach of Ibarra and Kim (J. ACM 22:463–468, 1975). Such an FPTAS is strongly polynomial. The obtained results outperform the previous polynomial approximation algorithms for this problem.

Worst case analysis of Max-Regret, Greedy and other heuristics for Multidimensional Assignment and Traveling Salesman Problems

Optimization heuristics are often compared with each other to determine which one performs best by means of worst-case performance ratio reflecting the quality of returned solution in the worst case. The domination number is a complement parameter indicating the quality of the heuristic in hand by determining how many feasible solutions are dominated by the heuristic solution. We prove that the Max-Regret heuristic introduced by Balas and Saltzman (Oper. Res. 39:150---161, 1991) finds the unique worst possible solution for some instances of the s-dimensional (s?3) assignment and asymmetric traveling salesman problems of each possible size. We show that the Triple Interchange heuristic (for s=3) also introduced by Balas and Saltzman and two new heuristics (Part and Recursive Opt Matching) have factorial domination numbers for the s-dimensional (s?3) assignment problem.

Approximation bounds for quadratic maximization and max-cut problems with semidefinite programming relaxation

In this paper, we consider a class of quadratic maximization problems. For a subclass of the problems, we show that the SDP relaxation approach yields an approximation solution with the worst-case performance ratio at least α = 0.87856 ⋯. In fact, the estimated worst-case performance ratio is dependent on the data of the problem with α being a uniform lower bound. In light of this new bound, we show that the actual worst-case performance ratio of the SDP relaxation approach (with the triangle inequalities added) is at least α + δ d if every weight is strictly positive, where δ d > 0 is a constant depending on the problem dimension and data.

Minimizing maximum completion time in a proportionate flow shop with one machine of different speed

A flow shop with identical machines is called a proportionate flow shop. In this paper, we consider the variant of the n-job, m-machine proportionate flow shop scheduling problem in which only one machine is different and job processing times are inversely proportional to machine speeds. The objective is to minimize maximum completion time. We describe some optimality conditions and show that the problem is NP-complete. We provide two heuristic procedures whose worst-case performance ratio is less than two. Extensive experiments with various sizes are conducted to show the performance of the proposed heuristics.

Performance ratios of the Karmarkar-Karp differencing method

We consider the multiprocessor scheduling problem in which one must schedule n independent tasks nonpreemptively on m identical, parallel machines, such that the completion time of the last task is minimal. For this well-studied problem the Largest Differencing Method of Karmarkar and Karp outperforms other existing polynomial-time approximation algorithms from an average-case perspective. For m ≥ 3 the worst-case performance of the Largest Differencing Method has remained a challenging open problem. In this paper we show that the worst-case performance ratio is bounded between \( frac{4}{3}-\frac{1}{3(m-1)}$ and $\frac{4}{3}-\frac{1}{3m}\). For fixed m we establish further refined bounds in terms of n.

Packing into the smallest square: Worst-case analysis of lower bounds

We address the problem of packing a given set of rectangles into the minimum size square. We consider three versions of the problem, arising when the rectangles (i) are squares; (ii) have a fixed orientation; (iii) can be rotated by 90 ∘. For each case we study lower bounds, and analyze their worst-case performance ratio. In addition, we evaluate through computational experiments their average performance on instances from the literature.

Bin‐packing problem with concave costs of bin utilization

AbstractWe consider a generalized one‐dimensional bin‐packing model where the cost of a bin is a nondecreasing concave function of the utilization of the bin. Four popular heuristics from the literature of the classical bin‐packing problem are studied: First Fit (FF), Best Fit (BF), First Fit Decreasing (FFD), and Best Fit Decreasing (BFD). We analyze their worst‐case performances when they are applied to our model. The absolute worst‐case performance ratio of FF and BF is shown to be exactly 2, and that of FFD and BFD is shown to be exactly 1.5. Computational experiments are also conducted to test the performance of these heuristics. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006

Performance guarantees for flowshop heuristics to minimize makespan

This paper reviews the known worst-case ratios and absolute performance guarantees of various flowshop heuristics to minimize makespan. The absolute and worst-case performance ratio bounds are compared using probabilistic analysis. The best absolute performance bound is shown to outperform the tightest worst-case ratio bound with 99% probability when a certain minimum number of jobs is present. Thus, probabilistic analysis may provide a bridge between the absolute and worst-case performance guarantees of heuristic algorithms.

Minimizing the total weighted completion time in a two-machine proportionate flow shop with different machine speeds

In this paper, we consider the problem of minimizing the total weighted completion time in an n-job, two-machine proportionate flow shop, where job processing times are inversely proportional to machine speeds. We show that the problem is NP-complete. We provide a tight lower bound to an optimal objective value and show that the worst-case performance ratio of the weighted shortest processing time rule is less than 2. We propose heuristic procedures to find an approximate solution for the problem. A computational study on randomly generated test problems is conducted to investigate the performance of the proposed heuristics.

Complex Quadratic Optimization and Semidefinite Programming

In this paper we study the approximation algorithms for a class of discrete quadratic optimization problems in the Hermitian complex form. A special case of the problem that we study corresponds to the max-3-cut model used in a recent paper of Goemans and Williamson J. Comput. System Sci., 68 (2004), pp. 442-470]. We first develop a closed-form formula to compute the probability of a complex-valued normally distributed bivariate random vector to be in a given angular region. This formula allows us to compute the expected value of a randomized (with a specific rounding rule) solution based on the optimal solution of the complex semidefinite programming relaxation problem. In particular, we present an $[m^2(1-\cos\frac{2\pi}{m})/8\pi]$-approximation algorithm, and then study the limit of that model, in which the problem remains NP-hard. We show that if the objective is to maximize a positive semidefinite Hermitian form, then the randomization-rounding procedure guarantees a worst-case performance ratio of $\pi/4 \approx 0.7854$, which is better than the ratio of $2/\pi \approx 0.6366$ for its counterpart in the real case due to Nesterov. Furthermore, if the objective matrix is real-valued positive semidefinite with nonpositive off-diagonal elements, then the performance ratio improves to 0.9349.

PREEMPTIVE SCHEDULING ON PARALLEL PROCESSORS WITH DUE DATES

The paper presents a priority algorithm for the maximum lateness problem with parallel identical processors, precedence constraints, and preemptions. The presented algorithm calculates the priority of each task by constructing a schedule for the set of its successors. The algorithm is motivated by comparison of its nonpreemptive counterpart with other algorithms for the problem with unit execution time tasks. It is shown that the presented algorithm constructs an optimal schedule for the problem with two processors and arbitrary precedence constraints, and for the problem with an arbitrary number of processors and precedence constraints in the form of an in-tree. This proof also indicates that the presented algorithm allows the worst-case performance ratio previously established for the so-called Muntz–Coffman algorithm for a particular case of the considered problem where all due dates are zero.

Dispatching vehicles in a mega container terminal

We consider a container terminal discharging and uploading containers to and from ships. The discharged containers are stored at prespecified storage locations in the terminal yard. Containers are moved between the ship area and the yard using a fleet of vehicles, each of which can carry one container at a time. The problem is to dispatch vehicles to the containers so as to minimize the total time it takes to serve a ship, which is the total time it takes to discharge all containers from the ship and upload new containers onto the ship. We develop easily implementable heuristic algorithms and identify both the absolute and asymptotic worst-case performance ratios of these heuristics. In simple settings, most of these algorithms are optimal, while in more general settings, we show, through numerical experiments, that these algorithms obtain near-optimal results for the dispatching problem.

An Improved Approximation Algorithm for Multicast k-Tree Routing

An improved approximation algorithm is presented in this paper for the multicast k-tree routing problem. The algorithm has a worst case performance ratio of (2.4 + ρ), where ρ is the best approximation ratio for the metric Steiner tree problem (and is about 1.55 so far). The previous best approximation algorithm for the multicast k-tree routing problem has a performance ratio of 4. Two techniques, weight averaging and tree partitioning, are developed to facilitate the algorithm design and analysis.

Flow shop vs. permutation shop with time delays

This paper considers the evaluation of the worst-case performance ratio between the best solution of the flow shop problem and the permutation flow shop with time delays considerations. It is observed that, even in the restricted case of two machines and unit execution time operations, the two models may generate different optimal values for the makespan. More specifically, it is shown that, in the two-machine case, the performance ratio between the best permutation schedule and the best flow shop schedule is bounded by 2. When the operations of the n jobs are restricted to be unit execution time, this ratio is reduced to (2-(3/n+2)) for the two-machine case, and is m for the m-machine case.

Machine scheduling with job delivery coordination

This paper considers together machine scheduling and finished product delivery. In particular, it addresses the situation in which jobs require different amounts of storage space during delivery. Three scenarios of the problem are discussed. For the problem in which jobs are processed on a single machine and delivered by a single vehicle to one customer area, we provide a proof of NP-hardness and a heuristic with worst-case analysis. The worst-case performance ratio for our heuristic is proven to be 5/3, and the bound is tight. For the problem in which jobs are processed by either one of two parallel machines and delivered by a single vehicle to one customer area, our heuristic could cause at most 100% error under the worst-case with the bound being tight. For the problem that considers jobs to be processed by a single machine and delivered by a single vehicle to two customer areas, we provide another heuristic that is 100% error bound.