A Heuristic Method for the Set Covering Problem

Public transport crew scheduling is a worldwide problem, which is NP-hard. This paper presents a new crew scheduling approach, called GRAVIG, which integrates grey relational analysis (GRA) into a Variable Iterated Greedy (VIG) algorithm. The GRA is served as a solver for the shift selection during the schedule construction process, which can be considered as a multiple attribute decision making (MADM) problem, since there are multiple static and dynamic criteria governing the efficiency of a shift to be selected into a schedule. Moreover, in the GRAVIG, a biased probability destruction strategy is elaborately devised to keep the ‘good’ shifts remained in the schedule without compromising the randomness. Experiments on eleven real-world crew scheduling problems show that the GRAVIG can generate high-quality solutions close to the lower bounds obtained by the CPLEX in terms of the number of shifts.

State space relaxation for set covering problems related to bus driver scheduling

The set covering problem has many diverse applications to problems arising in crew scheduling, facility location and other business areas. Since the problem is known to be hard to solve optimally, a number of approximate (heuristic) approaches have been designed for it. These approaches (with one exception) divide into two main groups, greedy heuristics and dual saturation heuristics. We use the concept of a Pareto optimal dual solution to show that an arbitrary dual saturation heuristic has the same worst-case performance guarantee as the two best known heuristics of that type. Moreover, this poor performance level is always attainable by those two heuristics.

Using swarm intelligence algorithms to detect influential individuals for influence maximization in social networks

Optimal Sensor Placement for Contamination Detection and Identification in Water Distribution Networks

Currently, the industry is experiencing an exponential increase in dealing with binary-based combinatorial problems. In this sense, metaheuristics have been a common trend in the field in order to design approaches to solve them successfully. Thus, a well-known strategy consists in the use of algorithms based on discrete swarms transformed to perform in binary environments. Following the No Free Lunch theorem, we are interested in testing the performance of the Fruit Fly Algorithm, this is a bio-inspired metaheuristic for deducing global optimization in continuous spaces, based on the foraging behavior of the fruit fly, which usually has much better sensory perception of smell and vision than any other species. On the other hand, the Set Coverage Problem is a well-known NP-hard problem with many practical applications, including production line balancing, utility installation, and crew scheduling in railroad and mass transit companies. In this paper, we propose different binarization methods for the Fruit Fly Algorithm, using S-shaped and V-shaped transfer functions and various discretization methods to make the algorithm work in a binary search space. We are motivated with this approach, because in this way we can deliver to future researchers interested in this area, a way to be able to work with continuous metaheuristics in binary domains. This new approach was tested on benchmark instances of the Set Coverage Problem and the computational results show that the proposed algorithm is robust enough to produce good results with low computational cost.

Optimization algorithms for the solution of the frictionless normal contact between rough surfaces

A channel scheduling problem for a given graph G(V,E) is to select a subset E′ of edges of E and assign channel to each one in E′ under the restriction that all edges of E′ are interference free. We introduce the notion of two sides approximation for the channel scheduling problem. A pair of parameters (f,g) controls the accuracy of approximation in this paper. A (f,g)-approximation satisfies \(\sum_{e\in E'}W(e)\ge{\sum_{e\in Opt}W(e)\over f}\) and \(\sum_{e\in E-E'}W(e)\le g\sum_{e\in E- Opt^*}W(e)\), where Opt * is the set of edges assigned channels in an optimal solution, and W(e) is the weight of edge e. An f-approximation satisfies \(\sum_{e\in E'}W(e)\ge{\sum_{e\in Opt}W(e)\over f}\). We show that a simple greedy algorithm can obtain an (O(1), O(1))-approximation for the single channel scheduling problem. In many cases, the greedy algorithms gives much more accurate result than the worst ratio. Furthermore, we develop an \(|E|^{O({1\over \epsilon})}\) time (1 − ε, O(1))-approximation algorithm for the single channel scheduling problem. We also show that a simple greedy algorithm can obtain an O(1)-approximation for the multi-channel scheduling problem which satisfies \(\sum_{e\in E'}W(e)\ge{\sum_{e\in Opt}W(e)\over \Omega(1)}\). We also develop a \(|E|^{O({d\over \epsilon})}\) time (1 − ε)-approximation algorithm the multi-channel scheduling problem, where d is the number of channels. This improves the existing approximation scheme for multi-channel scheduling problem with \(|E|^{O({d\over \epsilon^2})}\) time by Cheng et al. We also develop a polynomial time constant factor greedy approximation algorithm for the multi-channel scheduling that allows variate radii of interference among those nodes.

This study aims to solve the real-world multistage assignment problem. The proposed problem is composed of two stages of assignment: (1) different types of trucks are assigned to chicken farms to transport young chickens to egg farms, and (2) chicken farms are assigned to egg farms. Assigning different trucks to the egg farms and different egg farms to the chicken farms generates different costs and consumes different resources. The distance and the idle space in the truck have to be minimized, while constraints such as the minimum number of chickens needed for all egg farms and the longest time that chickens can be in the truck remain. This makes the problem a special case of the multistage assignment (S-MSA) problem. A mathematical model representing the problem was developed and solved to optimality using Lingo v.11 optimization software. Lingo v.11 can solve to optimality only small- and medium-sized test instances. To solve large-sized test instances, the differential evolution (DE) algorithm was designed. An excellent decoding method was developed to increase the search performance of DE. The proposed algorithm was tested with three randomly generated datasets (small, medium, and large test instances) and one real case study. Each dataset is composed of 12 problems, therefore we tested with 37 instances, including the case study. The results show that for small- and medium-sized test instances, DE has 0.03% and 0.05% higher cost than Lingo v.11. For large test instances, DE has 3.52% lower cost than Lingo v.11. Lingo v.11 uses an average computation time of 5.8, 103, and 4320 s for small, medium and large test instances, while DE uses 0.86, 1.68, and 8.79 s, which is, at most, 491 times less than Lingo v.11. Therefore, the proposed heuristics are an effective algorithm that can find a good solution while using less computation time.

In the minimum entropy set cover problem, one is given a collection of k sets which collectively cover an n-element ground set. A feasible solution of the problem is a partition of the ground set into parts such that each part is included in some of the k given sets. Such a partition defines a probability distribution, obtained by dividing each part size by n. The goal is to find a feasible solution minimizing the (binary) entropy of the corresponding distribution. Halperin and Karp have recently proved that the greedy algorithm always returns a solution whose cost is at most the optimum plus a constant. We improve their result by showing that the greedy algorithm approximates the minimum entropy set cover problem within an additive error of 1 nat =log 2 e bits $\simeq$1.4427 bits. Moreover, inspired by recent work by Feige, Lovasz and Tetali on the minimum sum set cover problem, we prove that no polynomial-time algorithm can achieve a better constant, unless P = NP. We also discuss some consequences for the related minimum entropy coloring problem.

This paper addresses the Minimum Linear Ordering Problem (MLOP): Given a nonnegative set function f on a finite set V, find a linear ordering on V such that the sum of the function values for all the suffixes is minimized. This problem generalizes well-known problems such as the Minimum Linear Arrangement, Min Sum Set Cover, Minimum Latency Set Cover, and Multiple Intents Ranking. Extending a result of Feige, Lovasz, and Tetali (2004) on Min Sum Set Cover, we show that the greedy algorithm provides a factor 4 approximate optimal solution when the cost function f is supermodular. We also present a factor 2 rounding algorithm for MLOP with a monotone submodular cost function, using the convexity of the Lovasz extension. These are among very few constant factor approximation algorithms for NP-hard minimization problems formulated in terms of submodular/supermodular functions. In contrast, when f is a symmetric submodular function, the problem has an information theoretic lower bound of 2 on the approximability.

Approximating the dense set-cover problem

A framework for the greedy algorithm

Greedy method, as an efficient computing tool, can be applied to various combinatorial or nonlinear optimization problems where finding the global optimum is difficult, if not computationally infeasible. A greedy algorithm has the nature of making the locally optimal choice at each stage and then solving the subproblems that arise later. It iteratively makes one greedy choice after another, reducing each given problem into a smaller one. In other words, a greedy algorithm never reconsiders its choices. Clearly, greedy method often fails to find the globally optimal solution. However, a greedy algorithm can be proven to yield the global optimum for a given class of problems such as Kruskal's algorithm and Prim's algorithm for finding minimum spanning tree, Dijkstra's algorithm for finding single-source shortest path, and the algorithm for finding optimum Huffman tree [5]. Even for some optimization problems proven to be NP hard, a greedy algorithm may generate near optimal solution with high probability if one exploits the problem structure properly. In this chapter, we focus on the optimization problems arising from plume detection, localization and tracking and provide convincing argument on the usefulness of greedy algorithms. Detection, identification, localization, tracking and prediction of chemical, biological or nuclear propagation is crucial to battlefield surveillance and homeland security. In addition, post-accident management for public protection relies critically on detecting and tracing dangerous gas leakages promptly. The determination of source origins and release rates is useful for the forecast of gas concentration in the atmosphere and for the management staff to prioritize off-site evacuation plans. A lot of research has been focused on detecting and localizing single or multiple plume sources with autonomous vehicles [11] or sensor networks such as [22] for a vapor-emitting source, [2] for a nuclear source, and [14, 15] for a chemical source. In [12] the plume detection and localization problem is formulated as abrupt change detection using sparse sensor measurements. The development of a large scale testbed has been reported in [8] for plume detection, identification and tracking. In [3] dense sensor coverage has been used for radioactive source detection while [26] showed that using three error-free intensity sensors, one can identify the plume origin to any desired accuracy with high probability. Although this approach offers an effective solution with linear complexity of the hypothesis space, a major limitation is that the continuous time dynamic model of plume propagation has to be in the product form. O pe n A cc es s D at ab as e w w w .in te ch w eb .o rg

With the rapid expansion of optical networks, the number of network cutover tasks has increased significantly. Manual scheduling of cutover tasks typically takes several days, leading to inefficient resource utilization. To address this issue, we propose systematic intelligent task scheduling methods of cutover tasks in optical networks for what we believe to be the first time. The rules and attributes of cutover tasks are abstracted into graphical representations or mathematical expressions. Then, the scheduling process employs layered topological scheduling methods based on greedy heuristic algorithm, maximal cliques and particle swarm optimization. The scheduling results are verified in the real-time resource library by the conflict verification method. When conflicts arise, the scheduling process is iterated to find a conflict-free schedule, and prior rules are updated. The field trial results of national optical networks demonstrated that our proposed intelligent schedule method realizes second-level scheduling without conflicts. Compared with the layered topological scheduling method based on greedy algorithm, the scheduling methods based on maximal clique and particle swarm optimization can reduce the conflict rate by 20% and 30%, enhance the ability of conflict adjustment by 12% and 43%, and improve scheduling efficiency by 12% and 9%, respectively. The proposed intelligent scheduling methods can also autonomously generate multi-period conflict-free schedules.