Graph Theory
This paper deals with a general variant of the reverse undesirable (obnoxious) center location problem on cycle graphs. Given a ‘selective’ subset of the vertices of the underlying cycle graph as location of the existing customers, the task is to modify the edge lengths within a given budget such that the minimum of distances between a predetermined undesirable facility location and the customer points is maximized under the perturbed edge lengths. We develop a combinatorial O(n log n) algorithm for the problem with continuous modifications. For the uniform-cost model, we solve this problem in linear time by an improved algorithm. Furthermore, exact solution methods are proposed for the problem with integer modifications.
The aim of this paper is to establish a fundamental theory of convex analysis for the sets and functions over a discrete domain. By introducing conjugate/biconjugate functions and a discrete duality notion for the cones over discrete domains, we study duals of optimization problems whose decision parameters are integers. In particular, we construct duality theory for integer linear programming, provide a discrete version of Slater’s condition that implies the strong duality and discuss the relationship between integrality and discrete convexity.
The pooling problem, also called the blending problem, is fundamental in production planning of petroleum. It can be formulated as an optimization problem similar with the minimum-cost flow problem. However, Alfaki and Haugland (J Glob Optim 56:897–916,2013) proved the strong NP-hardness of the pooling problem in general case. They also pointed out that it was an open problem to determine the computational complexity of the pooling problem with a fixed number of qualities. In this paper, we prove that the pooling problem is still strongly NP-hard even with only one quality. This means the quality is an essential difference between minimum-cost flow problem and the pooling problem. For solving large-scale pooling problems in real applications, we adopt the non-monotone strategy to improve the traditional successive linear programming method. Global convergence of the algorithm is established. The numerical experiments show that the non-monotone strategy is effective to push the algorithm to explore the global minimizer or provide a good local minimizer. Our results for real problems from factories show that the proposed algorithm is competitive to the one embedded in the famous commercial software Aspen PIMS.
The g-Good-Neighbor Conditional Diagnosability of Locally Twisted Cubes
In the work of Peng et al. (Appl Math Comput 218(21):10406–10412, 2012), a new measure was proposed for fault diagnosis of systems: namely g-good-neighbor conditional diagnosability, which requires that any fault-free vertex has at least g fault-free neighbors in the system. In this paper, we establish the g-good-neighbor conditional diagnosability of locally twisted cubes under the PMC model and the MM∗ model.
