Yazid M. Sharaiha

Heuristics for cardinality constrained portfolio optimisation

In this paper we consider the problem of finding the efficient frontier associated with the standard mean-variance portfolio optimisation model. We extend the standard model to include cardinality constraints that limit a portfolio to have a specified number of assets, and to impose limits on the proportion of the portfolio held in a given asset (if any of the asset is held). We illustrate the differences that arise in the shape of this efficient frontier when such constraints are present. We present three heuristic algorithms based upon genetic algorithms, tabu search and simulated annealing for finding the cardinality constrained efficient frontier. Computational results are presented for five data sets involving up to 225 assets.

Scheduling Aircraft Landings--The Static Case

In this paper, we consider the problem of scheduling aircraft (plane) landings at an airport. This problem is one of deciding a landing time for each plane such that each plane lands within a predetermined time window and that separation criteria between the landing of a plane and the landing of all successive planes are respected. We present a mixed-integer zero--one formulation of the problem for the single runway case and extend it to the multiple runway case. We strengthen the linear programming relaxations of these formulations by introducing additional constraints. Throughout, we discuss how our formulations can be used to model a number of issues (choice of objective function, precedence restrictions, restricting the number of landings in a given time period, runway workload balancing) commonly encountered in practice. The problem is solved optimally using linear programming-based tree search. We also present an effective heuristic algorithm for the problem. Computational results for both the heuristic and the optimal algorithm are presented for a number of test problems involving up to 50 planes and four runways.

Displacement problem and dynamically scheduling aircraft landings

In this paper we define a generic decision problem — the displacement problem. The displacement problem arises when we have to make a sequence of decisions and each new decision that must be made has an explicit link back to the previous decision that was made. This link is quantified by means of the displacement function. One situation where the displacement problem arises is that of dynamically scheduling aircraft landings at an airport. Here decisions about the landing times for aircraft (and the runways they land on) must be taken in a dynamic fashion as time passes and the operational environment changes. We illustrate the application of the displacement problem to the dynamic aircraft landing problem. Computational results are presented for a number of publicly available test problems involving up to 500 aircraft and five runways.

Comparison of Algorithms for the Degree Constrained Minimum Spanning Tree

The Degree Constrained Minimum Spanning Tree (DCMST) on a graph is the problem of generating a minimum spanning tree with constraints on the number of arcs that can be incident to vertices of the graph. In this paper we develop three heuristics for the DCMST, including simulated annealing, a genetic algorithm and a method based on problem space search. We propose alternative tree representations to facilitate the neighbourhood searches for the genetic algorithm. The tree representation that we use for the genetic algorithm can be generalised to other tree optimisation problems as well. We compare the computational performance of all of these approaches against the performance of an exact solution approach in the literature. In addition, we also develop a new exact solution approach based on the combinatorial structure of the problem. We test all of these approaches using standard problems taken from the literature and some new test problems that we generate.

A tabu search algorithm for the capacitated shortest spanning tree problem

The Capacitated Shortest Spanning Tree Problem consists of determining a shortest spanning tree in a vertex weighted graph such that the weight of every subtree linked to the root by an edge does not exceed a prescribed capacity. We propose a tabu search heuristic for this problem, as well as dynamic data structures developed to speed up the algorithm. Computational results on new randomly generated instances and on instances taken from the literature indicate that the proposed approach produces high-quality solutions within reasonable computing times.

Euclidean Ordering via Chamfer Distance Calculations

This paper studies the mapping between continuous and discrete distances. The continuous distance considered is the widely used Euclidean distance, whereas we consider as the discrete distance the chamfer distance based on 3×3 masks. A theoretical characterization of topological errors which arise during the approximation of Euclidean distances by discrete ones is presented. Optimal chamfer distance coefficients are characterized with respect to the topological ordering they induce, rather than the approximation of Euclidean distance values. We conclude the theoretical part by presenting a global upper bound for a topologically correct distance mapping, irrespective of the chamfer distance coefficients, and we identify the smallest coefficients associated with this bound. We illustrate the practical significance of these results by presenting a framework for the solution of a well-known problem, namely the Euclidean nearest-neighbor problem. This problem is formulated as a discrete optimization problem and solved accordingly using algorithmic graph theory and integer arithmetic.

A graph-theoretic approach to distance transformations

Abstract We present a novel graph-theoretic approach to the Distance Transformation (DT) problem. The binary digital image is considered as a graph and the DT problem reduces to a shortest path forest problem. An algorithm is presented which solves the chamfer DT, and the Euclidian DT for a given bound.

An Optimal Algorithm for the Straight Segment Approximation of Digital Arcs

Abstract In this paper, we define the straight segment approximation problem (SSAP) for a given digital arc as that of locating a minimum subset of vertices on the arc such that they form a connected sequence of digital straight segments. Sharaiha (Ph.D. thesis, Imperial College, London, 1991) introduced the compact chord property, and proved its equivalence to Rosenfeld′s chord property (IEEE Trans. Comput. C-23, 1974, 1264-1269). The SSAP is now constrained by the compact chord property, which offers a more convenient geometric representation than the chord property. We develop an O(n2) optimal algorithm for the solution of the SSAP using integer arithmetic. A relaxation of the problem is also presented such that the optimal number of vectors can be reduced according to a user definition. The original algorithm is adapted for the optimal solution of the relaxed problem. An extension to the relaxed problem is also addressed which finds a minimum level of relaxation such that the optimal number of vectors cannot be reduced.

Discrete Convexity, Straightness, and the 16-Neighborhood

In this paper, we extend some results in discrete geometry based on the 8-neighborhood to that of the 16-neighborhood, which now includes the chessboard and the knight moves. We first present some analogies between an 8-digital arc and a 16-digital arc as represented by shortest paths on the grid. We present a transformation which uniquely maps a 16-digital arc onto an 8-digital arc (and vice versa). The grid-intersect-quantization (GIQ) of real arcs is defined with the 16-neighborhood. This enables us to define a 16-digital straight segment. We then present two new distance functions which satisfy the metric properties and describe the extended neighborhood space. Based on these functions, we present some new results regarding discrete convexity and 16-digital straightness. In particular, we demonstrate the convexity of a 16-digital straight segment. Moreover, we define a new property for characterizing a digital straight segment in the 16-neighborhood space. In comparison to the 8-neighborhood space, the proposed 16-neighborhood coding scheme offers a more compact representation without any loss of information.

A minimum spanning tree approach to line image analysis

We propose a graph theoretic approach for extracting the skeleton of a binary line image. Unlike other thinning methods, emphasis is placed on the preservation of the topology of both the foreground and the background. Such conditions guarantee a relevant resulting structure that can be used as input for pattern recognition. Using the underlying graph structure, we can readily formulate this problem as an optimisation problem. Local information such as centrality is given by a distance transform operation. Global information such as location of a branch end is given via a minimum weighted spanning tree which spans all foreground pixels. The resulting structure is then characterised as a union of central paths between end points with their adjacency inter-relationships. Other image characteristics (e.g. width and length of the branches) are also provided. Computational results applied on real images illustrate the noise insensitivity of this method.