Hamid Mokhtar

Forwarding and optical indices of 4-regular circulant networks

An all-to-all routing in a graph G is a set of oriented paths of G, with exactly one path for each ordered pair of vertices. The load of an edge under an all-to-all routing R is the number of times it is used (in either direction) by paths of R, and the maximum load of an edge is denoted by π ( G , R ) . The edge-forwarding index π ( G ) is the minimum of π ( G , R ) over all possible all-to-all routings R, and the arc-forwarding index π ? ( G ) is defined similarly by taking direction into consideration, where an arc is an ordered pair of adjacent vertices. Denote by w ( G , R ) the minimum number of colours required to colour the paths of R such that any two paths having an edge in common receive distinct colours. The optical index w ( G ) is defined to be the minimum of w ( G , R ) over all possible R, and the directed optical index w ? ( G ) is defined similarly by requiring that any two paths having an arc in common receive distinct colours. In this paper we obtain lower and upper bounds on these four invariants for 4-regular circulant graphs with connection set { ? 1 , ? s } , 1 < s < n / 2 . We give approximation algorithms with performance ratio a small constant for the corresponding forwarding index and routing and wavelength assignment problems for some families of 4-regular circulant graphs.

Recursive cubes of rings as models for interconnection networks

We study recursive cubes of rings as models for interconnection networks. We first redefine each of them as a Cayley graph on the semidirect product of an elementary abelian group by a cyclic group in order to facilitate the study of them by using algebraic tools. We give an algorithm for computing shortest paths and the distance between any two vertices in recursive cubes of rings, and obtain the exact value of their diameters. We obtain sharp bounds on the Wiener index, vertex-forwarding index, edge-forwarding index and bisection width of recursive cubes of rings. The cube-connected cycles and cube-of-rings are special recursive cubes of rings, and hence all results obtained in the paper apply to these well-known networks.

The 2-allocation p-hub median problem and a modified Benders decomposition method for solving hub location problems

Abstract We study the uncapacitated 2-allocation p-hub median problem (U2ApHMP), which is a special case of the well-studied hub median problem. The hub median problem designs a hub network in which the location of p hubs needs to be decided (the hubs are fully interconnected). The other nodes (known as access nodes) in the hub median problem are then allocated to one or many hubs. In the U2ApHMP, each access node is allocated to exactly two hubs. We discuss how this problem provides an alternative network design option for well-known p-hub median problems. We show its relevance and usefulness in the context of survivable network design and show that it addresses network survivability, a feature that has often been largely overlooked in hub network design research to date. We show that U2ApHMP is NP-hard even for a fixed/known set of hubs. We propose a mathematical formulation and develop a modified Benders decomposition method for this problem. In this, we convert the corresponding subproblems to minimum cost network flow problems. This allows us to solve large instances efficiently. We believe that, while our resulting method solves the U2ApHMP efficiently, it is also generalisable and can potentially be employed for solving other classes and types of hub location problems too.

An intermodal hub location problem for container distribution in Indonesia

Abstract In this paper, we extend traditional hub location models for an intermodal network design on a sparse network structure. While traditional hub location problems have been employed for developing network designs for many specific applications, their general assumptions – such as full connectivity, uniform transfer mode, and direct connections between access nodes and hubs – restrict their direct applicability to real-world logistics problems in several ways. In many network design contexts, the usage of versatile transfer modes and hubs is required due to different pricing of modes and topological considerations. In this paper, we extend the traditional hub location problem by incorporating three transfer modes and two kinds of hubs. As an important additional modification, we do not assume that the underlying network is fully connected, or that hubs and access nodes are directly connected. The context for our modelling is intermodal container movements in an archipelago. We develop and formulate an intermodal hub location problem. We show that this problem is NP-hard. Furthermore, a dataset for intermodal hub location problem is provided, based on a real-world container distribution problem in Indonesia. This dataset involves three modes of transport and a sparse network structure. We perform computational experiments and analyse our computational results. Our model provides insights for decision making and determining pricing policies for the desired levels of network flow.

A Modified Benders Method for the Single- and Multiple Allocation P -Hub Median Problems

We consider the well-known uncapacitated p-hub median problem with multiple allocation (UMApHMP), and single allocation (USApHMP). These problems have received significant attention in the literature because while they are easy to state and understand, they are hard to solve. They also find practical applications in logistics and telecommunication network design. Due to the inherent complexity of these problems, we apply a modified Benders decomposition method to solve large instances of the UMApHMP and USApHMP. The Benders decomposition approach does, however, suffer from slow convergence mainly due to the high degeneracy of subproblems. To resolve this, we apply a novel method of accelerating Benders method. We improve the performance of the accelerated Benders method by more appropriately choosing parameters for generating cuts, and by solving subproblems more efficiently using minimum cost network flow algorithms. We implement our approach on well-known benchmark data sets in the literature and compare our computational results for our implementations of existing methods and commercial solvers. The computational results confirms that our approach is efficient and enables us to solve larger single- and multiple allocation hub median instances. We believe this paper is the first implementation of Benders method to solve USA\(p\)HMP and UMA\(p\)HMP.

Cube-Connected Circulants as Efficient Models for Interconnection Networks

We introduce cube-connected circulants as efficient models for communication networks. We give an algorithm for computing a shortest path between any pair of vertices in a cube-connected circulant. We give formulas for the diameter of a cube-connected circulant and the distance between any pair of vertices in such a graph. Then we give an embedding of cube-connected circulants into hypercubes, and an embedding of hypercubes into cube-connected circulants. We show cube-connected circulants outperform a few well-known network structures in several invariants.

A few families of Cayley graphs and their efficiency as communication networks

Cayley graphs are highly attractive structures for communication networks because of their many desirable properties, including vertex-transitivity and efficient routing algorithms. The families of circulants and cube-connected graphs are among the most popular Cayley graphs for efficient communication networks. The diameter, forwarding and optical indices, bisection width and Wiener index of a network are among the most important parameters to measure the efficiency of the network. Circulant graphs and, in particular, circulant graphs with small degrees are interesting models for communication networks. However, our knowledge of many of their parameters, including arc-forwarding index, edge-forwarding index, directed and undirected optical indices, are very limited, except for very few special cases. We study the family of circulant graphs of degree 4 and obtain lower and upper bounds for their forwarding and optical indices. We give approximation algorithms for the corresponding problems of the forwarding indices and optical indices with a small constant performance ratio. The family of recursive cubes of rings has received a lot of attention for communication networks, but many aspects of them have remained unknown. We study this family of graphs by redefining each of them as a Cayley graph on the semidirect product of an elementary abelian group by a cyclic group in order to facilitate the study of them by using algebraic tools. We give an algorithm for computing shortest paths and obtain the exact value of their diameters. We obtain sharp bounds on the Wiener index, vertex-forwarding index, edge-forwarding index and bisection width of recursive cubes of rings. The cube-connected cycles and cube-of-rings are special recursive cubes of rings, and our results apply to these well-known networks. We introduce cube-connected circulants as a new family of cube-connected Cayley graphs and study their efficiency for communication networks. We give an algorithm for computing shortest path routing and the exact value of the diameter of a cube-connected circulant. We observe that while recursive cubes of rings are special cube-connected circulants, these two families of cube-connected graphs have significantly different routing behaviours in general. Hence we develop results for ‘proportional’ graphs which will be useful in obtaining bounds for the edge-forwarding index of cube-connected circulants. We give sharp lower and upper bounds for the Wiener index, vertex-forwarding and edge-forwarding indices of cube-connected circulants. We study the embedding of cube-connected circulants into hypercubes and the embedding of hypercubes into cube-connected circulants. We show that cube-connected circulants outperform a few well-known network topologies in many aspects. Gratefully dedicated to