Multicolour paths in graphs: NP-hardness, algorithms, and applications on routing in WDM networks

详细信息    查看全文

• 作者：Rafael F. Santos ; Alessandro Andrioni…
• 关键词：Minimum paths in coloured graphs ; NP ; hardness ; Heuristics ; WDM networks
• 刊名：Journal of Combinatorial Optimization
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：33
• 期：2
• 页码：742-778
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Combinatorics; Convex and Discrete Geometry; Mathematical Modeling and Industrial Mathematics; Theory of Computation; Optimization; Operation Research/Decision Theory;
• 出版者：Springer US
• ISSN：1573-2886
• 卷排序：33

文摘

In this paper we study a problem of finding coloured paths of minimum weight in graphs. This problem has applications in WDM optical networks when high bandwidths are required to send data between a pair of nodes in the graph. Let \(G=(V,E)\) be a (directed) graph with a set of nodes V and a set of edges E in which each edge has an associated positive weight w(i, j), and let \(C = \{1, 2, \ldots , x\}\) be a set of x colours, \(x \in {\mathbb {N}}\). The function \(c : E \mapsto 2^C\) maps each edge of the graph G to a subset of colours. We say that edge e contains colours \(c(e) \subseteq C\). Given a positive integer \(k > 1\), a k-multicolour path is a path in G such that there exists a set of k colours \(K = \{c_1, \ldots , c_k\} \subseteq C\), with \(K \subseteq c(i,j)\) for each edge (i, j) in the path. The problem of finding one or more k-multicolour paths in a graph has applications in optical network and social network analysis. In the former case, the available wavelengths in the optical fibres are represented by colours in the edges and the objective is to connect two nodes through a path offering a minimum required bandwidth. For the latter case, the colours represent relations between elements and paths help identify structural properties in such networks. In this work we investigate the complexity of the multicolour path establishment problem. We show it is NP-hard and hard to approximate. Additionally, we develop Branch and Bound algorithms, ILPs, and heuristics for the problem. We then perform an experimental analysis of the developed algorithms to compare their performances.