Nowadays the term monochromatic and heterochromatic (or rainbow, multicolored) subgraphs of an edge-colored graph appeared frequently in literature, and many results on this topic have been obtained. In this paper, we survey results on this subject. We classify the results into the following categories: vertex-partitions by monochromatic subgraphs, such as cycles, paths, trees; vertex partition by some kinds of heterochromatic subgraphs; the computational complexity of these partition problems; some kinds of large monochromatic and heterochromatic subgraphs. We have to point out that there are a lot of results on Ramsey type problem of monochromatic and heterochromatic subgraphs. However, it is not our purpose to include them in this survey because this is slightly different from our topics and also contains too large amount of results to deal with together. There are also some interesting results on vertex-colored graphs, but we do not include them, either.

Monochromatic and Heterochromatic Subgraphs in Edge-Colored Graphs - A Survey

On the minimum label spanning tree problem

In this work, we consider the problem of maximizing the reliability of connections in mesh networks against failure scenarios in which multiple links may fail simultaneously. We consider the single-path connection problem as well as multiple-path (protected) connection problems. The problems are formulated as minimum-color path problems, where each link is associated with one or more colors, and each color corresponds to a given failure event Thus, when a certain color fails, all links which include that color will fail. In a single-path problem, by minimizing the number of colors on the path, the failure probability of the path can be minimized if all colors have the same probability of causing failures. In the case of two paths, where one path is a protection path, if all colors have the same probability of causing failures, the problem becomes that of finding two link-disjoint paths which either have a minimum total number of colors, or which have a minimum number of overlapping colors. By minimizing the total number of colors, the probability that a failure will occur on either of the paths is minimized. On the other hand, by minimizing the number of overlapping colors, the probability that a single failure event will cause both paths to fail simultaneously is minimized. The problems are proved to be NP-complete, and ILP formulations are developed. Heuristic algorithms are proposed for larger instances of the problems, and the heuristics are evaluated through simulation.

Minimum-color path problems for reliability in mesh networks

This article investigates complexity and approximability properties of combinatorial optimization problems yielded by the notion of Shared Risk Resource Group (SRRG). SRRG has been introduced in order to capture network survivability issues where a failure may break a whole set of resources, and has been formalized as colored graphs, where a set of resources is represented by a set of edges with same color. We consider here the analogous of classical problems such as determining paths or cuts with the minimum numbers of colors or color disjoint paths. These optimization problems are much more difficult than their counterparts in classical graph theory. In particular standard relationship such as the Max Flow - Min Cut equality do not hold any longer. In this article we identify cases where these problems are polynomial, for example when the edges of a given color form a connected subgraph, and otherwise give hardness and non approximability results for these problems.

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Shared Risk Resource Group : Complexity and Approximability Issues

Particle Swarm Optimization is a population-based method inspired by the social behaviour of individuals inside swarms in nature. Solutions of the problem are modelled as members of the swarm which fly in the solution space. The improvement of the swarm is obtained from the continuous movement of the particles that constitute the swarm submitted to the effect of inertia and the attraction of the members who lead the swarm. This work focuses on a recent Discrete Particle Swarm Optimization for combinatorial optimization, called Jumping Particle Swarm Optimization. Its effectiveness is illustrated on the minimum labelling Steiner tree problem: given an undirected labelled connected graph, the aim is to find a spanning tree covering a given subset of nodes, whose edges have the smallest number of distinct labels.

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Discrete Particle Swarm Optimization for the minimum labelling Steiner tree problem

The minimum labeling spanning trees

Routing paths in mobile ad hoc networks are easily disrupted due to the mobility of nodes. Therefore, it is desirable to find a routing path such that all edges in the path are long-lived. Based on the ad hoc on-demand distance vector routing (AODV) protocol, we propose an AODV-RSS (AODV with received signal strength) routing protocol such that connectivities of links in the route found can sustain longer. The AODV-RSS algorithm uses the received signal strength, RSS9 and the received signal strength changing rate, ARSS, to predict the link available time, LAT. Using the LAT as a constraint, our method will find out a satisfying routing path. Simulation results show that our routing algorithm can greatly improve the quality of the routing path in route connection time, and route reestablishment frequency.

Shing-Jiuan Leu

Papers

The minimum labeling spanning trees

Long-lived path routing with received signal strength for ad hoc networks