Shared Risk Resource Group Complexity and Approximability Issues

Abstract

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.

Methods for physical impairment constrained routing with selected protection in all-optical networks

“…Protection methods can be classified based on the offered protection level. For example, the concept of Shared Risk Groups (SRG) was introduced to handle problems that may affect not only a single component, but a set of resources [11,13,23]. However, in optical networks, probability of multiple failures is low.…”

Diverse routing in networks with star SRLGs

“…In the context of SRLG, basic network connectivity problems have been proven much more difficult to address than their usual counterparts. For instance, the problem of finding a "SRLG-shortest" st-path that is a path from node s to node t having the minimum number of risks has been proven N P -hard and hard to approximate in general (see [2]). However, the problem has been proven to be polynomial in two generic practical cases corresponding to localized failures: when all risks verify the star property [3] and when risks are of span 1 ; i.e.…”

“…However, the problem has been proven to be polynomial in two generic practical cases corresponding to localized failures: when all risks verify the star property [3] and when risks are of span 1 ; i.e. when a link is affected by at most one risk and links sharing a given risk form a connected component [2].…”

“…The problem has been proven to be polynomial in some specific cases of localized failures: when SRLGs have span 1 [2], and in a specific case of SRLG's having the star property [4] (this result also follows from [2]). …”

“…To state precisely our results we use the colored-graph model [2,7,8], also called labeled-graph model. In the colored-graph model, the network is modeled by a graph G = (V, E) and the set of SRLGs by a set of colors R .…”