On Approximation of New Optimization Methods for Assessing Network Vulnerability

Abstract

Assessing network vulnerability before potential disruptive events such as natural disasters or malicious attacks is vital for network planning and risk management. It enables us to seek and safeguard against most destructive scenarios in which the overall network connectivity falls dramatically. Existing vulnerability assessments mainly focus on investigating the inhomogeneous properties of graph elements, node degree for example, however, these measures and the corresponding heuristic solutions can provide neither an accurate evaluation over general network topologies, nor performance guarantees to large scale networks. To this end, in this paper, we investigate a measure called pairwise connectivity and formulate this vulnerability assessment problem as a new graph-theoretical optimization problem called β-disruptor, which aims to discover the set of critical node/edges, whose removal results in the maximum decline of the global pairwise connectivity. Our results consist of the NP-Completeness and inapproximability proof of this problem, an O(log n log log n) pseudo-approximation algorithm for detecting the set of critical nodes and an O(log 1.5 n) pseudoapproximation algorithm for detecting the set of critical edges. In addition, we devise an efficient heuristic algorithm and validate the performance of the our model and algorithms through extensive simulations.

Identifying critical nodes in undirected graphs: Complexity results and polynomial algorithms for the case of bounded treewidth

“…is N P -hard on general graphs even if c uv = 1 for all edges uv: this follows easily from [13,Theorem 1], where a slightly different problem (called β-edge disruptor problem) is considered. However, we show that the algorithm of Section 3 can be adapted to handle this variant of the CNP, provided that the concept of CCC is slightly extended.…”

“…In this case R is clearly a feasible solution to problem P i (S, α, m), thus f i (S, α, m) ≥ w(R) = f j (S, α, m). Now assume that f j (S ′ , α ′ , m) is finite for some α ′ as in (13). Then there exists R ′ such that S ′ ⊆ R ′ ⊆ V j \ (X j \ S ′ ), α ′ is the CCC of S ′ with respect to G[R ′ ], and m is the number of connected pairs in G[R ′ ].…”

An Overview of Algorithms for Network Survivability

“…The algebraic connectivity α could be used to approximate the sparsest cut γ as α/2 ≤ γ ≤ α(2δ − α) [4,21]. Dinh et al [23] investigated the notion of pairwise connectivity (the number of connected pairs, which bears similarities to the sparsest-cut problem), and proved that finding the smallest set of nodes/links whose removal degrades the pairwise connectivity to certain degree is NPcomplete.…”

Branch and cut algorithms for detecting critical nodes in undirected graphs

“…Dinh et al [6] focus on telecommunication networks; they propose algorithms for detecting in a directed graph what they call node-disruptors and arc-disruptors, i.e., sets of nodes and arcs to be deleted in order to minimize the number of directed connections surviving in the residual graph.…”