On DDoS Attack Related Minimum Cut Problems

Abstract

Distributed Denial of Service (DDoS) attack is one of the most preeminent threats in Internet. Despite considerable progress on this problem in recent years, a remaining challenge is to determine its hardness by adopting proper mathematical models. In this paper, we propose to use generalized minimum cut as basic tools to model various types of DDoS attacks. Particularly, we study two important extensions of the classical minimum cut problem, called Connectivity Preserving Minimum Cut (CPMC) and Threshold Minimum Cut (TMC), to model largescale DDoS attacks.In the CPMC problem, a minimum cut is sought to separate a source node from a destination node and meanwhile preserve the connectivity between the source and its partner node(s). The CPMC problem also has important applications in many other areas such as emergency responding, image processing, pattern recognition, and medical sciences. In the TMC problem, a minimum cut is sought to isolate a target node from a threshold number of partner nodes. TMC problem is an interesting special case of the network inhibition problem and finds applications in network security. We show that the general CPMC problem cannot be approximated within logn unless N P = P . We also show that a special case of the CPMC problem in planar graphs can be solved in polynomial time. The corollary of this result is that the network diversion problem in planar graphs is in P ; this settles a previously open problem. For the TMC problem, we show that the threshold minimum node cut (TMNC) problem can be approximated within a ratio of O( √ n) and the threshold minimum edge cut (TMEC) problem can be approximated within a ratio of O(log 2 n). As a consequence, we show that the related network inhibition problem and network interdiction problem cannot be approximated within any constant ratio unless N P ∩ δ>0 BP T IM E(2 n δ ). This settles another long standing open problem.