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In this paper, we study two important extensions of the classical minimum cutproblem, called {\em Connectivity Preserving Minimum Cut (CPMC)} problem and{\em Threshold Minimum Cut (TMC)} problem, which have important applications inlarge-scale DDoS attacks. In CPMC problem, a minimum cut is sought to separatea of source from a destination node and meanwhile preserve the connectivitybetween the source and its partner node(s). The CPMC problem also has importantapplications in many other areas such as emergency responding, imageprocessing, pattern recognition, and medical sciences. In TMC problem, aminimum cut is sought to isolate a target node from a threshold number ofpartner nodes. TMC problem is an important special case of network inhibitionproblem and has important applications in network security. We show that thegeneral CPMC problem cannot be approximated within $logn$ unless $NP=P$ hasquasi-polynomial algorithms. We also show that a special case of two group CPMCproblem in planar graphs can be solved in polynomial time. The corollary ofthis result is that the network diversion problem in planar graphs is in $P$, apreviously open problem. We show that the threshold minimum node cut (TMNC)problem can be approximated within ratio $O(\sqrt{n})$ and the thresholdminimum edge cut problem (TMEC) can be approximated within ratio$O(\log^2{n})$. \emph{We also answer another long standing open problem: thehardness of the network inhibition problem and network interdiction problem. Weshow that both of them cannot be approximated within any constant ratio. unless$NP \nsubseteq \cap_{\delta>0} BPTIME(2^{n^{\delta}})$.

On DDoS Attack Related Minimum Cut Problems