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In our cyber security model we define the concept of {\em penetration cost},which is the cost that must be paid in order to break into the next layer ofsecurity. Given a tree $T$ rooted at a vertex $r$, a {\em penetrating cost}edge function $c$ on $T$, a {\em target-acquisition} vertex function $p$ on$T$, the attacker's {\em budget} and the {\em game-over threshold} $B,G \in{\mathbb{Q}}^{+}$ respectively, we consider the problem of determining theexistence of a rooted subtree $T'$ of $T$ within the attacker's budget (thatis, the sum of the costs of the edges in $T'$ is less than or equal to $B$)with total acquisition value more than the game-over threshold (that is, thesum of the target values of the nodes in $T'$ is greater than or equal to $G$).We prove that the general version of this problem is intractable, but doesadmit a polynomial time approximation scheme. We also analyze the complexity ofthree restricted versions of the problems, where the penetration cost is theconstant function, integer-valued, and rational-valued among a given fixednumber of distinct values.

The complexity of cyber attacks in a new layered-security model and the maximum-weight, rooted-subtree problem