Extending Dijkstra's Algorithm to Maximize the Shortest Path by Node-Wise Limited Arc Interdiction

We consider the problem of computing shortest paths in a directed arc-weighted graph G = (V,A) in the presence of an adversary that can block (interdict), for each vertex v ∈ V, a given number p(v) of the arcs Aout(v) leaving v. We show that if all arc-weights are non-negative then the single-destination version of the problem can be solved by a natural extension of Dijkstra's algorithm in time $$O(|A|+|V|{\rm log}|V|+\Sigma_{\upsilon\in{V}\ \backslash \{t\}}(|A_{out}(\upsilon)|-p(\upsilon)){\rm log}(p(\upsilon)+1)).$$ Our result can be viewed as a polynomial algorithm for a special case of the network interdiction problem where the adversary's budget is node-wise limited. When the adversary can block a given number p of arcs distributed arbitrarily in the graph, the problem (p-most-vital-arcs problem) becomes NP-hard. This result is also closely related to so-called cyclic games. No polynomial algorithm computing the value of a cyclic game is known, though this problem belongs to both NP and coNP.