Stronger Hardness for Maximum Robust Flow and Randomized Network Interdiction

We study the following fundamental network optimization problem known as Maximum Robust Flow (MRF): A planner determines a flow on $s$-$t$-paths in a given capacitated network. Then, an adversary removes $k$ arcs from the network, interrupting all flow on paths containing a removed arc. The planner's goal is to maximize the value of the surviving flow, anticipating the adversary's response (i.e., a worst-case failure of $k$ arcs). It has long been known that MRF can be solved in polynomial time when $k = 1$ (Aneja et al., 2001), whereas it is $N\!P$-hard when $k$ is part of the input (Disser and Matuschke, 2020). However, the complexity of the problem for constant values of $k > 1$ has remained elusive, in part due to structure of the natural LP description preventing the use of the equivalence of optimization and separation. This paper introduces a reduction showing that the basic version of MRF described above encapsulates the seemingly much more general variant where the adversary's choices are constrained to $k$-cliques in a compatibility graph on the arcs of the network. As a consequence of this reduction, we are able to prove the following results: (1) MRF is $N\!P$-hard for any constant number $k > 1$ of failing arcs. (2) When $k$ is part of the input, MRF is $P^{N\!P[\log]}$-hard. (3) The integer version of MRF is $\Sigma_2^P$-hard.