Stronger Hardness for Maximum Robust Flow and Randomized Network Interdiction
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914144470433792 |
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| author | Matuschke, Jannik |
| author_facet | Matuschke, Jannik |
| contents | 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 $Σ_2^P$-hard. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_06505 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stronger Hardness for Maximum Robust Flow and Randomized Network Interdiction Matuschke, Jannik Discrete Mathematics 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 $Σ_2^P$-hard. |
| title | Stronger Hardness for Maximum Robust Flow and Randomized Network Interdiction |
| topic | Discrete Mathematics |
| url | https://arxiv.org/abs/2511.06505 |