Stronger Hardness for Maximum Robust Flow and Randomized Network Interdiction

TL;DR

This work settles the complexity of Maximum Robust Flow (MRF) and Randomized Network Interdiction (RNI) for constant interdiction budgets and in the general case. It introduces Maximum Robust Flow with Restricted Interdiction (MRF-R) and proves a polynomial reduction sequence MRF-R^* → MRF-M^* → MRF^*, preserving the interdiction budget and integral properties, to transfer hardness from restricted-interdiction variants to the standard problem. The authors establish NP-hardness for any fixed k>1 in the decision variant, coNP-hardness for the randomized interdiction counterpart, P^{NP[log]}-hardness when k is part of the input, and Σ_2^P-hardness for the integral version, placing MR F/RNI outside NP and coNP and suggesting strong limits on compact formulations. These results deepen our understanding of robustness in network flows and highlight the inherent complexity even for seemingly small interdiction budgets, with implications for LP formulations and the intractability of compact MILP encodings in this domain.

Abstract

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.

Figures (6)

• Figure 1: Overview of the reductions presented in this paper.
• Figure 2: Illustration of the reduction from Fractional Graph Coloring (graph on the left) to $\textup{MRF-R}^{\star}$ with $k = 2$ (digraph on the right). The compatibility graph of the constructed $\textup{MRF-R}^{\star}$ instance has the edges $\{a_{1v}, a_{3v}\}$, $\{a_{1w}, a_{2w}\}$, and $\{a_{2r}, a_{3r}\}$.
• Figure 3: From the $\textup{MRF-R}^{\star}$ instance with the digraph on the left and compatibility graph with edges $\{a_1, a_3\}, \{a_1, a_4\}, \{a_2, a_3\}$, we obtain the $\textup{MRF-M}^{\star}$ instance depicted on the right. The solid arcs represent the network $D'$, the dotted arcs indicate demands, i.e., a dotted arc $(s_i, t_i)$ represents a commodity $i$ with source $s_i$, sink $t_i$, and demand $d_i$ as indicated on the label (we omit commodity $0$). Capacities are $1$ except for the four arcs $(v^+_{a_1}, v^-_{a_1}), \dots, (v^+_{a_4}, v^-_{a_4})$, each of which has capacity $M = 4$.
• Figure 4: Illustration of the reduction from $\textup{MRF-M}^{\star}$ with digraph $D$ and four commodities to $\textup{MRF}$. Dashed arcs represent immune arcs, with the label indicating the capacity. Single-lined solid arcs represent regular arcs, with the label indicating the capacity. Double-lined arcs labeled $\ell \times c$ represent bundles of $\ell$ arcs with capacity $c$ each. The figure omits the internal nodes and arcs of the digraph $D$.
• Figure 5: Illustration of the construction from the proof of \ref{['lem:clique-flow']}. The figure on the left shows the compatibility graph $\bar{H}$ of the constructed instance, which consists of the graph $G$ from the clique instance $J^{\textsc{Clique}}$ with nodes $v_1, \dots, v_m$ and the compatibility graph $H'$ of the $\textup{MRF}$ instance $J^{\textsc{Flow}}$ (the edges of $G$ and $H'$ are omitted in the figure). The figure on the right shows the digraph $\bar{D}$ of the constructed instance, which consists of the digraph $D'$ of $J^{\textsc{Flow}}$ (the arcs of $D'$ are omitted in the figure) and the arcs $a_v$ for each node $v$ of $G$.

Theorems & Definitions (50)

• theorem 1
• theorem 2
• theorem 3
• theorem 4
• theorem 5
• theorem 6: lund1994hardness
• lemma 1
• proof
• theorem 7
• lemma 2