Improved Hardness and Approximations for Cardinality-Based Minimum $s$-$t$ Cuts Problems in Hypergraphs

In hypergraphs, an edge that crosses a cut (i.e., a bipartition of nodes) can be split in several ways, depending on how many nodes are placed on each side of the cut. A cardinality-based splitting function assigns a nonnegative cost of $w_i$ for each cut hyperedge $e$ with exactly $i$ nodes on the side of the cut that contains the minority of nodes from $e$. The cardinality-based minimum $s$-$t$ cut aims to find an $s$-$t$ cut with minimum total cost. We answer a recently posed open question by proving that the problem becomes NP-hard outside the submodular region shown by~\cite{veldt2022hypergraph}. Our result also holds for $r$-uniform hypergraphs with $r \geq 4$. Specifically for $4$-uniform hypergraphs we show that the problem is NP-hard for all $w_2 > 2$, and additionally prove that the No-Even-Split problem is NP-hard. We then turn our attention to approximation strategies and approximation hardness results in the non-submodular case. We design a strategy for projecting non-submodular penalties to the submodular region, which we prove gives the optimal approximation among all such projection strategies. We also show that alternative approaches are unlikely to provide improved guarantees, by showing matching approximation hardness bounds assuming the Unique Games Conjecture and asymptotically tight approximation hardness bounds assuming $\text{P} \neq \text{NP}$.