Improved Hardness Results of the Cardinality-Based Minimum s-t Cut Problem in Hypergraphs

In hypergraphs an edge that crosses a cut 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. Assuming the costs $w_i$ are polynomially bounded by the input size and $w_0=0$ and $w_1=1$, we show that if the costs satisfy $w_i > w_{i-j}+w_{j}$ for some $i \in \{2, \ldots \floor*{n/2}\}$ and $j \in \{1,\ldots,\floor*{i/2}\}$, then the problem becomes NP-hard. Our result also holds for $k$-uniform hypergraphs with $k \geq 4$. Additionally, we show that the \textsc{No-Even-Split} problem in $4$-uniform hypergraphs is NP-hard.