On the tractability and approximability of non-submodular cardinality-based $s$-$t$ cut problems in hypergraphs

TL;DR

This paper resolves the complexity of cardinality-based $s$-$t$ cuts in hypergraphs by linking them to Boolean VCSPs, proving NP-hardness for all non-submodular penalties except a degenerate zero-cost case, and establishing a clear tractability boundary at submodularity. It introduces optimal approximation strategies by projecting non-submodular penalties into the submodular region, and proves that this projection yields the best possible factors among projection-based methods. For 4-uniform hypergraphs, it also provides tight UGC-based hardness results that match the projection-based guarantees, highlighting fundamental limits. The work advances practical understanding for hypergraph clustering and related tasks where cut penalties are nearly submodular, while outlining avenues for stronger hardness results and improved downstream algorithms.

Abstract

A minimum $s$-$t$ cut in a hypergraph is a bipartition of vertices that separates two nodes $s$ and $t$ while minimizing a hypergraph cut function. The cardinality-based hypergraph cut function assigns a cut penalty to each hyperedge based on the number of nodes in the hyperedge that are on each side of the split. Previous work has shown that when hyperedge cut penalties are submodular, this problem can be reduced to a graph $s$-$t$ cut problem and hence solved in polynomial time. NP-hardness results are also known for a certain class of non-submodular penalties, though the complexity remained open in many parameter regimes. In this paper we highlight and leverage a connection to Valued Constraint Satisfaction Problems to show that the problem is NP-hard for all non-submodular hyperedge cut penalty, except for one trivial case where a 0-cost solution is always possible. 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 it is UGC-hard to obtain a better approximation in the simplest setting where all hyperedges have exactly 4 nodes.

Figures (7)

• Figure 1: Gadgets for reducing MaxCut to 4-CB Hyper-st-Cut when $w_2 < 1$ (left, first shown in veldt2022hypergraph) and $w_2 > 2$ (right).
• Figure 2: Submodular (shaded in blue) and NP-hard (shaded in gray) regions of $r$-CB Hyper-st-Cut for different values of $r$ and fixed $w_1 = 1$.
• Figure 3: Heatmaps of approximation guarantees obtained for $r$-CB Hyper-st-Cut when $r \in \{6,7\}$ for a grid of $(w_2,w_3)$ choices when $w_1=1$ is fixed, using four different techniques for projecting on the submodular region. Within the submodular region (shown by a white triangle), the approximation ratio is $1$ as no projection is required.
• Figure 4: Heatmaps showing the difference in approximation factors obtained using different types of projection techniques.
• Figure 5: An optimal PLC cover $\hat{\textbf{f}}$ (shown in solid blue) for the integer function $\textbf{w}$, with the discrete values of $\textbf{w}(i)$ marked in red. The blue line segments for each interval $[i,i+1]$ represent the linear pieces of $\hat{\textbf{f}}$, while the largest gap between $\hat{\textbf{f}}(i)$ and $\textbf{w}(i)$ (here at $i=3$) is denoted by the approximation bound $\rho$. The red dashed line interpolates $\textbf{w}$; the fact that it is not a PLC function indicates that $\textbf{w}$ does not define splitting penalties for a submodular function.

Theorems & Definitions (19)

• Theorem 2.1: Corollary 2 cohen2004complete
• Lemma 2.2
• proof
• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 4.1
• proof
• Definition 1
• Lemma 4.2