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

TL;DR

This paper resolves the complexity of cardinality-based hypergraph s-t cuts by showing NP-hardness for all non-submodular penalty vectors (except the trivial $w_1=0$ case) and extends the hardness to $r\ge 4$, including the $4$-uniform case with $w_2>2$ and the No-Even-Split variant. It then develops a projection-based approximation framework that replaces non-submodular penalties with nearby submodular ones, proving that this approach yields the best possible approximation factor among projection-based strategies. The authors further establish approximation-hardness bounds under the Unique Games Conjecture and, in the regime $P\neq NP$, asymptotically tight inapproximability results, along with convex-hull-based methods that compute an optimal projection in $O(q)$. Collectively, the results significantly sharpen the tractability and approximation landscape for hypergraph cardinality-based s-t cuts and connect them to VCSP dichotomies and LP relaxations. The methods have potential implications for hypergraph clustering and related partitioning tasks where non-submodular penalties capture meaningful cut penalties.

Abstract

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 $\text{P} \neq \text{NP}$.

Figures (8)

• Figure 1: Submodular (shaded in blue) and NP-hard (shaded in gray) regions of $\textsc{CBcut}(r, \mathbf{w})$ for different values of $r$ and fixed $w_1 = 1$.
• Figure 2: Heatmaps of approximation guarantees obtained for $\textsc{CBcut}(r, \mathbf{w})$ 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 3: Heatmaps showing the difference in approximation factors obtained using different types of projection techniques.
• Figure 4: Convex hull $h$ (shown in solid blue) for the integer function $\mathbf{w}$, with the discrete values of $\mathbf{w}(i)$ marked in red. The blue line segments for each interval $[i,i+1]$ represent the linear pieces of $h$, while the largest gap between $h(i)$ and $\mathbf{w}(i)$ (here at $i=3$) is denoted by the approximation bound $\rho$. The red dashed line interpolates $\mathbf{w}$; the fact that it is not a concave function indicates that $\mathbf{w}$ does not define splitting penalties for a submodular function.
• Figure 5: Integrality gap instance of $\textsc{EdgedCBcut}(4,(w_1=1,w_2<1))$

Theorems & Definitions (35)

• Theorem 1: Corollary 2 cohen2004complete
• Lemma 2
• proof
• Theorem 3
• proof : Direct proof of Theorem \ref{['thm:nphard']}
• Lemma 4
• proof
• proof : Proof of Theorem \ref{['thm:nphard']} via VSCP equivalence
• Lemma 5
• proof