严格不友好 $k$-划分：尖锐度阈值与基于 ETH 的下界
Strictly Unfriendly $k$-Partitions: Sharp Degree Thresholds and ETH-Based Lower Bounds
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中文总结 AI 辅助
该研究对严格不友好 k-划分问题进行了完整复杂度分类，确立最大度阈值（Δ≤2 可解，Δ=3 即 NP-hard 且 ETH-hard），并通过归约给出细粒度下界，揭示稀疏性代价。
中文摘要 AI 辅助
我们针对严格不友好 $k$-划分问题（$\text{SU}k\text{P}$）提出了完整的复杂度分类和细粒度分析，该问题询问能否将图的顶点划分为 $k$ 个类别，使得每个顶点在其他 $k-1$ 个类别中的邻居数严格多于其自身类别中的邻居数。我们首先建立了关于最大度 $\Delta$ 的尖锐可解性-不可解性阈值：对于 $k \in \{2, 3\}$，当 $\Delta \le 2$ 时，$\text{SU}k\text{P}$ 可在多项式时间内求解，但在次立方图（$\Delta = 3$）上立即变为 $\textbf{NP}$-难和 ETH-难，解决了先前工作中的度限制问题，并将次立方图确立为不可解性的精确前沿。此外，在指数时间假设（ETH）下，我们通过从 $(3,3)$-SAT 的直接归约建立了首个细粒度下界。在一般图上，我们为所有划分参数 $k \ge 2$ 建立了统一下界，揭示了显著复杂度收敛性，即尽管在构件构造上存在技术分歧，核心指数复杂度保持不变。在次立方图上，我们正式量化了“稀疏性代价”，推导出显式下界常数，以展示强制结构度限制如何降低归约效率。
英文摘要
We present a complete complexity classification and fine-grained analysis for the Strictly Unfriendly $k$-Partition problem ($\text{SU}k\text{P}$), which asks whether the vertices of a graph can be partitioned into $k$ classes such that every vertex has strictly more neighbors in each of the other $k-1$ classes than in its own. We first establish a sharp tractability-intractability threshold with respect to the maximum degree $Δ$: for $k \in \{2, 3\}$, $\text{SU}k\text{P}$ is solvable in polynomial time when $Δ\le 2$, but becomes $\mathbf{NP}$-hard and ETH-hard immediately on subcubic graphs ($Δ= 3$), resolving the degree limitations in prior work and establishing subcubic graphs as the precise frontier of intractability. Furthermore, under the Exponential Time Hypothesis (ETH), we establish the first fine-grained lower bounds via direct reductions from $(3,3)$-SAT. On general graphs, we establish a uniform lower bound across all partition parameters $k \ge 2$, revealing a striking complexity convergence where the core exponential complexity remains invariant despite technical divergences in gadget constructions. On subcubic graphs, we formally quantify the "cost of sparsity," deriving explicit lower bound constants to demonstrate how enforced structural degree restrictions degrade reduction efficiency.
发表机构
- School of Computing, National University of Singapore（新加坡国立大学计算机学院）
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