亚线性MPC中的固定阈值剥离：轮次-近似率权衡及应用
Fixed-Threshold Peeling in Sublinear MPC: Round-Approximation Tradeoffs and Applications
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中文总结 AI 辅助
本文在亚线性MPC模型中，针对依赖密度的边定向、着色、densest子图及k-核分解问题，提出固定阈值剥离方法，实现轮次-近似率权衡，突破了此前部分问题的轮次复杂度壁垒并改进了近似率。
中文摘要 AI 辅助
许多基础图问题可基于迭代剥离实现简单算法：反复移除当前度数低于固定阈值的所有顶点。该范式是依赖密度的边定向、依赖密度的着色、 densest subgraph（ densest子图）及k-核分解算法的基础。本文在亚线性MPC模型中研究这些问题，实现了如下轮次-近似率权衡：对于依赖密度的边定向，给定任意整数t>0，我们计算出最大出度至多为(2+ε)(t+1)α(G)的定向，其中α(G)表示图G的定向的最小可能最大出度，所需轮次为O(lg^(1/(t+2)) n · poly(lg lg n))；在poly(lg lg n)轮次范围内，该算法给出O(lg lg n / lg lg lg n)的近似率，改进了Ghaffari与Grunau[PODC 2025]近期工作的近似率。我们对依赖密度的着色也获得了类似改进。对于densest subgraph，我们在Õ(lg^(1/3) n) MPC轮次中获得(4+ε)近似率，在Õ(lg^(1/4) n) MPC轮次中获得(6+ε)近似率；这改进了Ghaffari、Lattanzi与Mitrović[ICML 2019]的Õ(√lg n)轮次复杂度，仅近似率略有增大，且是亚线性MPC模型中首个突破Θ(√lg n)轮次复杂度壁垒的O(1)近似率densest subgraph算法。对于k-核分解，给定任意整数t>0，我们在O(lg^(1/(t+2)) n · poly(lg lg n)) MPC轮次中计算出因子为(2+ε)(t+1)的近似核度，同样改进了Ghaffari、Lattanzi与Mitrović[ICML 2019]的Õ(√lg n)轮次复杂度，再次实现了轮次-近似率权衡。
英文摘要
A number of fundamental graph problems admit simple algorithms based on iterative peeling: repeatedly remove all vertices whose current degree is below a fixed threshold. This paradigm underlies algorithms for density-dependent edge orientation, density-dependent coloring, densest subgraph, and $k$-core decomposition. In this paper, we study these problems in the sub-linear MPC model and achieve the following round-approximation tradeoffs. For density-dependent edge orientation, given any integer $t > 0$, we compute an orientation with maximum out-degree at most $(2+ε)(t+1)α(G)$ in $O(\lg^{1/(t+2)} n \cdot \operatorname{poly}(\lg \lg n))$ rounds, where $α(G)$ denotes the minimum possible maximum out-degree of an orientation of $G$. In the $\operatorname{poly}(\lg\lg n)$-round regime, this gives an $O(\lg\lg n/\lg\lg\lg n)$-approximation, improving the approximation factor of the recent work by Ghaffari and Grunau [PODC 2025]. We obtain a similar improvement for density-dependent coloring. For densest subgraph, we obtain a $(4+ε)$-approximation in $\widetilde O(\lg^{1/3} n)$ MPC rounds and a $(6+ε)$-approximation in $\widetilde O(\lg^{1/4} n)$ MPC rounds. This improves the $\widetilde O(\sqrt{\lg n})$ round complexity of Ghaffari, Lattanzi, and Mitrović [ICML 2019] with a slightly larger approximation factor. This is the first $O(1)$-approximate algorithm for densest subgraph to break the $Θ(\sqrt{\lg n})$ round-complexity barrier in the sub-linear MPC model. For $k$-core decomposition, given any integer $t > 0$, we compute approximate coreness values within a factor of $(2+ε)(t+1)$ in $O(\lg^{1/(t+2)} n \cdot \operatorname{poly}(\lg \lg n))$ MPC rounds for any integer $t > 0$. This improves the $\widetilde O(\sqrt{\lg n})$ round complexity of Ghaffari, Lattanzi, and Mitrović [ICML 2019], again giving a round-approximation tradeoff.
