[论文解读] Augmented Sparsifiers for Generalized Hypergraph Cuts.

Austin R. Benson, Jon Kleinberg+1 more|arXiv (Cornell University)|Jul 16, 2020

Complexity and Algorithms in Graphs参考文献 60被引用 4

一句话总结

本文提出了一种稀疏化框架，通过使用基于基数的子模分裂函数对广义超图割进行近似，利用稀疏有向图实现更快的近似最小 $s$-$t$ 割算法。通过使用分段线性曲线近似凹函数，将每个超边减少至 $O(\varepsilon^{-1}|e|\log|e|)$ 条边，显著提升了密集还原方法的运行效率。

ABSTRACT

In recent years, hypergraph generalizations of many graph cut problems have been introduced and analyzed as a way to better explore and understand complex systems and datasets characterized by multiway relationships. Recent work has made use of a generalized hypergraph cut function which for a hypergraph $\mathcal{H} = (V,E)$ can be defined by associating each hyperedge $e \in E$ with a splitting function ${\bf w}_e$, which assigns a penalty to each way of separating the nodes of $e$. When each ${\bf w}_e$ is a submodular cardinality-based splitting function, meaning that ${\bf w}_e(S) = g(|S|)$ for some concave function $g$, previous work has shown that a generalized hypergraph cut problem can be reduced to a directed graph cut problem on an augmented node set. However, existing reduction procedures often result in a dense graph, even when the hypergraph is sparse, which leads to slow runtimes for algorithms that run on the reduced graph. We introduce a new framework of sparsifying hypergraph-to-graph reductions, where a hypergraph cut defined by submodular cardinality-based splitting functions is $(1+\varepsilon)$-approximated by a cut on a directed graph. Our techniques are based on approximating concave functions using piecewise linear curves. For $\varepsilon > 0$ we need at most $O(\varepsilon^{-1}|e| \log |e|)$ edges to reduce any hyperedge $e$, which leads to faster runtimes for approximating generalized hypergraph $s$-$t$ cut problems. For the machine learning heuristic of a clique splitting function, our approach requires only $O(|e| \varepsilon^{-1/2} \log \log \frac{1}{\varepsilon})$ edges. This sparsification leads to faster approximate min $s$-$t$ graph cut algorithms for certain classes of co-occurrence graphs. Finally, we apply our sparsification techniques to develop approximation algorithms for minimizing sums of cardinality-based submodular functions.

研究动机与目标

• 为解决现有超图到图的还原方法生成稠密图、从而拖慢 $s$-$t$ 割算法效率的问题。
• 开发一种稀疏化技术，以更少的边近似广义超图割，达到 $(1+\varepsilon)$ 精度。
• 通过降低超边复杂度，实现在共现图上更快的近似最小 $s$-$t$ 割计算。
• 为最小化基于基数的子模函数之和提供高效的近似算法。

提出的方法

• 使用分段线性曲线近似凹函数，以建模基于基数的子模分裂函数。
• 构建一个增强的有向图，其中每条超边被替换为大小为 $O(\varepsilon^{-1}|e|\log|e|)$ 的稀疏子图。
• 确保最终图在所有节点划分下，其超图割值与原图保持 $1+\varepsilon$ 因子内。
• 将该稀疏化方法应用于团分裂函数，实现每条超边仅需 $O(|e|\varepsilon^{-1/2}\log\log\frac{1}{\varepsilon})$ 条边。
• 利用稀疏图结构加速共现图上的最小 $s$-$t$ 割算法。
• 利用稀疏化还原方法，推导出最小化子模函数之和的近似算法。

实验结果

研究问题

• RQ1 我们能否在保持 $(1+\varepsilon)$ 近似割值的前提下，减少超图到图还原中的边数？
• RQ2 使用基于基数的子模分裂函数时，稀疏化一条超边所需的最少边数是多少？
• RQ3 对于如团分裂等特定函数，稀疏化性能如何变化？
• RQ4 在实际中，稀疏化还原能否加速近似最小 $s$-$t$ 割计算？
• RQ5 该稀疏化框架能否扩展至最小化子模函数之和的问题？

主要发现

• 所提出的方法将每条超边减少至 $O(\varepsilon^{-1}|e|\log|e|)$ 条边，同时保持广义超图割的 $(1+\varepsilon)$ 近似精度。
• 对于团分裂函数，边数减少至 $O(|e|\varepsilon^{-1/2}\log\log\frac{1}{\varepsilon})$，优于一般情况的边界。
• 由于图的稀疏性提升，该稀疏化方法显著加快了共现图上的近似最小 $s$-$t$ 割算法。
• 该框架可实现高效近似算法，用于最小化基于基数的子模函数之和。
• 使用凹函数的分段线性近似，可实现精确且可扩展的割近似。