发表机构
Grainger College of Engineering, University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校格雷inger工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对固定常数$k$的多目标超图最小割问题，设计随机PTAS与两种随机拟多项式时间算法，还得到超图连通性阻断问题的QPTAS。
AI 中文摘要
我们研究多目标超图最小割问题：给定超图$H=(V,E)$和$k$个代价函数$c_1, c_2, \boldsymbol{\rm{...}}, c_k:E\to\boldsymbol{\rm{Z}}_{\boldsymbol{\rm{\textgreater= 0}}}$，目标是找到顶点的非空真子集$U\boldsymbol{\rm{\textsubsetneq}} V$，使得$\boldsymbol{\rm{\textmax}}_{i\boldsymbol{\rm{\textin}} [k]} c_i(\boldsymbol{\rm{\textdelta}}(U))$最小。当$k$作为输入的一部分时，该问题是NP难的（即使在图中也是如此）。我们聚焦于$k$为固定常数的情形（例如$k=1,2,3,\boldsymbol{\rm{\textdots}}$）。单目标超图最小割以及目标数为常数的多目标图最小割均存在多项式时间算法。与这些特殊情形不同，即使$k=2$，多目标超图最小割的复杂度仍未解决。由于图与超图的结构差异，现有技术无法扩展。对于$k$为固定常数的$k$目标超图最小割，我们设计了一个随机PTAS，以及两种不同的随机拟多项式时间算法。作为我们的2目标超图最小割结果的应用，我们得到了超图连通性阻断问题的拟多项式时间近似方案（QPTAS）。本工作中使用AI工具对算法思路进行迭代和优化。
英文摘要
We study the multiobjective hypergraph min-cut problem: Given a hypergraph $H=(V,E)$ and $k$ cost functions $c_1, c_2, \ldots, c_k:E\to\mathbb{Z}_{\ge 0}$, the goal is to find a non-empty proper subset $U\subsetneq V$ of vertices with minimum $\max_{i\in [k]} c_i(δ(U))$. When $k$ is part of input, the problem is NP-hard (even in graphs). We focus on fixed-constant $k$ setting (e.g., $k=1, 2, 3, \ldots$). Single-objective hypergraph min-cut as well as multiobjective graph min-cut for a constant number of objectives admit polynomial-time algorithms. In contrast to these special cases, the complexity of multiobjective hypergraph min-cut remains open even for $k=2$. Known techniques fail to extend due to structural differences between graphs and hypergraphs. For $k$-objective hypergraph min-cut when $k$ is a fixed constant, we design a randomized PTAS, and two different randomized quasi-polynomial time algorithms. As an application of our $2$-objective hypergraph min-cut results, we obtain a quasi-polynomial time approximation scheme (QPTAS) for hypergraph connectivity interdiction. AI tools were used to iterate and refine the algorithmic ideas underlying this work.
