AI 中文总结
本文针对首个被研究的NP难问题完美公平三角形打包（PFTP），提出了基于匹配的确定性1/3近似算法与含最大权[1,2]-因子等技术的随机(16/47-ε)近似算法。
AI 中文摘要
本文研究完美公平三角形打包问题（缩写为PFTP），该问题将公平聚类中的公平准则融入最大权三角形打包问题。具体而言，输入是顶点数|V|=3n的边赋权完全图G=(V,E)，每个顶点被染为红色或蓝色。公平三角形是包含两种颜色顶点的三角形，PFTP要求将V划分为n个公平三角形，使得总边权最大化。据我们所知，这是第一篇研究PFTP的论文，PFTP是NP难问题。我们的主要贡献是一个运行时间为O(n³)的确定性1/3近似算法，以及一个运行时间为O(n⁴)的改进型随机(16/47-ε)近似算法，其中ε>0为固定小常数。确定性算法基于匹配，而随机算法采用了若干额外技术，包括最大权[1,2]-因子、随机破环过程和最大权匹配。
英文摘要
In this paper, we study the {\em perfect fair-triangle packing} problem (abbreviated as PFTP), which incorporates the fairness criterion from {\em fair clustering} into the {\em maximum-weight triangle packing} problem. Specifically, the input is an edge-weighted complete graph $G = (V, E)$ with $|V| = 3n$, where each vertex is colored red or blue. A {\em fair triangle} is a triangle containing vertices of both colors. PFTP asks for a partition of $V$ into $n$ fair triangles such that the total edge weight is maximized. To the best of our knowledge, this is the first paper to study PFTP. PFTP is NP-hard. Our main contributions are a deterministic $\frac 13$-approximation algorithm running in $O(n^3)$ time and an improved randomized $(\frac {16}{47}-ε)$-approximation algorithm running in $O(n^4)$ time, where $ε> 0$ is a fixed small constant. The deterministic algorithm is matching-based whereas the randomized algorithm employs several additional techniques, including maximum-weight $[1, 2]$-factor, a random cycle-breaking procedure, and maximum-weight matchings. Keywords: Triangle packing; fairness; approximation algorithms; randomized algorithms
