Approximation Algorithms for Perfect Fair-Triangle Packing

2026-08-07

AI Summary

This work introduces and investigates the fair maximum-weight triangle packing problem: given 3n vertices colored red and blue, the goal is to partition them into n vertex-disjoint triangles such that each triangle contains at least one red and one blue vertex, while maximizing the total edge weight. The problem is shown to be NP-hard. The authors propose two polynomial-time approximation algorithms: a deterministic 1/3-approximation algorithm based on matching and maximum-weight [1,2]-factor computation running in O(n³) time, and an improved randomized (16/47 − ε)-approximation algorithm combining random cycle-breaking with maximum-weight matching, which runs in O(n⁴) time. These results substantially advance the tractability frontier for this class of fair combinatorial optimization problems.

Abstract

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

Problem

• Triangle packing
• fairness
• approximation algorithms
• randomized algorithms

Innovation

• fair triangle packing
• approximation algorithms
• randomized algorithms
• fair clustering
• maximum-weight matching