An Improved Approximation Algorithm for Metric Triangle Packing

TL;DR

The paper tackles the Maximum Weight Metric Triangle Packing problem on edge-weighted metric complete graphs and achieves a deterministic $\,0.66835-\varepsilon$-approximation by combining three independently constructed triangle packings ($T_1$, $T_2$, $T_3$) and derandomizing the framework via conditional expectations. The core methods include a short cycle packing to guide the first packing, a maximum-weight matching-based second packing, and a novel randomized third packing that can be derandomized; a linear-programming trade-off solidifies the final approximation ratio. This work improves the previous best deterministic and randomized results, providing a practical polynomial-time algorithm with a tight analysis that surpasses the natural $2/3$ barrier. The approach has potential implications for related packing problems and motivates further exploration of deterministic techniques in cycle- and path-based packings.

Abstract

Given an edge-weighted metric complete graph with $n$ vertices, the maximum weight metric triangle packing problem is to find a set of $n/3$ vertex-disjoint triangles with the total weight of all triangles in the packing maximized. Several simple methods can lead to a 2/3-approximation ratio. However, this barrier is not easy to break. Chen et al. proposed a randomized approximation algorithm with an expected ratio of $(0.66768-\varepsilon)$ for any constant $\varepsilon>0$. In this paper, we improve the approximation ratio to $(0.66835-\varepsilon)$. Furthermore, we can derandomize our algorithm.

Figures (4)

• Figure 1: An illustration of the three kinds of triangles, where there are three cycles (the dotted edges) in $\mathcal{C}$ and three triangles (the solid edges) in $\mathcal{B}^*$: $t_1$ is an external triangle, $t_2$ is a partial-external triangle, and $t_3$ is an internal triangle
• Figure 2: An illustration of the oriented cycles (the dotted directed edges) and out-edges (the red dotted directed edges), where there are three out-edges from $t_1$ (an external triangle), one out-edge from $t_2$ (a partial-external triangle), and no out-edges from $t_3$ (an internal triangle)
• Figure 3: An illustration of the type-1 triangle $t_y$ and one of its out-edge $e=yz$, where the dotted directed edges are the edges of cycle $C$
• Figure 4: An illustration of the good triplet of a type-2 triangle $t=xyz$, where the directed dotted edges are the edges of $\mathcal{C}$ and the red directed dotted edge $xx'$ is an out-edge in $E_t$ such that $w(xx')\leq\frac{1}{2}(1-\tau)w(t)$

Theorems & Definitions (26)

• Lemma 1: DBLP:journals/jco/ChenCLWZ21
• Lemma 2
• Lemma 3
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7