Given an edge-weighted complete graph G on 3n vertices, the maximum-weight triangle packing problem asks for a collection of n vertex-disjoint triangles in G such that the total weight of edges in these n triangles is maximized. Although the problem has been extensively studied in the literature, it is surprising that prior to this work, no nontrivial approximation algorithm had been designed and analyzed for its metric case, where the edge weights in the input graph satisfy the triangle inequality. In this paper, we design the first nontrivial polynomial-time approximation algorithm for the maximum-weight metric triangle packing problem. Our algorithm is randomized and achieves an expected approximation ratio of \(0.66768 - \epsilon \) for any constant \(\epsilon > 0\).

给定一个在3 n个顶点上具有边加权的完整图G，最大权重三角形堆积问题要求在G中收集n个不相交的三角形，以使这n个边的总权重三角形最大化。尽管该问题已在文献中进行了广泛研究，但令人惊讶的是，在此工作之前，还没有针对其度量情况设计和分析非平凡的近似算法，其中输入图中的边权重满足三角形不等式。在本文中，我们设计了第一个非平凡的多项式时间近似算法来解决最大权重度量三角形堆积问题。我们的算法是随机的，并且对于任何常数\（\ epsilon> 0 \）都能达到预期的近似值\（0.66768-\ epsilon \）。