Given an edge-weighted (metric/general) complete graph with $n$ vertices, the maximum weight (metric/general) $k$-cycle/path packing problem is to find a set of $\frac{n}{k}$ vertex-disjoint $k$-cycles/paths such that the total weight is maximized. In this paper, we consider approximation algorithms. For metric $k$-cycle packing, we improve the previous approximation ratio from $3/5$ to $7/10$ for $k=5$, and from $7/8\cdot(1-1/k)^2$ for $k>5$ to $(7/8-0.125/k)(1-1/k)$ for constant odd $k>5$ and to $7/8\cdot (1-1/k+\frac{1}{k(k-1)})$ for even $k>5$. For metric $k$-path packing, we improve the approximation ratio from $7/8\cdot (1-1/k)$ to $\frac{27k^2-48k+16}{32k^2-36k-24}$ for even $10\geq k\geq 6$. For the case of $k=4$, we improve the approximation ratio from $3/4$ to $5/6$ for metric 4-cycle packing, from $2/3$ to $3/4$ for general 4-cycle packing, and from $3/4$ to $14/17$ for metric 4-path packing.

翻译：给定一个具有 $n$ 个顶点的边加权（度量/一般）完全图，最大权重（度量/一般）$k$-圈/路径打包问题旨在找到一组 $\frac{n}{k}$ 个顶点不相交的 $k$-圈/路径，使得总权重最大化。本文研究该问题的近似算法。对于度量 $k$-圈打包问题，当 $k=5$ 时，我们将先前的近似比从 $3/5$ 提升至 $7/10$；当 $k>5$ 时，对于常数奇数 $k>5$，近似比从 $7/8\cdot(1-1/k)^2$ 提升至 $(7/8-0.125/k)(1-1/k)$，对于偶数 $k>5$，提升至 $7/8\cdot (1-1/k+\frac{1}{k(k-1)})$。对于度量 $k$-路径打包问题，当 $10\geq k\geq 6$ 且 $k$ 为偶数时，近似比从 $7/8\cdot (1-1/k)$ 提升至 $\frac{27k^2-48k+16}{32k^2-36k-24}$。对于 $k=4$ 的情形，我们将度量 4-圈打包的近似比从 $3/4$ 提升至 $5/6$，将一般 4-圈打包的近似比从 $2/3$ 提升至 $3/4$，并将度量 4-路径打包的近似比从 $3/4$ 提升至 $14/17$。