Approximation algorithms for cycle and path partitions in complete graphs

Abstract

Given an edge-weighted (metric/general) complete graph with n vertices, where $n\mathrm{mod}k=0$, the maximum weight (metric/general) k-cycle/path partition 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 partition, we improve the previous approximation ratio from $\frac{3}{5}$ to $\frac{7}{10}$ for $k=5$, and from $\frac{7}{8}{(1-\frac{1}{k})}^2$ for $k>5$ to $(\frac{7}{8}-\frac{1}{8k})(1-\frac{1}{k})$ for constant odd $k>5$ and to $\frac{7}{8}(1-\frac{1}{k}+\frac{1}{k(k-1)})$ for even $k>5$. For metric k-path partition, we improve the approximation ratio from $\frac{7}{8}(1-\frac{1}{k})$ to $\frac{27k^2-48k+16}{32k^2-36k-24}$ for $k\in \{6,8,10\}$. For the case of $k=4$, we improve the approximation ratio from $\frac{3}{4}$ to $\frac{5}{6}$ for metric 4-cycle partition, from $\frac{2}{3}$ to $\frac{3}{4}$ for general 4-cycle partition, and from $\frac{3}{4}$ to $\frac{14}{17}$ for metric 4-path partition.