Improved Approximation Algorithms for Cycle and Path Packings

Jingyang Zhao, Mingyu Xiao

Abstract: 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 \(10\ge k\ge 6\). 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.