Improved Approximation Algorithms for Cycle and Path Packings

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 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 center dot (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 center dot (1 - 1/k + 1 k(k-1)) for even k > 5. For metric k-path packing, we improve the approximation ratio from 7/8 center dot (1 - 1/k) to 27k(2)- 48k+16/32k(2)-36k-24 for even 10 >= k >= 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.