Approximation algorithms for cycle and path partitions in complete graphs

Jingyang Zhao, Mingyu Xiao

Abstract: Given an edge-weighted (metric/general) complete graph with n vertices, where nmodk=0, the maximum weight (metric/general) k-cycle/path partition problem is to find a set of nk 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 35 to 710 for k=5, and from 78(1−1k)2 for k>5 to (78−18k)(1−1k) for constant odd k>5 and to 78(1−1k+1k(k−1)) for even k>5. For metric k-path partition, we improve the approximation ratio from 78(1−1k) to 27k2−48k+1632k2−36k−24 for k∈{6,8,10}. For the case of k=4, we improve the approximation ratio from 34 to 56 for metric 4-cycle partition, from 23 to 34 for general 4-cycle partition, and from 34 to 1417 for metric 4-path partition.