New Approximation Algorithms for the Minimum Cycle Cover Problem

Given an undirected weighted graph G = (V, E) with nonnegative weight function obeying the triangle inequality, a set {C-1, C-2, ... , C-k} of cycles is called a cycle cover if V subset of U-i=1(k) V (C-i) and its cost is given by the maximum weight of the cycles. The Minimum Cycle Cover Problem aims to find a cycle cover of cost at most lambda with the minimum number of cycles. An O(n(2)) 24/5-approximation algorithm and an O(n(5)) 14/3-approximation algorithm are given by Yu and Liu (Theor Comput Sci 654:45-58, 2016). However, the original proofs for approximation ratios are incomplete. In this paper we first present a corrected simplified analysis on the 24/5-approximation algorithm. Then we give a new O(n(3)) approximation algorithm that achieves the same ratio 14/3 and has much simpler proof on the approximation ratio. Moreover, we derive an improved 32/7-approximation algorithm that runs in O(n(5)).