Improved approximation algorithms for some min-max and minimum cycle cover problems

Given an undirected weighted graph G=(V,E), a set {C1,C2,…,Ck} of cycles is called a cycle cover of the vertex subset V′ if V′⊆∪i=1kV(Ci) and its cost is given by the maximum weight of the cycles. The Min-Max Cycle Cover Problem (MMCCP) is to find a minimum cost cycle cover of V with at most k cycles. The Rooted Min-Max Cycle Cover Problem (RMMCCP) is to find a minimum cost cycle cover of V∖D with at most k cycles, each of which contains one vertex in D. The Minimum Cycle Cover Problem (MCCP) aims to find a cycle cover of V of cost at most λ with the minimum number of cycles. We propose approximation algorithms for MMCCP and RMMCCP with performance ratios 5 and 6, respectively. These results improve the previous algorithms in term of both approximation ratios and running times. For MCCP we obtain a 143-approximation algorithm that has the same time complexity as the previous best 5-approximation algorithm. Moreover, we transform a ρ-approximation algorithm for TSP into approximation algorithms for MMCCP, RMMCCP and MCCP with ratios 4ρ, 4ρ+1 and 4ρ, respectively.