Improved Approximation Algorithms for Min-Max and Minimum Vehicle Routing Problems

Given an undirected weighted graph \(G=(V,E)\), a set \(C_1,C_2,\) \(\ldots ,C_k\) of cycles is called a cycle cover of \(V'\) if \(V' \subset \cup _{i=1}^k V(C_i)\) and its cost is 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 \(V\setminus D\) with at most k cycles and each cycle contains one vertex in D. The Minimum Cycle Cover Problem(MCCP) aims to find a cycle cover of V of cost at most \(\lambda \) with minimum number of cycles. We propose approximation algorithms for the MMCCP, RMCCP and MCCP with ratios 5, 6 and 24/5, respectively. Our results improve the previous algorithms in term of both approximation ratios and running times. Moreover, we transform a \(\rho \)-approximation algorithm for the TSP into approximation algorithms for the MMCCP, RMCCP and MCCP with ratios \(4\rho \), \(4\rho +1\) and \(4\rho \), respectively.