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

Abstract

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.