Approximation algorithms for some min–max postmen cover problems

We investigate two min–max k-postmen cover problems. The first is the Min–Max Rural Postmen Cover Problem (RPC), in which we are given an undirected weighted graph and the objective is to find at most k closed walks, covering a required subset of edges, to minimize the weight of the maximum weight closed walk. The other is called the Min–Max Chinese Postmen Cover Problem, in which the goal is to find at most k closed walks, covering all the edges of an undirected weighted graph, to minimize the weight of the maximum weight closed walk. For both problems we propose the first constant-factor approximation algorithms with ratios 10 and 4, respectively. For the Metric RPC, a special case of the RPC with the edge weights obeying the triangle inequality, we obtain an improved 6-approximation algorithm by a matching-based approach. For the Min–Max Rural Postmen Walk Cover Problem (RPWC), a variant of the RPC with the closed walks replaced by (open) walks, we give a 5-approximation algorithm that improves on the previous 7-approximation algorithm. If k is fixed, we devise improved approximation algorithms for the Metric RPC and the RPWC with ratios $$4+\epsilon $$ 4 + ϵ and $$3+\epsilon $$ 3 + ϵ , respectively, where $$\epsilon >0$$ ϵ > 0 is an arbitrary small constant. The latter result improves on the existing $$(4+\epsilon )$$ ( 4 + ϵ ) -approximation algorithm. Moreover, we develop a $$(3+\epsilon )$$ ( 3 + ϵ ) -approximation algorithm for a special case of the RPC with fixed k, improving on the previous $$(4+\epsilon )$$ ( 4 + ϵ ) -approximation algorithm.