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Approximation Algorithms for the Min–Max Mixed Rural Postmen Cover Problem and Its Variants

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Abstract

In this work, we introduce a multi-vehicle (or multi-postman) extension of the classical Mixed Rural Postman Problem, which we call the Min–Max Mixed Rural Postmen Cover Problem (MRPCP). The MRPCP is defined on a mixed graph \(G=(V,E,A)\), where V is the vertex set, E denotes the (undirected) edge set and A represents the (directed) arc set. Let \(F\subseteq E\) (\(H\subseteq A\)) be the set of required edges (required arcs). There is a nonnegative weight associated with each edge and arc. The objective is to determine no more than k closed walks to cover all the required edges in F and all the required arcs in H such that the weight of the maximum weight closed walk is minimized. By replacing closed walks with (open) walks in the MRPCP, we obtain the Min–Max Mixed Rural Postmen Walk Cover Problem (MRPWCP). The Min–Max Mixed Chinese Postmen Cover Problem (MCPCP) is a special case of the MRPCP where \(F=E\) and \(H=A\). The Min–Max Stacker Crane Cover Problem (SCCP) is another special case of the MRPCP where \(F=\emptyset \) and \(H=A\) For the MRPCP with the input graph satisfying the weakly symmetric condition, i.e., for each arc there exists a parallel edge whose weight is not greater than this arc, we devise a \(\frac{27}{4}\)-approximation algorithm. This algorithm achieves an approximation ratio of \(\frac{33}{5}\) for the SCCP with the weakly symmetric condition. Moreover, we obtain the first 5-approximation algorithm (4-approximation algorithm) for the MRPWCP (MCPCP) with the weakly symmetric condition.

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Notes

  1. For the reverse direction, we simply replace each non-required edge \([u',v']\in E_{1}\), where \(u'\) (\(v'\)) is introduced by \(u\in V(F\cup H)\) (\(v\in V(F\cup H)\)), by a shortest undirected path P[u, v] in G.

  2. The equality is due to \(V=V(F\cup H)\) and the third property of reduced graphs and the inequality follows by our assumption in Sect. 2.

  3. Strictly speaking, we should replace \(L_j\) with \(L_j^i\). However, we omit the index i for simplicity.

  4. In fact, we need to orient the parallel edges such that their directions are opposite to the corresponding arcs in \(A(W){\setminus } \{e\}\).

  5. \(G(L_{j})\) satisfies the weakly symmetric condition by the fifth property of reduced graphs. Moreover, \(H_j=A_{j}\) follows from the fact that \(G(L_{j})\) satisfies the second property of reduced graphs.

  6. For example, if \(u=f_i\) corresponding to the edge \(e_i=[v_i^1,v_i^2]\), \(v\in V_j^2{\setminus } D(F'_j)\) and \({\hat{w}}[u,v]=l[v_i^1,v]\), we add \(P[v_i^1,v]\) to \(F_j\cup H_j\).

  7. By definition, only edges in \(F_j\) may have odd degree in \({\tilde{Q}}_{j}^{0}\). Moreover, the two end vertices of each edge in \(F_j\) have the same parity in \({\tilde{Q}}_{j}^{0}\).

  8. Since \(i\le k\), the first two steps of Algorithm TourAllocation take \(O(k^2)\) time. The last two steps of Algorithm TourAllocation take \(O(\sum _{j=1}^i (|E(C_j) |+|A(C_j) |))=O(|E |+ |A |)=O(n^2)\) time.

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Acknowledgements

The authors are very grateful to the anonymous referees for their valuable and constructive comments, which significantly improve the presentation of the paper. This research is supported by the National Natural Science Foundation of China under Grant Nos. 11671135, 11871213 and the Natural Science Foundation of Shanghai under Grant No. 19ZR1411800.

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  1. School of Mathematics, East China University of Science and Technology, 130 Meilong Road, Shanghai, 200237, China

    Liting Huang, Wei Yu & Zhaohui Liu

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LH and WY wrote the main manuscript. All authors reviewed the manuscript.

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Huang, L., Yu, W. & Liu, Z. Approximation Algorithms for the Min–Max Mixed Rural Postmen Cover Problem and Its Variants. Algorithmica 86, 1135–1162 (2024). https://doi.org/10.1007/s00453-023-01187-z

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