Approximation algorithms for solving the vertex-traversing-constrained mixed Chinese postman problem

In this paper, we address the vertex-traversing-constrained mixed Chinese postman problem (the VtcMCP problem), which is a further generalization of the Chinese postman problem, and this new problem has many practical applications in real life. Specifically, given a connected mixed graph \(G=(V, E\cup A; w,b)\) with length function \(w(\cdot )\) on edges and arcs and traversal function \(b(\cdot )\) on vertices, we are asked to determine a tour traversing each link (i.e., either edge or arc) at least once and each vertex v at most b(v) times, the objective is to minimize the total length of such a tour, where \(n=|V|\) is the number of vertices and \(m=|E\cup A|\) is the number of links of G, respectively. We obtain the following four main results. (1) Given any two constants \(\beta \ge 1\) and \(\alpha \ge 1\), we prove that there is no polynomial-time algorithm with approximation ratios \((1,\beta )\) or \((\alpha , 1)\) for solving the VtcMCP problem, where an (h, k)-approximation algorithm for solving the VtcMCP problem is one algorithm that produces a solution with violating the vertex-traversing constraints by at most a ratio of h and with costing at most k times the optimal value; (2) We design a (3, 2)-approximation algorithm \({{\mathcal {A}}}\) to solve the VtcMCP problem in time \(O(m^{2}\log n)\); (3) We prove the fact that this algorithm \({{\mathcal {A}}}\) in (2) is indeed an exact algorithm to optimally solve the VtcMCP problem for the case \(E=\emptyset \); (4) We present an exact algorithm to optimally solve the VtcMCP problem in time \(O(m^{3})\) for the case \(A=\emptyset \).