Approximation Algorithms for Solving the k-Chinese Postman Problem Under Interdiction Budget Constraints

Abstract

In this paper, we address the k-Chinese postman problem under interdiction budget constraints (the k-CPIBC problem, for short), which is a further generalization of the k-Chinese postman problem and has many practical applications in real life. Specifically, given a weighted graph $G=(V,E;w,c;v_{1})$ equipped with a weight function $w:E\rightarrow {\mathbb {R}}^{+}$ that satisfies the triangle inequality, an interdiction cost function $c:E\rightarrow {\mathbb {Z}}^{+}$, a fixed depot $v_{1}\in V$, an integer $k\in {\mathbb {Z}}^{+}$ and a budget $B\in {\mathbb {N}}$, we are asked to find a subset $S_{k}\subseteq E$ such that $c(S_{k})=\sum _{e\in S_{k}}c(e)\leqslant B$ and that the subgraph $G\backslash S_{k}$ is connected, the objective is to minimize the value $\min _{\mathcal{C}_{E\backslash S_{k}}} \max \{w(C_{i})\,\vert \,C_{i}\in \mathcal{C}_{E\backslash S_{k}}\}$ among such all aforementioned subsets $S_{k}$, where $\mathcal{C}_{E\backslash S_{k}}$ is a set of k-tours (of $G\backslash S_{k}$) starting and ending at the depot $v_{1}$, jointly traversing each edge in $G\backslash S_{k}$ at least once, and $w(C_{i})=\sum _{e\in C_{i}}w(e)$ for each tour $C_{i}\in \mathcal{C}_{E\backslash S_{k}}$. We obtain the following main results: (1) Given an $\alpha$-approximation algorithm to solve the minimization knapsack problem, we design an $(\alpha +\beta )$-approximation algorithm to solve the k-CPIBC problem, where $\beta$ $=$ $\frac{7}{2}-\frac{1}{k}-\lfloor \frac{1}{k}\rfloor$. (2) We present a $\beta$-approximation algorithm to solve the special version of the k-CPIBC problem, where c(e) $\equiv$ 1 for each edge e in G and $\beta$ is defined in (1).