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

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 -> R+ that satisfies the triangle inequality, an interdiction cost function c : E -> Z(+), a fixed depot v(1) is an element of V, an integer k is an element of Z(+) and a budget B is an element of N, we are asked to find a subset S-k subset of E such that c(S-k) = Sigma(e is an element of Sk) c(e) <= B and that the subgraph G\S-k is connected, the objective is to minimize the value minC(E\Sk) max{w(C-i) vertical bar C-i is an element of C-E\Sk} among such all aforementioned subsets S-k, where C-E\S-k is a set of k-tours (of G\S-k) starting and ending at the depot v1, jointly traversing each edge in G\S-k at least once, and w(C-i) = Sigma(e is an element of Ci) w(e) for each tour C-i is an element of C-E\Sk. We obtain the following main results: (1) Given an alpha-approximation algorithm to solve theminimization knapsack problem, we design an (alpha + beta)-approximation algorithm to solve the k-CPIBC problem, where beta = 7/2 - 1/k - left perpendicular1/kRIGHT perpendicular. (2) We present a beta-approximation algorithm to solve the special version of the k-CPIBC problem, where c(e) = 1 for each edge e in G and beta is defined in (1).