Approximation algorithms for solving the constrained arc routing problem in mixed graphs

Given a mixed graph G with vertex set V, let E and A denote the sets of edges and arcs, respectively. We use Q+ and Z+ to denote the sets of positive rational numbers and positive integers, respectively. For any connected mixed graph G=(V,E∪A;w;l,u) with a length function w:E∪A→Q+ and two integer functions l,u:E∪A→Z+ satisfying l(e)⩽u(e) for each e∈E∪A, we are asked to determine a minimum length tour T traversing each e∈E∪A at least l(e) and at most u(e) times. This new constrained arc routing problem generalizes the mixed Chinese postman problem. Let n=|V| and m=|E∪A| denote the number of vertices and edges (including arcs), respectively. Using network flow techniques, we design a (1+1/l0)-approximation algorithm in time O(n2m3logn) to solve this constrained arc routing problem such that l(e)<u(e) holds for each edge e∈E,l(e)⩽u(e) holds for each arc e∈A and l0=min{l(e)|e∈E}. In addition, we present two optimal combinatorial algorithms in times O(n3) and O(nm2logn) to solve this problem for the cases A=∅ and E=∅, respectively.