An LP-based approximation algorithm for the generalized traveling salesman path problem

The traveling salesman problem (TSP) is one of the classic research topics in the field of operations research, graph theory and computer science. In this paper, we propose a generalized model of traveling salesman problem, denoted by generalized traveling salesman path problem. Let G = (V, E, c) be a weighted complete graph, in which c is a nonnegative metric cost function on edge set E, i.e., c : E -> R+. The traveling salesman path problem aims to find a Hamiltonian path in G with minimum cost. Compared to the traveling salesman path problem, we are given extra vertex subset V ' and edge subset E' in the problem proposed in this paper; its goal is to construct a path which traverses all the edges in E' while only needs to visit each vertex in V ' exactly once. Based on integer programming, we give a mathematical model of the problem, and design a 1+root 5 2 -approximation algorithm for the problem by combining linear programming rounding strategy and a special graph structure.