Approximation algorithms for covering a graph by vertex‐disjoint paths of maximum total weight

We consider the problem of covering a weighted graph G = (V, E) by a set of vertex‐disjoint paths, such that the total weight of these paths is maximized. This problem is clearly NP‐complete, since it contains the Hamiltonian path problem as a special case. Three approximation algorithms for this problem are presented, exhibiting a complexity‐performance trade‐off. First, we develop an algorithm for covering undirected graphs. The time complexity of this algorithm is O(|E|log|E|), and its performance‐ratio is ½. Second, we present an algorithm for covering undirected graphs, whose performance‐ratio is ⅔. This algorithm uses a maximum weight matching algorithm as a subroutine, which dominates the overall complexity of our algorithm. Finally, we develop an algorithm for covering directed graphs, whose performanceratio is ⅔. This algorithm uses a maximum weight bipartite matching algorithm as a subroutine, which dominates the overall complexity of the algorithm.