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 contain's 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(IEllogIEi), and its performance-ratio is ~. Second, we present an algorithm for covering undirected graphs, whose performance-ratio is j. 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 performance­ ratio is j. This algorithm uses a maximum weight bipartite matching algorithm as a subroutine, which dominates the overall complexity of the algorithm.