Improved approximation algorithms for the k-path partition problem

The k-path partition problem (kPP), defined on a graph \(G=(V,E)\), is a well-known NP-hard problem when \(k\ge 3\). The goal of the kPP is to find a minimum collection of vertex-disjoint paths to cover all the vertices in G such that the number of vertices on each path is no more than k. In this paper, we give two approximation algorithms for the kPP. The first one, called Algorithm 1, uses an algorithm for the (0,1)-weighted maximum traveling salesman problem as a subroutine. When G is undirected, the approximation ratio of Algorithm 1 is \(\frac{k+12}{7} -\frac{6}{7k} \), which improves on the previous best-known approximation algorithm for every \(k\ge 7\). When G is directed, Algorithm 1 is a \(\left( \frac{k+6}{4} -\frac{3}{4k}\right) \)-approximation algorithm, which improves the existing best available approximation algorithm for every \(k\ge 10\). Our second algorithm, i.e. Algorithm 2, is a local search algorithm tailored for the kPP in undirected graphs with small k. Algorithm 2 improves on the approximation ratios of the best available algorithm for every \(k=4,5,6\). Combined with Algorithms 1 and 2, we have improved the approximation ratio for the kPP in undirected graphs for each \(k\ge 4\) as well as the approximation ratio for the kPP in directed graphs for each \(k\ge 10\). As for the negative side, we show that for any \(\epsilon >0\) it is NP-hard to approximate the kPP (with k being part of the input) within the ratio \(O(k^{1-\epsilon })\), which implies that Algorithm 1 is asymptotically optimal.