Improved approximation algorithms for the k-path partition problem - Li, Shiming; Yu, Wei; Liu, Zhaohui (Journal of global optimization, 2024)
The k -path partition problem (kPP), defined on a graph G = ( V , E ) , is a well-known NP-hard problem when k ≥ 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 k + 12 7 - 6 7 k , which improves on the previous best-known approximation algorithm for every k ≥ 7 . When G is directed, Algorithm 1 is a k + 6 4 - 3 4 k -approximation algorithm, which improves the existing best available approximation algorithm for every k ≥ 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 ≥ 4 as well as the approximation ratio for the kPP in directed graphs for each k ≥ 10 . As for the negative side, we show that for any ϵ > 0 it is NP-hard to approximate the kPP (with k being part of the input) within the ratio O ( k 1 - ϵ ) , which implies that Algorithm 1 is asymptotically optimal.