Improved approximation algorithms for weighted 2-path partitions

We investigate two NP-complete vertex partition problems on edge-weighted complete graphs with 3k vertices. The first problem asks to partition the graph into k vertex disjoint paths of length 2 (referred to as 2-paths) such that the total weight of the paths is maximized. We present a cubic time approximation algorithm that computes a 2-path partition whose total weight is at least.5833 of the weight of an optimal partition, improving upon the (.5265−ϵ)-approximation algorithm of Tanahashi and Chen (2010). Restricting the input to graphs with edge weights in {0,1}, we present a.75 approximation algorithm improving upon the.55-approximation algorithm of Hassin and Schneider (2013). Combining this algorithm with a previously known approximation algorithm for the 3-SET PACKING problem, we obtain a.6-approximation algorithm for the problem of partitioning a {0,1}-edge-weighted graph into k vertex disjoint triangles of maximum total weight. The best known approximation algorithm for general weights is due to Chen and Tanahashi (2009) and achieves an approximation ratio of.5257.