A local search 4/3-approximation algorithm for the minimum 3-path partition problem

Given a graph $$G = (V, E)$$ , the 3-path partition problem is to find minimum collection of vertex-disjoint paths each order at most 3 cover all vertices V. The previous best approximation algorithm for has performance ratio 13/9, which based on simple local search strategy. We propose more involved and show by an amortized analysis that it 4/3-approximation; we also design instance illustrate tight.