On the complexity of the edge-disjoint min-min problem in undirected graphs

The min-min problem of finding a disjoint-path pair with the length of the shorter path minimized is known to be NP-complete and admits no K-approximation for any K >; 1 in the general case [1]. In this paper, we show that Bhatia et al [2]´s NP-complete proof, a claim of correction to Xu et al´s proof [1], for the edge-disjoint min-min problem in undirected graphs is incorrect by giving a counter example that is an unsatisfiable 3SAT instance but classified as a satisfiable 3SAT instance in Bhatia et al´s proof [2]. We then give a correct proof of NP-completeness of this problem in undirected graphs.