The complexity of recognizing linear systems with certain integrality properties

Abstract

Let A be a 0 − 1 matrix with precisely two 1’s in each column and let 1 be the all-one vector. We show that the problems of deciding whether the linear system $${A{\bf x} \ge {\bf 1}, {\bf x}\ge {\bf 0}}$$ (1) defines an integral polyhedron, (2) is totally dual integral (TDI), and (3) box-totally dual integral (box-TDI) are all co-NP-complete, thereby confirming the conjecture on NP-hardness of recognizing TDI systems made by Edmonds and Giles in 1984.