The complexity of recognizing linear systems with certain integrality properties
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 Ax >= 1, x >= 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.
linear system polyhedron total dual integrality np-hardness