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 (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. Supported in part by NSA grant H98230-05-1-0081 and NSF grants DMS-0556091 and ITR-0326387. Supported in part by the Research Grants Council of Hong Kong and Seed Funding for Basic Research of HKU.