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) is 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. Key words: linear system, polyhedron, total dual integrality, NP-hardness. 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. 1 

• Type: Journal
• Year: 2008
• Where: MP
• Authors: Guoli Ding, Li Feng, Wenan Zang
• Keywords: Integral, Linear System, Linear System Ax, MP 2008