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.
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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.
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Ding, G., Feng, L. & Zang, W. The complexity of recognizing linear systems with certain integrality properties. Math. Program. 114, 321–334 (2008). https://doi.org/10.1007/s10107-007-0103-y
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DOI: https://doi.org/10.1007/s10107-007-0103-y