On Sub-determinants and the Diameter of Polyhedra |
| |
Authors: | Nicolas Bonifas Marco Di Summa Friedrich Eisenbrand Nicolai Hähnle Martin Niemeier |
| |
Institution: | 1. LIX, école Polytechnique, Palaiseau, France 2. IBM, Gentilly, France 3. Dipartimento di Matematica, Università di Padova, Padua, Italy 4. école Polytechnique Fédérale de Lausanne, Lausanne, Switzerland 5. University of Bonn, Bonn, Germany 6. Technische Universit?t Berlin, Berlin, Germany
|
| |
Abstract: | We derive a new upper bound on the diameter of a polyhedron \(P = \{x {\in } {\mathbb {R}}^n :Ax\le b\}\) , where \(A \in {\mathbb {Z}}^{m\times n}\) . The bound is polynomial in \(n\) and the largest absolute value of a sub-determinant of \(A\) , denoted by \(\Delta \) . More precisely, we show that the diameter of \(P\) is bounded by \(O(\Delta ^2 n^4\log n\Delta )\) . If \(P\) is bounded, then we show that the diameter of \(P\) is at most \(O(\Delta ^2 n^{3.5}\log n\Delta )\) . For the special case in which \(A\) is a totally unimodular matrix, the bounds are \(O(n^4\log n)\) and \(O(n^{3.5}\log n)\) respectively. This improves over the previous best bound of \(O(m^{16}n^3(\log mn)^3)\) due to Dyer and Frieze (Math Program 64:1–16, 1994). |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|