TY - GEN
T1 - Random-edge is slower than random-facet on abstract cubes
AU - Hansen, Thomas Dueholm
AU - Zwick, Uri
N1 - Funding Information:
Thomas Dueholm Hansen was supported by the Carlsberg Foundation, grant no. CF14-0617. Uri Zwick was supported by BSF grant no. 2012338 and by the The Israeli Centers of Research Excellence (I-CORE) program, (Center No. 4/11)
PY - 2016/8/1
Y1 - 2016/8/1
N2 - Random-Edge and Random-Facet are two very natural randomized pivoting rules for the simplex algorithm. The behavior of Random-Facet is fairly well understood. It performs an expected sub-exponential number of pivoting steps on any linear program, or more generally, on any Acyclic Unique Sink Orientation (AUSO) of an arbitrary polytope, making it the fastest known pivoting rule for the simplex algorithm. The behavior of Random-Edge is much less understood. We show that in the AUSO setting, Random-Edge is slower than Random-Facet. To do that, we construct AUSOs of the n-dimensional hypercube on which Random-Edge performs an expected number of 2Ω(√n log n) steps. This improves on a 2√(3√ n) lower bound of Matoušek and Szabó. As Random-Facet performs an expected number of 2O(√n) steps on any n-dimensional AUSO, this established our result. Improving our 2Ω(√n log n) lower bound seems to require radically new techniques.
AB - Random-Edge and Random-Facet are two very natural randomized pivoting rules for the simplex algorithm. The behavior of Random-Facet is fairly well understood. It performs an expected sub-exponential number of pivoting steps on any linear program, or more generally, on any Acyclic Unique Sink Orientation (AUSO) of an arbitrary polytope, making it the fastest known pivoting rule for the simplex algorithm. The behavior of Random-Edge is much less understood. We show that in the AUSO setting, Random-Edge is slower than Random-Facet. To do that, we construct AUSOs of the n-dimensional hypercube on which Random-Edge performs an expected number of 2Ω(√n log n) steps. This improves on a 2√(3√ n) lower bound of Matoušek and Szabó. As Random-Facet performs an expected number of 2O(√n) steps on any n-dimensional AUSO, this established our result. Improving our 2Ω(√n log n) lower bound seems to require radically new techniques.
KW - Acyclic unique sink orientations
KW - Linear programming
KW - Pivoting rules
KW - The simplex algorithm
UR - http://www.scopus.com/inward/record.url?scp=85012873117&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.ICALP.2016.51
DO - 10.4230/LIPIcs.ICALP.2016.51
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AN - SCOPUS:85012873117
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016
A2 - Rabani, Yuval
A2 - Chatzigiannakis, Ioannis
A2 - Sangiorgi, Davide
A2 - Mitzenmacher, Michael
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016
Y2 - 12 July 2016 through 15 July 2016
ER -