TY - GEN
T1 - On isoperimetric profiles and computational complexity
AU - Hrubeš, Pavel
AU - Yehudayoff, Amir
N1 - Funding Information: For the first author, the research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Program (FP7/2007-2013) / ERC grant agreement no. 339691. The Institute of Mathematics is supported by RVO:67985840. For the second author, the research is supported by the ISF and BSF
PY - 2016/8/1
Y1 - 2016/8/1
N2 - The isoperimetric profile of a graph is a function that measures, for an integer k, the size of the smallest edge boundary over all sets of vertices of size k. We observe a connection between isoperimetric profiles and computational complexity. We illustrate this connection by an example from communication complexity, but our main result is in algebraic complexity. We prove a sharp super-polynomial separation between monotone arithmetic circuits and monotone arithmetic branching programs. This shows that the classical simulation of arithmetic circuits by arithmetic branching programs by Valiant, Skyum, Berkowitz, and Rackoff (1983) cannot be improved, as long as it preserves monotonicity. A key ingredient in the proof is an accurate analysis of the isoperimetric profile of finite full binary trees. We show that the isoperimetric profile of a full binary tree constantly fluctuates between one and almost the depth of the tree.
AB - The isoperimetric profile of a graph is a function that measures, for an integer k, the size of the smallest edge boundary over all sets of vertices of size k. We observe a connection between isoperimetric profiles and computational complexity. We illustrate this connection by an example from communication complexity, but our main result is in algebraic complexity. We prove a sharp super-polynomial separation between monotone arithmetic circuits and monotone arithmetic branching programs. This shows that the classical simulation of arithmetic circuits by arithmetic branching programs by Valiant, Skyum, Berkowitz, and Rackoff (1983) cannot be improved, as long as it preserves monotonicity. A key ingredient in the proof is an accurate analysis of the isoperimetric profile of finite full binary trees. We show that the isoperimetric profile of a full binary tree constantly fluctuates between one and almost the depth of the tree.
KW - Communication complexity
KW - Isoperimetry
KW - Monotone computation
KW - Separations
UR - http://www.scopus.com/inward/record.url?scp=85012886781&partnerID=8YFLogxK
U2 - https://doi.org/10.4230/LIPIcs.ICALP.2016.89
DO - https://doi.org/10.4230/LIPIcs.ICALP.2016.89
M3 - منشور من مؤتمر
T3 - Leibniz International Proceedings in Informatics, LIPIcs
SP - 1
BT - 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016
A2 - Rabani, Yuval
A2 - Chatzigiannakis, Ioannis
A2 - Sangiorgi, Davide
A2 - Mitzenmacher, Michael
T2 - 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016
Y2 - 12 July 2016 through 15 July 2016
ER -