TY - GEN
T1 - Average case lower bounds for monotone switching networks
AU - Filmus, Yuval
AU - Pitassi, Toniann
AU - Robere, Robert
AU - Cook, Stephen A.
PY - 2013
Y1 - 2013
N2 - An approximate computation of a Boolean function by a circuit or switching network is a computation in which the function is computed correctly on the majority of the inputs (rather than on all inputs). Besides being interesting in their own right, lower bounds for approximate computation have proved useful in many subareas of complexity theory, such as cryptography and derandomization. Lower bounds for approximate computation are also known as correlation bounds or average case hardness. In this paper, we obtain the first average case monotone depth lower bounds for a function in monotone P. We tolerate errors that are asymptotically the best possible for monotone circuits. Specifically, we prove average case exponential lower bounds on the size of monotone switching networks for the GEN function. As a corollary, we separate the monotone NC hierarchy in the case of errors - A result which was previously only known for exact computations. Our proof extends and simplifies the Fourier analytic technique due to Potechin [21], and further developed by Chan and Potechin [8]. As a corollary of our main lower bound, we prove that the communication complexity approach for monotone depth lower bounds does not naturally generalize to the average case setting.
AB - An approximate computation of a Boolean function by a circuit or switching network is a computation in which the function is computed correctly on the majority of the inputs (rather than on all inputs). Besides being interesting in their own right, lower bounds for approximate computation have proved useful in many subareas of complexity theory, such as cryptography and derandomization. Lower bounds for approximate computation are also known as correlation bounds or average case hardness. In this paper, we obtain the first average case monotone depth lower bounds for a function in monotone P. We tolerate errors that are asymptotically the best possible for monotone circuits. Specifically, we prove average case exponential lower bounds on the size of monotone switching networks for the GEN function. As a corollary, we separate the monotone NC hierarchy in the case of errors - A result which was previously only known for exact computations. Our proof extends and simplifies the Fourier analytic technique due to Potechin [21], and further developed by Chan and Potechin [8]. As a corollary of our main lower bound, we prove that the communication complexity approach for monotone depth lower bounds does not naturally generalize to the average case setting.
KW - Switching-networks
UR - http://www.scopus.com/inward/record.url?scp=84893508957&partnerID=8YFLogxK
U2 - 10.1109/FOCS.2013.70
DO - 10.1109/FOCS.2013.70
M3 - ???researchoutput.researchoutputtypes.contributiontobookanthology.conference???
AN - SCOPUS:84893508957
SN - 9780769551357
T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
SP - 598
EP - 607
BT - Proceedings - 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013
T2 - 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013
Y2 - 27 October 2013 through 29 October 2013
ER -