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Summary: On the power of algebraic branching programs of width two
Eric Allender
Department of Computer Science
Rutgers University
Piscataway, NJ 08855, USA
allender@cs.rutgers.edu
Fengming Wang
Department of Computer Science
Rutgers University
Piscataway, NJ 08855, USA
fengming@cs.rutgers.edu
May 21, 2011
Abstract
We show that there are families of polynomials having small depth-two arithmetic circuits that cannot
be expressed by algebraic branching programs of width two. This clarifies the complexity of the problem
of computing the product of a sequence of two-by-two matrices, which arises in several settings.
1 Introduction
The nth Iterated Matrix Multiplication polynomial of degree d, denoted IMMd,n is the multilinear polyno-
mial with d2n variables that is the result of multiplying n d-by-d matrices of indeterminates. This family
plays a central role in the study of algebraic complexity. Ben-Or and Cleve showed that IMM3,n is complete
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