Abstract
We prove a lower bound of Ω(n2/log2n) on the size of any syntactically multilinear arithmetic circuit computing some explicit multilinear polynomial f(x1,...,xn). Our approach expands and improves upon a result of Raz, Shpilka and Yehudayoff ([34]), who proved a lower bound of Ω(n4/3/log2n) for the same polynomial. Our improvement follows from an asymptotically optimal lower bound for a generalized version of Galvin's problem in extremal set theory. A special case of our combinatorial result implies, for every n, a tight Ω(n) lower bound on the minimum size of a family F of subsets of cardinality 2n of a set X of size 4n, so that any subset of X of size 2n has intersection of size exactly n with some member of F. This settles a problem of Galvin up to a constant factor, extending results of Frankl and Rodl [15] and Enomoto et al. [12], who proved in 1987 the above statement (with a tight constant) for odd values of n, leaving the even case open.
Highlights
An arithmetic circuit is one of the most natural and standard computational models for computing multivariate polynomials
Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 11:2 Unbalancing Sets and Lower Bounds for Multilinear Arithmetic Circuits multivariate polynomials, and in some sense, they can be thought of as algebraic analogs of boolean circuits
A sum gate computes the polynomial which is the sum of the polynomials computed at its children and a product gate computes the polynomial which is the product of the polynomials at its children
Summary
An arithmetic circuit is one of the most natural and standard computational models for computing multivariate polynomials. Such circuits provide a succinct representation of. 11:2 Unbalancing Sets and Lower Bounds for Multilinear Arithmetic Circuits multivariate polynomials, and in some sense, they can be thought of as algebraic analogs of boolean circuits. A circuit can have one or more vertices of out degree zero, known as the output gates. It is not hard to show (see, e.g., [7]) that a random polynomial of degree d = poly(n) in n variables cannot be computed by an arithmetic circuit of size poly(n) with overwhelmingly high probability. One subclass which has received a lot of attention in the last two decades and will be the focus of this paper is the class of multilinear arithmetic circuits
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