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### Revision(s):

Revision #1 to TR04-005 | 14th February 2005 00:00

Revision #1
Authors: Stasys Jukna
Accepted on: 14th February 2005 00:00
Keywords:

Abstract:

### Paper:

TR04-005 | 19th January 2004 00:00

#### On Graph Complexity

TR04-005
Authors: Stasys Jukna
Publication: 20th January 2004 08:26
Keywords:

Abstract:

A boolean circuit $f(x_1,\ldots,x_n)$ \emph{represents} a graph $G$
on $n$ vertices if for every input vector $a\in\{0,1\}^n$ with
precisely two $1$'s in, say, positions $i$ and $j$, $f(a)=1$
precisely when $i$ and $j$ are adjacent in $G$; on inputs with more
or less than two $1$'s the circuit can output arbitrary values.

We consider several types of boolean circuits (depth-$3$ circuits and
boolean formulas) and show that some explicit graphs cannot be
represented by small circuits. As a consequence we obtain that
an explicit boolean function in $2m$ variables cannot be computed
as an OR of fewer than $2^{\Omega(m)}$ products of linear forms
over $GF(2)$. Lower bounds for this model obtainable by previously
known (algebraic) arguments do not exceed $2^{O(\sqrt{m})}$.

We conclude with a graph-theoretic problem whose solution would have
some intriguing consequences in computational complexity.

### Comment(s):

Comment #1 to TR04-005 | 31st August 2009 13:40

#### Journal version

Authors: Stasys Jukna
Accepted on: 31st August 2009 13:40
Keywords:

Comment:

Journal version published in: Combinatorics, Probability & Computing 15 (2006) 855-876.

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