Discrete Applied Mathematics 159 (2011) 1–14
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Discrete Applied Mathematics
journal homepage: www.elsevier.com/locate/dam
On the expressive power of CNF formulas of bounded tree- and
clique-width
Irenée Briquel
a
, Pascal Koiran
a
, Klaus Meer
b,∗
a
Laboratoire de l’Informatique du Parallélisme, ENS Lyon, France
b
Lehrstuhl Theoretische Informatik, BTU Cottbus, Germany
a r t i c l e i n f o
Article history:
Received 2 November 2009
Received in revised form 27 August 2010
Accepted 6 September 2010
Available online 16 October 2010
Keywords:
Expressive power of polynomials
Permanent function
Conjunctive normal form formulas
Tree- and clique-width
Valiant’s complexity theory for polynomial
families
a b s t r a c t
We study representations of polynomials over a field K from the point of view of their
expressive power. Three important examples for the paper are polynomials arising as
permanents of bounded tree-width matrices, polynomials given via arithmetic formulas,
and families of so called CNF polynomials. The latter arise in a canonical way from families
of Boolean formulas in conjunctive normal form. To each such CNF formula there is a
canonically attached incidence graph. Of particular interest to us are CNF polynomials
arising from formulas with an incidence graph of bounded tree- or clique-width.
We show that the class of polynomials arising from families of polynomial size
CNF formulas of bounded tree-width is the same as those represented by polynomial
size arithmetic formulas, or permanents of bounded tree-width matrices of polynomial
size. Then, applying arguments from communication complexity we show that general
permanent polynomials cannot be expressed by CNF polynomials of bounded tree-width.
We give a similar result in the case where the clique-width of the incidence graph
is bounded, but for this we need to rely on the widely believed complexity theoretic
assumption #P ̸⊆ FP/poly.
© 2010 Elsevier B.V. All rights reserved.
1. Introduction
An active field of research in complexity is devoted to the design of efficient algorithms for subclasses of problems which
in full generality are likely hard to solve. It is common in this area to define such subclasses via bounding some significant
problem parameters. Typical such parameters are the tree- and clique-width if a graph structure is involved in the problem’s
description.
At the center of the present paper stand problems related to families of polynomials. These families are given in a
particular manner through certain Boolean formulas in conjunctive normal form, CNF formulas for short.
More precisely, we consider a Boolean CNF formula ϕ representing a function from {0, 1}
n
→ {0, 1}. If no confusion
can arise we denote this function again by ϕ. For n variables x
1
, . . . , x
n
ranging over a field K and an e ∈ {0, 1}
n
define the
monomial x
e
:= x
e
1
1
· ··· · x
e
n
n
, where x
0
i
:= 1 and x
1
i
:= x
i
. Now define a function f : K
n
→ K by
f (x) =
e∈{0,1}
n
ϕ(e)· x
e
for x ∈ K
n
. (∗)
The function f is a kind of enumerating polynomial for ϕ. We are interested in the question of how expressive such a
representation of polynomials by CNF formulas is, and under which additional conditions the polynomial f (x) in (∗) can
∗
Corresponding author. Tel.: +49 355 693883; fax: +49 355 693810.
E-mail addresses: irenee.briquel@ens-lyon.fr (I. Briquel), pascal.koiran@ens-lyon.fr (P. Koiran), meer@informatik.tu-cottbus.de (K. Meer).
0166-218X/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.dam.2010.09.007