11 research outputs found
The Iteration Number of the Weisfeiler-Leman Algorithm
We prove new upper and lower bounds on the number of iterations the
-dimensional Weisfeiler-Leman algorithm (-WL) requires until
stabilization. For , we show that -WL stabilizes after at most
iterations (where denotes the number of vertices of the
input structures), obtaining the first improvement over the trivial upper bound
of and extending a previous upper bound of for
[Lichter et al., LICS 2019].
We complement our upper bounds by constructing -ary relational structures
on which -WL requires at least iterations to stabilize. This
improves over a previous lower bound of [Berkholz,
Nordstr\"{o}m, LICS 2016].
We also investigate tradeoffs between the dimension and the iteration number
of WL, and show that -WL, where , can
simulate the -WL algorithm using only many iterations, but still requires at least
iterations for any (that is sufficiently smaller than ).
The number of iterations required by -WL to distinguish two structures
corresponds to the quantifier rank of a sentence distinguishing them in the -variable fragment of first-order logic with counting
quantifiers. Hence, our results also imply new upper and lower bounds on the
quantifier rank required in the logic , as well as tradeoffs between
variable number and quantifier rank.Comment: 30 pages, 1 figure, full version of a paper accepted at LICS 2023;
second version improves the presentation of the result
The Weisfeiler-Leman Dimension of Planar Graphs is at most 3
We prove that the Weisfeiler-Leman (WL) dimension of the class of all finite
planar graphs is at most 3. In particular, every finite planar graph is
definable in first-order logic with counting using at most 4 variables. The
previously best known upper bounds for the dimension and number of variables
were 14 and 15, respectively.
First we show that, for dimension 3 and higher, the WL-algorithm correctly
tests isomorphism of graphs in a minor-closed class whenever it determines the
orbits of the automorphism group of any arc-colored 3-connected graph belonging
to this class.
Then we prove that, apart from several exceptional graphs (which have
WL-dimension at most 2), the individualization of two correctly chosen vertices
of a colored 3-connected planar graph followed by the 1-dimensional
WL-algorithm produces the discrete vertex partition. This implies that the
3-dimensional WL-algorithm determines the orbits of a colored 3-connected
planar graph.
As a byproduct of the proof, we get a classification of the 3-connected
planar graphs with fixing number 3.Comment: 34 pages, 3 figures, extended version of LICS 2017 pape
The Iteration Number of Colour Refinement
The Colour Refinement procedure and its generalisation to higher dimensions, the Weisfeiler-Leman algorithm, are central subroutines in approaches to the graph isomorphism problem. In an iterative fashion, Colour Refinement computes a colouring of the vertices of its input graph.
A trivial upper bound on the iteration number of Colour Refinement on graphs of order n is n-1. We show that this bound is tight. More precisely, we prove via explicit constructions that there are infinitely many graphs G on which Colour Refinement takes |G|-1 iterations to stabilise. Modifying the infinite families that we present, we show that for every natural number n ? 10, there are graphs on n vertices on which Colour Refinement requires at least n-2 iterations to reach stabilisation
Cutting Planes Width and the Complexity of Graph Isomorphism Refutations
The width complexity measure plays a central role in Resolution and other propositional proof systems like Polynomial Calculus (under the name of degree). The study of width lower bounds is the most extended method for proving size lower bounds, and it is known that for these systems, proofs with small width also imply the existence of proofs with small size. Not much has been studied, however, about the width parameter in the Cutting Planes (CP) proof system, a measure that was introduced by Dantchev and Martin in 2011 under the name of CP cutwidth.
In this paper, we study the width complexity of CP refutations of graph isomorphism formulas. For a pair of non-isomorphic graphs G and H, we show a direct connection between the Weisfeiler-Leman differentiation number WL(G, H) of the graphs and the width of a CP refutation for the corresponding isomorphism formula Iso(G, H). In particular, we show that if WL(G, H) ? k, then there is a CP refutation of Iso(G, H) with width k, and if WL(G, H) > k, then there are no CP refutations of Iso(G, H) with width k-2. Similar results are known for other proof systems, like Resolution, Sherali-Adams, or Polynomial Calculus. We also obtain polynomial-size CP refutations from our width bound for isomorphism formulas for graphs with constant WL-dimension
Recommended from our members
Proof Complexity and Beyond
Proof complexity is a multi-disciplinary intellectual endeavor that addresses questions of the general form âhow difficult is it to prove certain mathematical facts?â The current workshop focused on recent advances in our understanding of logic-based proof systems and on connections to algorithms, geometry and combinatorics research, such as the analysis of approximation algorithms, or the size of linear or semidefinite programming formulations of combinatorial optimization problems, to name just two important examples
From Quantifier Depth to Quantifier Number: Separating Structures with k Variables
Given two -element structures, and , which can
be distinguished by a sentence of -variable first-order logic
(), what is the minimum such that there is guaranteed to
be a sentence with at most quantifiers, such
that but ? We present
various results related to this question obtained by using the recently
introduced QVT games. In particular, we show that when we limit the number of
variables, there can be an exponential gap between the quantifier depth and the
quantifier number needed to separate two structures. Through the lens of this
question, we will highlight some difficulties that arise in analysing the QVT
game and some techniques which can help to overcome them. As a consequence, we
show that is exponentially more succinct than
. We also show, in the setting of the existential-positive
fragment, how to lift quantifier depth lower bounds to quantifier number lower
bounds. This leads to almost tight bounds.Comment: 53 pages, 8 figures; added new result on the relative succinctness of
finite variable logi
Lifting with Simple Gadgets and Applications to Circuit and Proof Complexity
We significantly strengthen and generalize the theorem lifting
Nullstellensatz degree to monotone span program size by Pitassi and Robere
(2018) so that it works for any gadget with high enough rank, in particular,
for useful gadgets such as equality and greater-than. We apply our generalized
theorem to solve two open problems:
* We present the first result that demonstrates a separation in proof power
for cutting planes with unbounded versus polynomially bounded coefficients.
Specifically, we exhibit CNF formulas that can be refuted in quadratic length
and constant line space in cutting planes with unbounded coefficients, but for
which there are no refutations in subexponential length and subpolynomial line
space if coefficients are restricted to be of polynomial magnitude.
* We give the first explicit separation between monotone Boolean formulas and
monotone real formulas. Specifically, we give an explicit family of functions
that can be computed with monotone real formulas of nearly linear size but
require monotone Boolean formulas of exponential size. Previously only a
non-explicit separation was known.
An important technical ingredient, which may be of independent interest, is
that we show that the Nullstellensatz degree of refuting the pebbling formula
over a DAG G over any field coincides exactly with the reversible pebbling
price of G. In particular, this implies that the standard decision tree
complexity and the parity decision tree complexity of the corresponding
falsified clause search problem are equal
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum