30 research outputs found
Graph Isomorphism for unit square graphs
In the past decades for more and more graph classes the Graph Isomorphism
Problem was shown to be solvable in polynomial time. An interesting family of
graph classes arises from intersection graphs of geometric objects. In this
work we show that the Graph Isomorphism Problem for unit square graphs,
intersection graphs of axis-parallel unit squares in the plane, can be solved
in polynomial time. Since the recognition problem for this class of graphs is
NP-hard we can not rely on standard techniques for geometric graphs based on
constructing a canonical realization. Instead, we develop new techniques which
combine structural insights into the class of unit square graphs with
understanding of the automorphism group of such graphs. For the latter we
introduce a generalization of bounded degree graphs which is used to capture
the main structure of unit square graphs. Using group theoretic algorithms we
obtain sufficient information to solve the isomorphism problem for unit square
graphs.Comment: 31 pages, 6 figure
Canonisation and Definability for Graphs of Bounded Rank Width
We prove that the combinatorial Weisfeiler-Leman algorithm of dimension
is a complete isomorphism test for the class of all graphs of rank
width at most . Rank width is a graph invariant that, similarly to tree
width, measures the width of a certain style of hierarchical decomposition of
graphs; it is equivalent to clique width. It was known that isomorphism of
graphs of rank width is decidable in polynomial time (Grohe and Schweitzer,
FOCS 2015), but the best previously known algorithm has a running time
for a non-elementary function . Our result yields an isomorphism
test for graphs of rank width running in time . Another
consequence of our result is the first polynomial time canonisation algorithm
for graphs of bounded rank width. Our second main result is that fixed-point
logic with counting captures polynomial time on all graph classes of bounded
rank width.Comment: 32 page
Hypergraph Isomorphism for Groups with Restricted Composition Factors
We consider the isomorphism problem for hypergraphs taking as input two hypergraphs over the same set of vertices V and a permutation group ? over domain V, and asking whether there is a permutation ? ? ? that proves the two hypergraphs to be isomorphic. We show that for input groups, all of whose composition factors are isomorphic to a subgroup of the symmetric group on d points, this problem can be solved in time (n+m)^O((log d)^c) for some absolute constant c where n denotes the number of vertices and m the number of hyperedges. In particular, this gives the currently fastest isomorphism test for hypergraphs in general. The previous best algorithm for the above problem due to Schweitzer and Wiebking (STOC 2019) runs in time n^O(d)m^O(1).
As an application of this result, we obtain, for example, an algorithm testing isomorphism of graphs excluding K_{3,h} as a minor in time n^O((log h)^c). In particular, this gives an isomorphism test for graphs of Euler genus at most g running in time n^O((log g)^c)
The Power of the Weisfeiler-Leman Algorithm to Decompose Graphs
The Weisfeiler-Leman procedure is a widely-used approach for graph
isomorphism testing that works by iteratively computing an
isomorphism-invariant coloring of vertex tuples. Meanwhile, a fundamental tool
in structural graph theory, which is often exploited in approaches to tackle
the graph isomorphism problem, is the decomposition into 2- and 3-connected
components.
We prove that the 2-dimensional Weisfeiler-Leman algorithm implicitly
computes the decomposition of a graph into its 3-connected components. Thus,
the dimension of the algorithm needed to distinguish two given graphs is at
most the dimension required to distinguish the corresponding decompositions
into 3-connected components (assuming it is at least 2).
This result implies that for k >= 2, the k-dimensional algorithm
distinguishes k-separators, i.e., k-tuples of vertices that separate the graph,
from other vertex k-tuples. As a byproduct, we also obtain insights about the
connectivity of constituent graphs of association schemes.
In an application of the results, we show the new upper bound of k on the
Weisfeiler-Leman dimension of graphs of treewidth at most k. Using a
construction by Cai, F\"urer, and Immerman, we also provide a new lower bound
that is asymptotically tight up to a factor of 2.Comment: 30 pages, 4 figures, full version of a paper accepted at MFCS 201
Benchmark Graphs for Practical Graph Isomorphism
The state-of-the-art solvers for the graph isomorphism problem can readily
solve generic instances with tens of thousands of vertices. Indeed, experiments
show that on inputs without particular combinatorial structure the algorithms
scale almost linearly. In fact, it is non-trivial to create challenging
instances for such solvers and the number of difficult benchmark graphs
available is quite limited. We describe a construction to efficiently generate
small instances for the graph isomorphism problem that are difficult or even
infeasible for said solvers. Up to this point the only other available
instances posing challenges for isomorphism solvers were certain incidence
structures of combinatorial objects (such as projective planes, Hadamard
matrices, Latin squares, etc.). Experiments show that starting from 1500
vertices our new instances are several orders of magnitude more difficult on
comparable input sizes. More importantly, our method is generic and efficient
in the sense that one can quickly create many isomorphism instances on a
desired number of vertices. In contrast to this, said combinatorial objects are
rare and difficult to generate and with the new construction it is possible to
generate an abundance of instances of arbitrary size. Our construction hinges
on the multipedes of Gurevich and Shelah and the Cai-F\"{u}rer-Immerman gadgets
that realize a certain abelian automorphism group and have repeatedly played a
role in the context of graph isomorphism. Exploring limits of such
constructions, we also explain that there are group theoretic obstructions to
generalizing the construction with non-abelian gadgets.Comment: 32 page
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
An Improved Isomorphism Test for Bounded-Tree-Width Graphs
We give a new fpt algorithm testing isomorphism of n-vertex graphs of tree width k in time 2^{k polylog(k)} poly n, improving the fpt algorithm due to Lokshtanov, Pilipczuk, Pilipczuk, and Saurabh (FOCS 2014), which runs in time 2^{O(k^5 log k)}poly n. Based on an improved version of the isomorphism-invariant graph decomposition technique introduced by Lokshtanov et al., we prove restrictions on the structure of the automorphism groups of graphs of tree width k. Our algorithm then makes heavy use of the group theoretic techniques introduced by Luks (JCSS 1982) in his isomorphism test for bounded degree graphs and Babai (STOC 2016) in his quasipolynomial isomorphism test. In fact, we even use Babai\u27s algorithm as a black box in one place. We give a second algorithm which, at the price of a slightly worse run time 2^{O(k^2 log k)}poly n, avoids the use of Babai\u27s algorithm and, more importantly, has the additional benefit that it can also be used as a canonization algorithm
Computing Square Colorings on Bounded-Treewidth and Planar Graphs
A square coloring of a graph is a coloring of the square of ,
that is, a coloring of the vertices of such that any two vertices that are
at distance at most in receive different colors. We investigate the
complexity of finding a square coloring with a given number of colors. We
show that the problem is polynomial-time solvable on graphs of bounded
treewidth by presenting an algorithm with running time for graphs of treewidth at most . The somewhat
unusual exponent in the running time is essentially
optimal: we show that for any , there is no algorithm with running
time unless the
Exponential-Time Hypothesis (ETH) fails.
We also show that the square coloring problem is NP-hard on planar graphs for
any fixed number of colors. Our main algorithmic result is showing
that the problem (when the number of colors is part of the input) can be
solved in subexponential time on planar graphs. The
result follows from the combination of two algorithms. If the number of
colors is small (), then we can exploit a treewidth bound on the
square of the graph to solve the problem in time . If
the number of colors is large (), then an algorithm based on
protrusion decompositions and building on our result for the bounded-treewidth
case solves the problem in time .Comment: 72 pages, 15 figures, full version of a paper accepted at SODA 202