1,085 research outputs found
Combinatorial refinement on circulant graphs
The combinatorial refinement techniques have proven to be an efficient
approach to isomorphism testing for particular classes of graphs. If the number
of refinement rounds is small, this puts the corresponding isomorphism problem
in a low-complexity class. We investigate the round complexity of the
2-dimensional Weisfeiler-Leman algorithm on circulant graphs, i.e. on Cayley
graphs of the cyclic group , and prove that the number of rounds
until stabilization is bounded by , where is
the number of divisors of . As a particular consequence, isomorphism can be
tested in NC for connected circulant graphs of order with an odd
prime, and vertex degree smaller than .
We also show that the color refinement method (also known as the
1-dimensional Weisfeiler-Leman algorithm) computes a canonical labeling for
every non-trivial circulant graph with a prime number of vertices after
individualization of two appropriately chosen vertices. Thus, the canonical
labeling problem for this class of graphs has at most the same complexity as
color refinement, which results in a time bound of . Moreover, this provides a first example where a sophisticated approach to
isomorphism testing put forward by Tinhofer has a real practical meaning.Comment: 19 pages, 1 figur
An exponential lower bound for Individualization-Refinement algorithms for Graph Isomorphism
The individualization-refinement paradigm provides a strong toolbox for
testing isomorphism of two graphs and indeed, the currently fastest
implementations of isomorphism solvers all follow this approach. While these
solvers are fast in practice, from a theoretical point of view, no general
lower bounds concerning the worst case complexity of these tools are known. In
fact, it is an open question whether individualization-refinement algorithms
can achieve upper bounds on the running time similar to the more theoretical
techniques based on a group theoretic approach.
In this work we give a negative answer to this question and construct a
family of graphs on which algorithms based on the individualization-refinement
paradigm require exponential time. Contrary to a previous construction of
Miyazaki, that only applies to a specific implementation within the
individualization-refinement framework, our construction is immune to changing
the cell selector, or adding various heuristic invariants to the algorithm.
Furthermore, our graphs also provide exponential lower bounds in the case when
the -dimensional Weisfeiler-Leman algorithm is used to replace the standard
color refinement operator and the arguments even work when the entire
automorphism group of the inputs is initially provided to the algorithm.Comment: 21 page
An adaptive prefix-assignment technique for symmetry reduction
This paper presents a technique for symmetry reduction that adaptively
assigns a prefix of variables in a system of constraints so that the generated
prefix-assignments are pairwise nonisomorphic under the action of the symmetry
group of the system. The technique is based on McKay's canonical extension
framework [J.~Algorithms 26 (1998), no.~2, 306--324]. Among key features of the
technique are (i) adaptability---the prefix sequence can be user-prescribed and
truncated for compatibility with the group of symmetries; (ii)
parallelizability---prefix-assignments can be processed in parallel
independently of each other; (iii) versatility---the method is applicable
whenever the group of symmetries can be concisely represented as the
automorphism group of a vertex-colored graph; and (iv) implementability---the
method can be implemented relying on a canonical labeling map for
vertex-colored graphs as the only nontrivial subroutine. To demonstrate the
practical applicability of our technique, we have prepared an experimental
open-source implementation of the technique and carry out a set of experiments
that demonstrate ability to reduce symmetry on hard instances. Furthermore, we
demonstrate that the implementation effectively parallelizes to compute
clusters with multiple nodes via a message-passing interface.Comment: Updated manuscript submitted for revie
Isomorphism test for digraphs with weighted edges
Colour refinement is at the heart of all the most efficient graph isomorphism software packages. In this paper we present a method for extending the applicability of refinement algorithms to directed graphs with weighted edges. We use Traces as a reference software, but the proposed solution is easily transferrable to any other refinement-based graph isomorphism tool in the literature. We substantiate the claim that the performances of the original algorithm remain substantially unchanged by showing experiments for some classes of benchmark graphs
A Generic Framework for Engineering Graph Canonization Algorithms
The state-of-the-art tools for practical graph canonization are all based on
the individualization-refinement paradigm, and their difference is primarily in
the choice of heuristics they include and in the actual tool implementation. It
is thus not possible to make a direct comparison of how individual algorithmic
ideas affect the performance on different graph classes.
We present an algorithmic software framework that facilitates implementation
of heuristics as independent extensions to a common core algorithm. It
therefore becomes easy to perform a detailed comparison of the performance and
behaviour of different algorithmic ideas. Implementations are provided of a
range of algorithms for tree traversal, target cell selection, and node
invariant, including choices from the literature and new variations. The
framework readily supports extraction and visualization of detailed data from
separate algorithm executions for subsequent analysis and development of new
heuristics.
Using collections of different graph classes we investigate the effect of
varying the selections of heuristics, often revealing exactly which individual
algorithmic choice is responsible for particularly good or bad performance. On
several benchmark collections, including a newly proposed class of difficult
instances, we additionally find that our implementation performs better than
the current state-of-the-art tools
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