5 research outputs found
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
Engineering a Preprocessor for Symmetry Detection
State-of-the-art solvers for symmetry detection in combinatorial objects are becoming increasingly sophisticated software libraries. Most of the solvers were initially designed with inputs from combinatorics in mind (nauty, bliss, Traces, dejavu). They excel at dealing with a complicated core of the input. Others focus on practical instances that exhibit sparsity. They excel at dealing with comparatively easy but extremely large substructures of the input (saucy). In practice, these differences manifest in significantly diverging performances on different types of graph classes.
We engineer a preprocessor for symmetry detection. The result is a tool designed to shrink sparse, large substructures of the input graph. On most of the practical instances, the preprocessor improves the overall running time significantly for many of the state-of-the-art solvers. At the same time, our benchmarks show that the additional overhead is negligible.
Overall we obtain single algorithms with competitive performance across all benchmark graphs. As such, the preprocessor bridges the disparity between solvers that focus on combinatorial graphs and large practical graphs. In fact, on most of the practical instances the combined setup significantly outperforms previous state-of-the-art
Towards an Isomorphism Dichotomy for Hereditary Graph Classes
In this paper we resolve the complexity of the isomorphism problem on all but
finitely many of the graph classes characterized by two forbidden induced
subgraphs. To this end we develop new techniques applicable for the structural
and algorithmic analysis of graphs. First, we develop a methodology to show
isomorphism completeness of the isomorphism problem on graph classes by
providing a general framework unifying various reduction techniques. Second, we
generalize the concept of the modular decomposition to colored graphs, allowing
for non-standard decompositions. We show that, given a suitable decomposition
functor, the graph isomorphism problem reduces to checking isomorphism of
colored prime graphs. Third, we extend the techniques of bounded color valence
and hypergraph isomorphism on hypergraphs of bounded color size as follows. We
say a colored graph has generalized color valence at most k if, after removing
all vertices in color classes of size at most k, for each color class C every
vertex has at most k neighbors in C or at most k non-neighbors in C. We show
that isomorphism of graphs of bounded generalized color valence can be solved
in polynomial time.Comment: 37 pages, 4 figure