Search Space Contraction in Canonical Labeling of Graphs

📝 Abstract

The individualization-refinement paradigm for computing a canonical labeling and the automorphism group of a graph is investigated. A new algorithmic design aimed at reducing the size of the associated search space is introduced, and a new tool, named “Traces”, is presented, together with experimental results and comparisons with existing software, such as McKay’s “nauty”. It is shown that the approach presented here leads to a huge reduction in the search space, thereby making computation feasible for several classes of graphs which are hard for all the main canonical labeling tools in the literature.

💡 Analysis

The individualization-refinement paradigm for computing a canonical labeling and the automorphism group of a graph is investigated. A new algorithmic design aimed at reducing the size of the associated search space is introduced, and a new tool, named “Traces”, is presented, together with experimental results and comparisons with existing software, such as McKay’s “nauty”. It is shown that the approach presented here leads to a huge reduction in the search space, thereby making computation feasible for several classes of graphs which are hard for all the main canonical labeling tools in the literature.

📄 Content

Search Space Contraction in Canonical Labeling of Graphs Adolfo Piperno Dipartimento di Informatica, La Sapienza Università di Roma, Via Salaria 113, I-00198 Roma Abstract The individualization-refinement paradigm for computing a canonical labeling and the automorphism group of a graph is investigated. A new algorithmic design aimed at reducing the size of the associ- ated search space is introduced, and a new tool, named Traces, is presented, together with experimental results and comparisons with existing software, such as McKay’s nauty. It is shown that the approach presented here leads to a huge reduction in the search space, thereby making computation feasible for several classes of graphs which are hard for all the main canonical labeling tools in the literature. Key words: (Practical) graph isomorphism, canonical labeling, partition refinement, automorphism group computation. 1. Introduction A canonical labeling (or canonical form) of a graph G is a graph G ′ — isomorphic to G — representing the whole isomorphism class of G. In terms of computational complexity the theoretical status of canonical labeling (CL) is still unsettled, since an efficient algorithm for CL would imply an efficient algorithm for the graph isomorphism problem (GI). In practice, however, CL algorithms are widely used, as they enable (possibly large) sequences of graphs coming from both combinatorial problems and industrial applications to be checked for iso- morphism by simply comparing their canonical forms. The literature on methods for approaching GI and CL displays a peculiar “separation” be- tween theoretical and practical studies. On the theoretical side, besides papers substantiating the thesis that GI is not NP-complete [28, 37] (a survey is in [1]), there are a large number of noteworthy pieces of mathematics showing the existence of polynomial solutions of GI for significant classes of graphs. While moderately exponential solutions have been provided for the general problem of graph isomorphism [3, 6], polynomial algorithms exist for pla- nar graphs [18, 17], graphs of bounded genus [15], graphs with colored vertices and bounded Email address: piperno@di.uniroma1.it (Adolfo Piperno). URL: http://www.dsi.uniroma1.it/ ∼piperno/pers/Traces.html (Adolfo Piperno). Preprint submitted to Elsevier November 26, 2024 arXiv:0804.4881v2 [cs.DS] 26 Jan 2011 color-classes [2], graphs with bounded multiplicity of eigenvalues [5], graphs of bounded va- lence [26], and more (see [4]). On the practical side, there are some noteworthy pieces of software which originate from the outstanding tool nauty [29]. nauty was introduced in the 1980s by McKay [31] and has become a standard in the area of canonical labeling and determination of the automorphism group of a graph. Moreover, it has been incorporated into more general mathematical soft- ware tools such as GAP [16] and MAGMA [27]. It is important to observe that, with the exception of planar graphs, none of the polyno- mial algorithms mentioned above has been implemented in software, as noted by Junttila and Kaski in [19]. A reasonable justification for this absence would seem to be that, given a class C of graphs for which an efficient algorithm for isomorphism testing exists, nauty is usually able to process almost all the graphs in C in a considerably smaller number of steps than that established by the theoretical bound (with respect to the number of vertices). However, there exist graphs in C for which nauty exhibits an exponential behavior, as shown by the series of graphs constructed by Miyazaki in [33]: all these graphs are 3-regular and have color-class size equal to 4, hence they intersect two classes of graphs for which polynomial solutions for GI exist. A further distinctive feature of Miyazaki’s graphs is that the size of their automorphism group is quite large. This contrasts with the fact that graphs which are hard for nauty usually have a high degree of regularity but a small automorphism group. In recent years the tools saucy [13, 12] and bliss [19, 20] have been introduced, aimed at handling large sparse graphs coming either from the satisfiability problem (SAT), or from in- dustrial applications. Like nauty these are general purpose devices implementing backtrack algorithms based on the so called individualization-refinement technique, however they dif- fer from nauty in respect of the data structures and heuristics used. We briefly recall here that the key of the individualization-refinement technique is the notion of equitable parti- tion, a coloring of vertices of a graph such that any two vertices with the same color have the same number of neighbors in each color class. A vertex is individualized by assigning to it a fresh color, the coloring so obtained is refined when a new equitable partition (finer than the initial one) is produced. A backtrack search on the space of all possible individualizations, along with some initial assumptions that allow the associated tree to be

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