A Faster Isomorphism Test for Graphs of Small Degree

Grohe, Martin; Neuen, Daniel; Schweitzer, Pascal

doi:10.1109/focs.2018.00018

Cited by 20 publications

(59 citation statements)

References 31 publications

(72 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This approach has established that on many graph classes Graph Isomorphism is polynomial-time solvable, but that on many others the problem remains GI-complete. We refer to [16] for a survey, but some recent examples include a polynomial-time algorithm for unit square graphs [26], a complexity dichotomy for H-induced-minor-free graphs [3] and a polynomial-time algorithm for graphs of bounded maximum degree [18] (improving on the runtime of previous polynomial-time algorithms on graphs of bounded maximum degree [2,25]).…”

Section: Introductionmentioning

confidence: 99%

Graph Isomorphism for $$(H_1,H_2)$$-Free Graphs: An Almost Complete Dichotomy

Bonamy

Dabrowski

Johnson

et al. 2019

Lecture Notes in Computer Science

View full text Add to dashboard Cite

We resolve the computational complexity of Graph Isomorphism for classes of graphs characterized by two forbidden induced subgraphs H1 and H2 for all but six pairs (H1, H2). Schweitzer had previously shown that the number of open cases was finite, but without specifying the open cases. Grohe and Schweitzer proved that Graph Isomorphism is polynomial-time solvable on graph classes of bounded clique-width. Our work combines known results such as these with new results. By exploiting a relationship between Graph Isomorphism and cliquewidth, we simultaneously reduce the number of open cases for boundedness of clique-width for (H1, H2)-free graphs to five.Later, Colbourn [10] proved that Graph Isomorphism is polynomial-time solvable even for the class of permutation graphs, which form a superclass of the class of P 4 -free graphs. Classifying the case where F G has size 2 is much more difficult than the size-1 case. Kratsch and Schweitzer [22] initiated this classification. Schweitzer [30] later extended the results of [22] and proved that only a finite number of cases remain open. This leads to our research question: Is it possible to determine the computational complexity of Graph Isomorphism for (H 1 , H 2 )-free graphs 5 for all pairs H 1 , H 2 ? The analogous research question for H-induced-minor-free graphs was fully answered by Belmonte, Otachi and Schweitzer [3], who also determined all graphs H for which the class of H-induced-minor-free graphs has bounded clique-width. Similar classifications for Graph Isomorphism [28] and boundedness of clique-width [15] are also known for H-minor-free graphs. Lokshtanov et al. [23] recently gave an FPT algorithm for Graph Isomorphism with parameter k on graph classes of treewidth at most k, and this has since been improved by Grohe et al. [19]. Whether an FPT algorithm exists when parameterized by clique-width is still open. Grohe and Schweitzer [20] proved membership of XP. Theorem 2 ([20]). For every constant c, Graph Isomorphism is polynomial-time solvable on graphs of clique-width at most c.Grohe and Neuen [17] have since improved this result by showing that the more general Canonisation problem is also in XP when parameterized by clique-width. 5 A graph is (H1, H2)-free if it has no induced subgraph isomorphic to H1 or H2.

show abstract

Section: Introductionmentioning

confidence: 99%

Graph Isomorphism for $$(H_1,H_2)$$-Free Graphs: An Almost Complete Dichotomy

Bonamy

Dabrowski

Johnson

et al. 2019

Lecture Notes in Computer Science

View full text Add to dashboard Cite

show abstract

“…However, concurrently to our work, he extended his result to canonization [Bab19]. For results building on his algorithm [GNS18], there are still no known canonization versions. Also for the isomorphism problem for groups, nearly all of the most recent results seem not to provide canonical forms [BCQ12, LW12, RW15, GQ15, GQ17, BMW17].…”

Section: Introductionmentioning

confidence: 84%

“…Especially the use of canonical generating sets simplifies the task to adapt theorems used for isomorphism to adequate canonization variants. Naturally, the bounded degree isomorphism algorithm of [GNS18] should then be amenable to canonization. However, this remains as future work.…”

Section: Outlook and Open Questionsmentioning

confidence: 99%

A unifying method for the design of algorithms canonizing combinatorial objects

Schweitzer

Wiebking

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

Self Cite

View full text Add to dashboard Cite

We devise a unified framework for the design of canonization algorithms. Using hereditarily finite sets, we define a general notion of combinatorial objects that includes graphs, hypergraphs, relational structures, codes, permutation groups, tree decompositions, and so on.Our approach allows for a systematic transfer of the techniques that have been developed for isomorphism testing to canonization. We use it to design a canonization algorithm for combinatorial objects in general. This result gives new fastest canonization algorithms with an asymptotic running time matching the best known isomorphism algorithm for the following types of objects: hypergraphs, hypergraphs of bounded color class size, permutation groups (up to permutational isomorphism) and codes that are explicitly given (up to code equivalence).

show abstract

“…It is unclear that graph isomorphism is in P or NP-complete (many researchers believe that it lies between P and NPcomplete). However, it is known that graph isomorphism can be solved in polynomial time if the maximum degree of input graphs is bounded by a constant [6].…”

Section: Unordered Treesmentioning

confidence: 99%

Tree Edit Distance with Variables. Measuring the Similarity between Mathematical Formulas

Akutsu,

Mori,

Nakamura

et al. 2021

Preprint

View full text Add to dashboard Cite

In this article, we propose tree edit distance with variables, which is an extension of the tree edit distance to handle trees with variables and has a potential application to measuring the similarity between mathematical formulas, especially, those appearing in mathematical models of biological systems. We analyze the computational complexities of several variants of this new model. In particular, we show that the problem is NP-complete for ordered trees. We also show for unordered trees that the problem of deciding whether or not the distance is 0 is graph isomorphism complete but can be solved in polynomial time if the maximum outdegree of input trees is bounded by a constant. This distance model is then extended for measuring the difference/similarity between two systems of differential equations, for which results of preliminary computational experiments using biological models are provided.

show abstract

A Faster Isomorphism Test for Graphs of Small Degree

Cited by 20 publications

References 31 publications

Graph Isomorphism for $$(H_1,H_2)$$-Free Graphs: An Almost Complete Dichotomy

Graph Isomorphism for $$(H_1,H_2)$$-Free Graphs: An Almost Complete Dichotomy

A unifying method for the design of algorithms canonizing combinatorial objects

Tree Edit Distance with Variables. Measuring the Similarity between Mathematical Formulas

Contact Info

Product

Resources

About