Canonical labeling of graphs

Babai, László; Luks, Eugene M.

doi:10.1145/800061.808746

Cited by 325 publications

(455 citation statements)

References 22 publications

Supporting

Mentioning

452

Contrasting

Order By: Relevance

“…Though the fastest known graph isomorphism algorithm for general graphs has running time 2 O( √ n log n) [6], polynomial-time algorithms are known for many interesting subclasses, e.g. bounded degree graphs [19], bounded genus graphs [21], and bounded eigenvalue multiplicity graphs [5].…”

Section: Introductionmentioning

confidence: 99%

Approximate Graph Isomorphism

Arvind

Köbler

Kuhnert

et al. 2012

Mathematical Foundations of Computer Science 2012

View full text Add to dashboard Cite

Abstract. We study optimization versions of Graph Isomorphism. Given two graphs G1, G2, we are interested in finding a bijection π from V (G1) to V (G2) that maximizes the number of matches (edges mapped to edges or non-edges mapped to non-edges). We give an n O(log n) time approximation scheme that for any constant factor α < 1, computes an α-approximation. We prove this by combining the n O(log n) time additive error approximation algorithm of Arora et al. [Math. Program., 92, 2002] with a simple averaging algorithm. We also consider the corresponding minimization problem (of mismatches) and prove that it is NP-hard to α-approximate for any constant factor α. Further, we show that it is also NP-hard to approximate the maximum number of edges mapped to edges beyond a factor of 0.94. We also explore these optimization problems for bounded color class graphs which is a well studied tractable special case of Graph Isomorphism. Surprisingly, the bounded color class case turns out to be harder than the uncolored case in the approximate setting.

show abstract

Section: Introductionmentioning

confidence: 99%

Approximate Graph Isomorphism

Arvind

Köbler

Kuhnert

et al. 2012

Mathematical Foundations of Computer Science 2012

View full text Add to dashboard Cite

show abstract

“…Indeed, we do not know if the problem is fixed parameter tractable. The following result is the best we know which follows easily by applying known techniques [6]. Proof Sketch.…”

Section: Discussionmentioning

confidence: 81%

“…This latter condition is easy to enforce. However, since the above reduction blows up the size of the vertex set in the bipartite encoding, the Zemlyachenko-Luks-Babai graph isomorphism algorithm [3,5,6,25] that runs in time c √ n log n , where n is the size of the vertex set of the graph, does not yield a similar algorithm for hypergraph isomorphism. We note here that the best known hypergraph isomorphism test due to Luks [16] has running time c n .…”

Section: Introductionmentioning

confidence: 99%

Colored Hypergraph Isomorphism is Fixed Parameter Tractable

et al. 2013

View full text Add to dashboard Cite

We describe a fixed parameter tractable (fpt) algorithm for Colored Hypergraph Isomorphism which has running time 2 O(b) N O(1) , where the parameter b is the maximum size of the color classes of the given hypergraphs and N is the input size. We also describe fpt algorithms for certain permutation group problems that are used as subroutines in our algorithm. Keywords and phrases IntroductionA hypergraph is an ordered pair X = (V, E) where V is the vertex set and E ⊆ 2 V is the edge set. Two hypergraphs X = (V, E) and X = (V , E ) are said to be isomorphic,Given two hypergraphs X and X , the decision problem Hypergraph Isomorphism (HI) asks whether X ∼ = X . Graph Isomorphism (GI) is obviously polynomial-time reducible to HI. Conversely, HI is also known to be polynomial-time reducible to GI: Given a pair of hypergraphs X = (V, E) and X = (V , E ) as instance for HI, the reduced instance of GI consists of two corresponding bipartite graphs Y and Y defined as follows. The graph Y has vertex set V E and edge set E(Y ) = {{v, e} | v ∈ V, e ∈ E and v ∈ e}, and Y is defined similarly. Here, C D denotes the disjoint union of the sets C and D. It is easy to verify that Y ∼ = Y if and only if X ∼ = X assuming that V can be mapped only to V and E can be mapped only to E . This latter condition is easy to enforce. However, since the above reduction blows up the size of the vertex set in the bipartite encoding, the Zemlyachenko-Luks-Babai graph isomorphism algorithm [3,5,6,25] that runs in time c √ n log n , where n is the size of the vertex set of the graph, does not yield a similar algorithm for hypergraph isomorphism. We note here that the best known hypergraph isomorphism test due to Luks [16] has running time c n . Recently, Babai and Codenotti [4] have shown a 2Õ (k 2 √ n) isomorphism testing algorithm for hypergraphs with hyperedges of size bounded by k.Motivated by this situation, we explore the same algorithmic problem for bounded color class hypergraphs. The input to Colored Hypergraph Isomorphism (CHI) is a pair

show abstract

“…Max degree O(m) O(n ck ) [4] Genus O(f (k)m) [19] O(n ck ) [13,23] Treewidth O(f (k)n) [6] O(n k+4.5 ) [5] Rooted tree distance width O(kn 2 Parameterized complexity (see [9]) studies these function classes in a multivariate analysis of algorithms, motivated by the much better scalability of O(f (k)n c ) algorithms, so-called fixed-parameter tractable algorithms. In the case of Graph Isomorphism for a large number of parameters only O(n f (k) ) algorithms are known.…”

Section: Comp Of the Parametermentioning

confidence: 99%

Isomorphism for Graphs of Bounded Feedback Vertex Set Number

Kratsch

Schweitzer

2010

Lecture Notes in Computer Science

View full text Add to dashboard Cite

This paper presents an O(n 2 ) algorithm for deciding isomorphism of graphs that have bounded feedback vertex set number. This number is defined as the minimum number of vertex deletions required to obtain a forest. Our result implies that Graph Isomorphism is fixedparameter tractable with respect to the feedback vertex set number. Central to the algorithm is a new technique consisting of an application of reduction rules that produce an isomorphisminvariant outcome, interleaved with the creation of increasingly large partial isomorphisms.

show abstract

Canonical labeling of graphs

Cited by 325 publications

References 22 publications

Approximate Graph Isomorphism

Approximate Graph Isomorphism

Colored Hypergraph Isomorphism is Fixed Parameter Tractable

Isomorphism for Graphs of Bounded Feedback Vertex Set Number

Contact Info

Product

Resources

About