Results 1  10
of
34
Completeness results for Graph Isomorphism
, 2002
"... We prove that the graph isomorphism problem restricted to trees and to colored graphs with color multiplicities 2 and 3 is manyone complete for several complexity classes within NC². In particular we show that tree isomorphism, when trees are encoded as strings, is NC¹hard under AC0reductions ..."
Abstract

Cited by 22 (9 self)
 Add to MetaCart
We prove that the graph isomorphism problem restricted to trees and to colored graphs with color multiplicities 2 and 3 is manyone complete for several complexity classes within NC². In particular we show that tree isomorphism, when trees are encoded as strings, is NC¹hard under AC0reductions. NC¹completeness thus follows from Buss's NC¹ upper bound. By contrast, we prove that testing isomorphism of two trees encoded as pointer lists is Lcomplete. Concerning colored graphs we show that the isomorphism problem for graphs with color multiplicities 2 and 3 is complete for symmetric logarithmic space SL under manyone reductions. This result improves the existing upper bounds for the problem. We also show that the graph automorphism problem for colored graphs with color classes of size 2 is equivalent to deciding whether a graph has more than a single connected component and we prove that for color classes of size 3 the graph automorphism problem is contained in SL.
Engineering an Efficient Canonical Labeling Tool for Large and Sparse Graphs
"... The problem of canonically labeling a graph is studied. Within the general framework of backtracking algorithms based on individualization and refinement, data structures, subroutines, and pruning heuristics especially for fast handling of large and sparse graphs are developed. Experiments indicate ..."
Abstract

Cited by 16 (1 self)
 Add to MetaCart
The problem of canonically labeling a graph is studied. Within the general framework of backtracking algorithms based on individualization and refinement, data structures, subroutines, and pruning heuristics especially for fast handling of large and sparse graphs are developed. Experiments indicate that the algorithm implementation in most cases clearly outperforms existing stateoftheart tools.
Equivalence and isomorphism for Boolean constraint satisfaction
 In Proceedings of the 16th Annual Conference of the EACSL (CSL 2002
, 2002
"... Abstract. A Boolean constraint satisfaction instance is a conjunction of constraint applications, where the allowed constraints are drawn from a fixed set C of Boolean functions. We consider the problem of determining whether two given constraint satisfaction instances are equivalent and prove a Dic ..."
Abstract

Cited by 12 (6 self)
 Add to MetaCart
Abstract. A Boolean constraint satisfaction instance is a conjunction of constraint applications, where the allowed constraints are drawn from a fixed set C of Boolean functions. We consider the problem of determining whether two given constraint satisfaction instances are equivalent and prove a Dichotomy Theorem by showing that for all sets C of allowed constraints, this problem is either polynomialtime solvable or coNPcomplete, and we give a simple criterion to determine which case holds. A more general problem addressed in this paper is the isomorphism problem, the problem of determining whether there exists a renaming of the variables that makes two given constraint satisfaction instances equivalent in the above sense. We prove that this problem is coNPhard if the corresponding equivalence problem is coNPhard, and polynomialtime manyone reducible to the graph isomorphism problem in all other cases.
Planar graph isomorphism is in logspace
 In IEEE Conference on Computational Complexity
, 2009
"... Abstract. We show that the isomorphism of 3connected planar graphs can be decided in deterministic logspace. This improves the previously known bound UL ∩ coUL of [13]. 1 ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
Abstract. We show that the isomorphism of 3connected planar graphs can be decided in deterministic logspace. This improves the previously known bound UL ∩ coUL of [13]. 1
Equivalence Problems for Boolean Constraint Satisfaction
, 2001
"... A Boolean constraint satisfaction instance is a conjunction of constraint applications, where the allowed constraints are drawn from a fi xed set C of Boolean functions. We consider the problem of determining whether two given constraint satisfaction instances are equivalent in the sense that they p ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
A Boolean constraint satisfaction instance is a conjunction of constraint applications, where the allowed constraints are drawn from a fi xed set C of Boolean functions. We consider the problem of determining whether two given constraint satisfaction instances are equivalent in the sense that they possess the same sets of satisfying assignments. We prove a Dichotomy Theorem by showing that for all sets C of allowed constraints, this problem is either polynomialtime solvable or coNPcomplete, and we give a simple criterion to determine which case holds. Another equivalence problem...
The isomorphism problem for ktrees is complete for logspace
 PROCEEDINGS OF 34TH INTERNATIONAL SYMPOSIUM MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE (MFCS), NUMBER 5734 IN LNCS
, 2009
"... We show that ktree isomorphism can be decided in logarithmic space by giving a logspace canonical labeling algorithm. This improves over the previous StUL upper bound and matches the lower bound. As a consequence, the isomorphism, the automorphism, as well as the canonization problem for ktrees ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
We show that ktree isomorphism can be decided in logarithmic space by giving a logspace canonical labeling algorithm. This improves over the previous StUL upper bound and matches the lower bound. As a consequence, the isomorphism, the automorphism, as well as the canonization problem for ktrees are all complete for deterministic logspace. We also show that even simple structural properties of ktrees are complete for logspace.
On graph isomorphism for restricted graph classes
 In
, 2006
"... Abstract. Graph isomorphism (GI) is one of the few remaining problems in NP whose complexity status couldn’t be solved by classifying it as being either NPcomplete or solvable in P. Nevertheless, efficient (polynomialtime or even NC) algorithms for restricted versions of GI have been found over th ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Abstract. Graph isomorphism (GI) is one of the few remaining problems in NP whose complexity status couldn’t be solved by classifying it as being either NPcomplete or solvable in P. Nevertheless, efficient (polynomialtime or even NC) algorithms for restricted versions of GI have been found over the last four decades. Depending on the graph class, the design and analysis of algorithms for GI use tools from various fields, such as combinatorics, algebra and logic. In this paper, we collect several complexity results on graph isomorphism testing and related algorithmic problems for restricted graph classes from the literature. Further, we provide some new complexity bounds (as well as easier proofs of some known results) and highlight some open questions. 1
The Isomorphism Problem for kTrees is Complete for Logspace
, 2012
"... We show that, for k constant, ktree isomorphism can be decided in logarithmic space by giving an O(k log n) space canonical labeling algorithm. The algorithm computes a unique tree decomposition, uses colors to fully encode the structure of the original graph in the decomposition tree and invokes L ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
We show that, for k constant, ktree isomorphism can be decided in logarithmic space by giving an O(k log n) space canonical labeling algorithm. The algorithm computes a unique tree decomposition, uses colors to fully encode the structure of the original graph in the decomposition tree and invokes Lindell’s tree canonization algorithm. As a consequence, the isomorphism, the automorphism, as well as the canonization problem for ktrees are all complete for deterministic logspace. Completeness for logspace holds even for simple structural properties of ktrees. We also show that a variant of our canonical labeling algorithm runs in time O((k + 1)! n), where n is the number of vertices, yielding the fastest known FPT algorithm for ktree isomorphism.
A logspace algorithm for canonization of planar graphs
, 2008
"... Planar graph canonization is known to be hard for L this directly follows from Lhardness of treecanonization [Lin92]. We give a logspace algorithm for planar graph canonization. This gives completeness for logspace under AC 0 manyone reductions and improves the previously known upper bound of A ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Planar graph canonization is known to be hard for L this directly follows from Lhardness of treecanonization [Lin92]. We give a logspace algorithm for planar graph canonization. This gives completeness for logspace under AC 0 manyone reductions and improves the previously known upper bound of AC 1 [MR91]. A planar graph can be decomposed into biconnected components. We give a logspace procedure for the decomposition of a biconnected planar graph into a triconnected component tree. The canonization process is based on these decomposition steps. 1