Graph Isomorphism, Color Refinement, and Compactness

Color refinement is a classical technique used to show that two given graphs G and H are non-isomorphic; it is very efficient, although it does not succeed on all graphs. We call a graph G amenable to color refinement if the color refinement procedure succeeds in distinguishing G from any non-isomor...

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Vydáno v:	Computational complexity Ročník 26; číslo 3; s. 627 - 685
Hlavní autoři:	Arvind, V., Köbler, Johannes, Rattan, Gaurav, Verbitsky, Oleg
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Cham Springer International Publishing 01.09.2017 Springer Nature B.V
Témata:	Algorithm Analysis and Problem Complexity Automorphisms Color Combinatorial analysis Computational Mathematics and Numerical Analysis Computer Science Graph theory Graphs Isomorphism Linear programming color refinement linear programming relaxation Graph Isomorphism 68Q25 Analysis of algorithms and problem complexity polytope of fractional automorphisms
ISSN:	1016-3328, 1420-8954
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Abstract	Color refinement is a classical technique used to show that two given graphs G and H are non-isomorphic; it is very efficient, although it does not succeed on all graphs. We call a graph G amenable to color refinement if the color refinement procedure succeeds in distinguishing G from any non-isomorphic graph H . Babai et al. (SIAM J Comput 9(3):628–635, 1980 ) have shown that random graphs are amenable with high probability. We determine the exact range of applicability of color refinement by showing that amenable graphs are recognizable in time O ( ( n + m ) log n ) , where n and m denote the number of vertices and the number of edges in the input graph. We use our characterization of amenable graphs to analyze the approach to Graph Isomorphism based on the notion of compact graphs . A graph is called compact if the polytope of its fractional automorphisms is integral. Tinhofer (Discrete Appl Math 30(2–3):253–264, 1991 ) noted that isomorphism testing for compact graphs can be done quite efficiently by linear programming. However, the problem of characterizing compact graphs and recognizing them in polynomial time remains an open question. Our results in this direction are summarized below: We show that all amenable graphs are compact. In other words, the applicability range for Tinhofer’s linear programming approach to isomorphism testing is at least as large as for the combinatorial approach based on color refinement. Exploring the relationship between color refinement and compactness further, we study related combinatorial and algebraic graph properties introduced by Tinhofer and Godsil. We show that the corresponding classes of graphs form a hierarchy, and we prove that recognizing each of these graph classes is P-hard. In particular, this gives a first complexity lower bound for recognizing compact graphs.
AbstractList	Color refinement is a classical technique used to show that two given graphs G and H are non-isomorphic; it is very efficient, although it does not succeed on all graphs. We call a graph G amenable to color refinement if the color refinement procedure succeeds in distinguishing G from any non-isomorphic graph H . Babai et al. (SIAM J Comput 9(3):628–635, 1980 ) have shown that random graphs are amenable with high probability. We determine the exact range of applicability of color refinement by showing that amenable graphs are recognizable in time O ( ( n + m ) log n ) , where n and m denote the number of vertices and the number of edges in the input graph. We use our characterization of amenable graphs to analyze the approach to Graph Isomorphism based on the notion of compact graphs . A graph is called compact if the polytope of its fractional automorphisms is integral. Tinhofer (Discrete Appl Math 30(2–3):253–264, 1991 ) noted that isomorphism testing for compact graphs can be done quite efficiently by linear programming. However, the problem of characterizing compact graphs and recognizing them in polynomial time remains an open question. Our results in this direction are summarized below: We show that all amenable graphs are compact. In other words, the applicability range for Tinhofer’s linear programming approach to isomorphism testing is at least as large as for the combinatorial approach based on color refinement. Exploring the relationship between color refinement and compactness further, we study related combinatorial and algebraic graph properties introduced by Tinhofer and Godsil. We show that the corresponding classes of graphs form a hierarchy, and we prove that recognizing each of these graph classes is P-hard. In particular, this gives a first complexity lower bound for recognizing compact graphs. Color refinement is a classical technique used to show that two given graphs G and H are non-isomorphic; it is very efficient, although it does not succeed on all graphs. We call a graph Gamenable to color refinement if the color refinement procedure succeeds in distinguishing G from any non-isomorphic graph H. Babai et al. (SIAM J Comput 9(3):628–635, 1980) have shown that random graphs are amenable with high probability. We determine the exact range of applicability of color refinement by showing that amenable graphs are recognizable in time O ( ( n + m ) log n ) , where n and m denote the number of vertices and the number of edges in the input graph. We use our characterization of amenable graphs to analyze the approach to Graph Isomorphism based on the notion of compact graphs. A graph is called compact if the polytope of its fractional automorphisms is integral. Tinhofer (Discrete Appl Math 30(2–3):253–264, 1991) noted that isomorphism testing for compact graphs can be done quite efficiently by linear programming. However, the problem of characterizing compact graphs and recognizing them in polynomial time remains an open question. Our results in this direction are summarized below: ○ We show that all amenable graphs are compact. In other words, the applicability range for Tinhofer’s linear programming approach to isomorphism testing is at least as large as for the combinatorial approach based on color refinement. ○ Exploring the relationship between color refinement and compactness further, we study related combinatorial and algebraic graph properties introduced by Tinhofer and Godsil. We show that the corresponding classes of graphs form a hierarchy, and we prove that recognizing each of these graph classes is P -hard. In particular, this gives a first complexity lower bound for recognizing compact graphs.
Author	Rattan, Gaurav Köbler, Johannes Verbitsky, Oleg Arvind, V.
Author_xml	– sequence: 1 givenname: V. surname: Arvind fullname: Arvind, V. email: arvind@imsc.res.in organization: The Institute of Mathematical Sciences – sequence: 2 givenname: Johannes surname: Köbler fullname: Köbler, Johannes organization: Institut für Informatik, Humboldt Universität zu Berlin – sequence: 3 givenname: Gaurav surname: Rattan fullname: Rattan, Gaurav organization: The Institute of Mathematical Sciences – sequence: 4 givenname: Oleg surname: Verbitsky fullname: Verbitsky, Oleg organization: Institut für Informatik, Humboldt Universität zu Berlin
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Cites_doi	10.1017/jsl.2015.28 10.1007/s004939970004 10.1145/1008354.1008356 10.1007/s10114-004-0485-1 10.1016/0095-8956(82)90042-9 10.1016/0166-218X(88)90100-X 10.1007/978-3-319-22177-9_26 10.1007/978-3-662-48054-0_3 10.1016/S0012-365X(00)00152-7 10.1016/0166-218X(91)90049-3 10.1007/978-1-4612-0333-9 10.1016/S0024-3795(97)83595-1 10.1007/978-1-4612-4478-3_5 10.1016/j.disopt.2014.01.004 10.1016/0012-365X(94)90241-0 10.2307/2371086 10.1016/S0012-365X(99)90071-7 10.1007/978-3-642-40450-4_13 10.1007/BF02240204 10.1016/0024-3795(88)90239-X 10.7155/jgaa.00229 10.1016/S0012-365X(99)00381-7 10.1016/0024-3795(88)90054-7 10.1137/0209047 10.1007/978-94-015-8937-6_3 10.1609/aaai.v28i1.8992 10.1007/978-3-662-48057-1_25 10.1137/120867834 10.1007/BF01305232 10.2140/pjm.1975.56.143 10.1017/CBO9780511721182 10.1016/0304-3975(82)90016-0 10.1007/978-3-662-44777-2_42 10.1016/S0095-8956(76)80006-8
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References_xml	– reference: László Babai & Ludek Kučera (1979). Canonical labelling of graphs in linear average time. In Proceedings of the 20th Annual Symposium on Foundations of Computer Science, 39–46. – reference: JohannesKöblerUweSchöningJacoboToránThe graph isomorphism problem: its structural complexity1993BostonMA: Birkhäuser.0813.68103 – reference: Boris Yu. Weisfeiler & Andrei A. Leman (1968). A reduction of a graph to a canonical form and an algebra arising during this reduction. Nauchno-Technicheskaya Informatsia, Seriya 229, 12–16. In Russian. – reference: Andreas Krebs & Oleg Verbitsky (2015). Universal covers, color refinement, and two-variable logic with counting quantifiers: Lower bounds for the depth. In Proceedings of the 30-th ACM/IEEE Annual Symposium on Logic in Computer Science (LICS), 689–700. IEEE Computer Society. – reference: Ramana MotakuriV.Scheinerman EdwardR.UllmanDanielFractional isomorphism of graphsDiscrete Mathematics19941321-3247265129738510.1016/0012-365X(94)90241-00808.05077 – reference: Frank ThomsonLeightonFinite common coverings of graphsJournal of Combinatorial Theory, Series B198233323123869336210.1016/0095-8956(82)90042-90488.05033 – reference: HasslerWhitneyCongruent graphs and connectivity of graphsAmerican Journal of Mathematics193254150168150688110.2307/23710860003.32804 – reference: Christoph Berkholz, Paul Bonsma & Martin Grohe (2013). Tight lower and upper bounds for the complexity of canonical colour refinement. In Proceedings of 21st Annual European Symposium on Algorithms (ESA), volume 8125 of Lecture Notes in Computer Science, 145–156. 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Snippet	Color refinement is a classical technique used to show that two given graphs G and H are non-isomorphic; it is very efficient, although it does not succeed on... Color refinement is a classical technique used to show that two given graphs G and H are non-isomorphic; it is very efficient, although it does not succeed on...
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SubjectTerms	Algorithm Analysis and Problem Complexity Automorphisms Color Combinatorial analysis Computational Mathematics and Numerical Analysis Computer Science Graph theory Graphs Isomorphism Linear programming
Title	Graph Isomorphism, Color Refinement, and Compactness
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