{"title":"有界秩宽度图的规范化和可定义性","authors":"Martin Grohe, Daniel Neuen","doi":"https://dl.acm.org/doi/10.1145/3568025","DOIUrl":null,"url":null,"abstract":"<p>We prove that the combinatorial Weisfeiler-Leman algorithm of dimension (3<i>k</i>+4) is a complete isomorphism test for the class of all graphs of rank width at most <i>k</i>. Rank width is a graph invariant that, similarly to tree width, measures the width of a certain style of hierarchical decomposition of graphs; it is equivalent to clique width.</p><p>It was known that isomorphism of graphs of rank width <i>k</i> is decidable in polynomial time (Grohe and Schweitzer, FOCS 2015), but the best previously known algorithm has a running time <i>n<sup>f(k)</sup></i> for a non-elementary function <i>f</i>. Our result yields an isomorphism test for graphs of rank width <i>k</i> running in time <i>n<sup>O(k)</sup></i>. Another consequence of our result is the first polynomial-time canonisation algorithm for graphs of bounded rank width.</p><p>Our second main result is that fixed-point logic with counting captures polynomial time on all graph classes of bounded rank width.</p>","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"37 7","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2023-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Canonisation and Definability for Graphs of Bounded Rank Width\",\"authors\":\"Martin Grohe, Daniel Neuen\",\"doi\":\"https://dl.acm.org/doi/10.1145/3568025\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We prove that the combinatorial Weisfeiler-Leman algorithm of dimension (3<i>k</i>+4) is a complete isomorphism test for the class of all graphs of rank width at most <i>k</i>. Rank width is a graph invariant that, similarly to tree width, measures the width of a certain style of hierarchical decomposition of graphs; it is equivalent to clique width.</p><p>It was known that isomorphism of graphs of rank width <i>k</i> is decidable in polynomial time (Grohe and Schweitzer, FOCS 2015), but the best previously known algorithm has a running time <i>n<sup>f(k)</sup></i> for a non-elementary function <i>f</i>. Our result yields an isomorphism test for graphs of rank width <i>k</i> running in time <i>n<sup>O(k)</sup></i>. Another consequence of our result is the first polynomial-time canonisation algorithm for graphs of bounded rank width.</p><p>Our second main result is that fixed-point logic with counting captures polynomial time on all graph classes of bounded rank width.</p>\",\"PeriodicalId\":50916,\"journal\":{\"name\":\"ACM Transactions on Computational Logic\",\"volume\":\"37 7\",\"pages\":\"\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2023-01-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACM Transactions on Computational Logic\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://doi.org/https://dl.acm.org/doi/10.1145/3568025\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACM Transactions on Computational Logic","FirstCategoryId":"94","ListUrlMain":"https://doi.org/https://dl.acm.org/doi/10.1145/3568025","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
Canonisation and Definability for Graphs of Bounded Rank Width
We prove that the combinatorial Weisfeiler-Leman algorithm of dimension (3k+4) is a complete isomorphism test for the class of all graphs of rank width at most k. Rank width is a graph invariant that, similarly to tree width, measures the width of a certain style of hierarchical decomposition of graphs; it is equivalent to clique width.
It was known that isomorphism of graphs of rank width k is decidable in polynomial time (Grohe and Schweitzer, FOCS 2015), but the best previously known algorithm has a running time nf(k) for a non-elementary function f. Our result yields an isomorphism test for graphs of rank width k running in time nO(k). Another consequence of our result is the first polynomial-time canonisation algorithm for graphs of bounded rank width.
Our second main result is that fixed-point logic with counting captures polynomial time on all graph classes of bounded rank width.
期刊介绍：
TOCL welcomes submissions related to all aspects of logic as it pertains to topics in computer science. This area has a great tradition in computer science. Several researchers who earned the ACM Turing award have also contributed to this field, namely Edgar Codd (relational database systems), Stephen Cook (complexity of logical theories), Edsger W. Dijkstra, Robert W. Floyd, Tony Hoare, Amir Pnueli, Dana Scott, Edmond M. Clarke, Allen E. Emerson, and Joseph Sifakis (program logics, program derivation and verification, programming languages semantics), Robin Milner (interactive theorem proving, concurrency calculi, and functional programming), and John McCarthy (functional programming and logics in AI).
Logic continues to play an important role in computer science and has permeated several of its areas, including artificial intelligence, computational complexity, database systems, and programming languages.
The Editorial Board of this journal seeks and hopes to attract high-quality submissions in all the above-mentioned areas of computational logic so that TOCL becomes the standard reference in the field.
Both theoretical and applied papers are sought. Submissions showing novel use of logic in computer science are especially welcome.
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