On the Parallel Complexity of Group Isomorphism via Weisfeiler–Leman

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Grochow","affiliation":[{"name":"University of Colorado Boulder","address":{"name":"University of Colorado Boulder, Boulder, USA","@type":"PostalAddress"},"@type":"Organization"}],"@type":"Person"},{"name":"Michael Levet","affiliation":[{"name":"College of Charleston","address":{"name":"College of Charleston, Charleston, USA","@type":"PostalAddress"},"@type":"Organization"}],"email":"levetm@cofc.edu","@type":"Person"}],"keywords":"Group Isomorphism, Graph Isomorphism, Weisfeiler–Leman, Descriptive Complexity","description":"In this paper, we show that the constant-dimensional Weisfeiler–Leman algorithm for groups (Brachter & Schweitzer, LICS 2020) can be fruitfully used to improve parallel complexity upper bounds on isomorphism testing for several families of groups. In particular, we show:\n \n \n \n \n \n \n \n \n \n \n We finally consider the count-free Weisfeiler–Leman algorithm, where we show that count-free WL is unable to even distinguish Abelian groups in polynomial-time. Nonetheless, we use count-free WL in tandem with bounded non-determinism and limited counting to obtain a new upper bound of \n \n \n \n $$\\beta _{1}\\textsf {MAC}^{0}(\\textsf {FOLL})$$\n \n for isomorphism testing of Abelian groups. This improves upon the previous \n \n \n \n $$\\textsf {TC}^{0}(\\textsf {FOLL})$$\n \n upper bound due to Chattopadhyay, Torán, & Wagner (ACM Trans. Comput. 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c-article-identifiers--cite-list"> <li class="c-article-identifiers__item"> <span class="c-chapter-book-details__meta"> pp 234–247</span> </li> <li class="c-article-identifiers__item c-article-identifiers__item--cite"> <a href="#citeas" data-track="click" data-track-action="cite this chapter" data-track-category="chapter body" data-track-label="link">Cite this conference paper</a> </li> </ul> <div class="app-article-masthead__buttons" data-track-context="masthead"> </div> </div> <div class="app-article-masthead__brand app-article-masthead__brand--no-border app-article-masthead__conference-link"> <a href="/book/10.1007/978-3-031-43587-4" class="app-article-masthead__conference-link app-article-masthead__journal-link" data-track="click" data-track-action="book homepage" data-track-label="link"> <picture> <source type="image/webp" media="(min-width: 768px)" width="120" height="182" srcset="https://media.springernature.com/w120/springer-static/cover-hires/book/978-3-031-43587-4?as=webp, https://media.springernature.com/w316/springer-static/cover-hires/book/978-3-031-43587-4?as=webp 2x"> <img width="72" height="109" src="https://media.springernature.com/w72/springer-static/cover-hires/book/978-3-031-43587-4?as=webp" srcset="https://media.springernature.com/w144/springer-static/cover-hires/book/978-3-031-43587-4?as=webp 2x" alt=""> </picture> <span class="app-article-masthead__journal-title ">Fundamentals of Computation Theory</span> </a> <span class="app-article-masthead__conference-info">(FCT 2023) </span> </div> </div> </div> </section> </div> <div class="c-article-main u-container u-mt-24 u-mb-32 l-with-sidebar" id="main-content" data-component="article-container"> <main class="js-main-column u-serif c-chapter-body" data-track-component="chapter"> <article lang="en"> <div class="c-article-header"> <header> <div class="app-overview-section"> <ul class="c-article-author-list c-article-author-list--short" data-test="authors-list" data-component-authors-activator="authors-list"><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Joshua_A_-Grochow" data-author-popup="auth-Joshua_A_-Grochow">Joshua A. Grochow</a><sup class="u-js-hide"><a href="#Aff9">9</a></sup> & </li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Michael-Levet" data-author-popup="auth-Michael-Levet" data-corresp-id="c1">Michael Levet<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-mail-medium"></use></svg></a><sup class="u-js-hide"><a href="#Aff10">10</a></sup> </li></ul> <div class="app-overview-section__separator app-overview-section__book-series"> <div class="app-book-series-listing"> <div> <svg class="app-book-series-listing__icon" width="24" height="24" aria-hidden="true" focusable="false"><use href="#icon-eds-i-book-series-medium"></use></svg> </div> <div> <p data-test="series-link"> <span class="app-book-series-listing__description">Part of the book series:</span> <a href="https://www.springer.com/series/558" data-track="click" data-track-action="open book series" data-track-label="link">Lecture Notes in Computer Science</a> ((LNCS,volume 14292)) </p> </div> </div> </div> <div class="app-book-series-listing app-overview-section__conf-series"> <div> <svg class="app-book-series-listing__icon" width="24" height="24" aria-hidden="true" focusable="false"><use href="#icon-eds-i-conference-series-medium"></use></svg> </div> <div> <p class="app-book-series-listing__description" data-test="conference-series-link">Included in the following conference series:</p> <ul class="app-book-series-listing__list" data-component="data-book-show-more"> <li class="app-book-series-listing__item"><a href="https://link.springer.com/conference/fct" data-test="conference-series-link" data-track="click" data-track-action="open conference" data-track-label="link">International Symposium on Fundamentals of Computation Theory</a></li> </ul> </div> </div> <div class="app-overview-section__separator" data-test="article-metrics"> <div id="altmetric-container"> <ul class="app-article-metrics-bar u-list-reset" data-test="article-metrics"> <li class="app-article-metrics-bar__item" data-test="access-count"> <p class="app-article-metrics-bar__count"><svg class="u-icon app-article-metrics-bar__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-accesses-medium"></use> </svg>261 <span class="app-article-metrics-bar__label">Accesses</span></p> </li> <li class="app-article-metrics-bar__item" data-test="citation-count"> <p class="app-article-metrics-bar__count"><svg class="u-icon app-article-metrics-bar__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-citations-medium"></use> </svg>2 <span class="app-article-metrics-bar__label"> <a href="http://citations.springer.com/item?doi=10.1007/978-3-031-43587-4_17" target="_blank" rel="noopener" title="Visit Springer Citations for full citation details" data-track="click" data-track-action="Citation count" data-track-label="link">Citations</a> </span></p> </li> </ul> </div> </div> </div> </header> </div> <div data-article-body="true" data-track-component="chapter body" class="c-article-body"> <section aria-labelledby="Abs1" data-title="Abstract" lang="en"><div class="c-article-section" id="Abs1-section"><h2 id="Abs1" class="c-article-section__title js-section-title js-c-reading-companion-sections-item"><span class="c-article-section__title-number"> </span>Abstract</h2><div class="c-article-section__content" id="Abs1-content"><p>In this paper, we show that the constant-dimensional Weisfeiler–Leman algorithm for groups (Brachter & Schweitzer, LICS 2020) can be fruitfully used to improve parallel complexity upper bounds on isomorphism testing for several families of groups. In particular, we show:</p><ul class="u-list-style-dash"> <li> <p>Groups with an Abelian normal Hall subgroup whose complement is <i>O</i>(1)-generated are identified by constant-dimensional Weisfeiler–Leman using only a constant number of rounds. This places isomorphism testing for this family of groups into <span class="mathjax-tex">$\textsf {L}$</span>; the previous upper bound for isomorphism testing was <span class="mathjax-tex">$\textsf{P}$</span> (Qiao, Sarma, & Tang, STACS 2011).</p> </li> <li> <p>We use the individualize-and-refine paradigm to obtain a <span class="mathjax-tex">$\textsf {quasiSAC}^{1}$</span> isomorphism test for groups without Abelian normal subgroups, previously only known to be in <span class="mathjax-tex">$\textsf{P}$</span> (Babai, Codenotti, & Qiao, ICALP 2012).</p> </li> <li> <p>We extend a result of Brachter & Schweitzer (ESA, 2022) on direct products of groups to the parallel setting. Namely, we also show that Weisfeiler–Leman can identify direct products in parallel, provided it can identify each of the indecomposable direct factors in parallel. They previously showed the analogous result for <span class="mathjax-tex">$\textsf{P}$</span>.</p> </li> </ul> <p>We finally consider the count-free Weisfeiler–Leman algorithm, where we show that count-free WL is unable to even distinguish Abelian groups in polynomial-time. Nonetheless, we use count-free WL in tandem with bounded non-determinism and limited counting to obtain a new upper bound of <span class="mathjax-tex">$\beta _{1}\textsf {MAC}^{0}(\textsf {FOLL})$</span> for isomorphism testing of Abelian groups. This improves upon the previous <span class="mathjax-tex">$\textsf {TC}^{0}(\textsf {FOLL})$</span> upper bound due to Chattopadhyay, Torán, & Wagner (<i>ACM Trans. Comput. Theory</i>, 2013).</p></div></div></section><div class="c-article-section__content c-article-section__content--separator"><p>ML thanks Keith Kearnes for helpful discussions, which led to a better understanding of the Hella-style pebble game. ML also wishes to thank Richard Lipton for helpful discussions regarding previous results. We wish to thank J. Brachter and P. Schweitzer for helpful feedback. JAG was partially supported by NSF award DMS-1750319 and NSF CAREER award CCF-2047756 and during this work. ML was partially supported by J. 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On the Parallel Complexity of Group Isomorphism via Weisfeiler–Leman | SpringerLink