{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T17:11:29Z","timestamp":1760202689123,"version":"3.41.0"},"reference-count":24,"publisher":"Association for Computing Machinery (ACM)","issue":"4","license":[{"start":{"date-parts":[[2014,8,1]],"date-time":"2014-08-01T00:00:00Z","timestamp":1406851200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004963","name":"Seventh Framework Programme","doi-asserted-by":"publisher","award":["238381"],"award-info":[{"award-number":["238381"]}],"id":[{"id":"10.13039\/501100004963","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Comput. Theory"],"published-print":{"date-parts":[[2014,8]]},"abstract":"<jats:p>In 1990, Subramanian [1990] defined the complexity class CC as the set of problems log-space reducible to the comparator circuit value problem (CCV). He and Mayr showed that NL \u2286 CC \u2286 P, and proved that in addition to CCV several other problems are complete for CC, including the stable marriage problem, and finding the lexicographically first maximal matching in a bipartite graph. Although the class has not received much attention since then, we are interested in CC because we conjecture that it is incomparable with the parallel class NC which also satisfies NL \u2286 NC \u2286 P, and note that this conjecture implies that none of the CC-complete problems has an efficient polylog time parallel algorithm. We provide evidence for our conjecture by giving oracle settings in which relativized CC and relativized NC are incomparable.<\/jats:p>\n          <jats:p>We give several alternative definitions of CC, including (among others) the class of problems computed by uniform polynomial-size families of comparator circuits supplied with copies of the input and its negation, the class of problems AC0-reducible to Ccv, and the class of problems computed by uniform AC0 circuits with AXccv gates. We also give a machine model for CC, which corresponds to its characterization as log-space uniform polynomial-size families of comparator circuits. These various characterizations show that CC is a robust class. Our techniques also show that the corresponding function class FCC is closed under composition. The main technical tool we employ is universal comparator circuits.<\/jats:p>\n          <jats:p>Other results include a simpler proof of NL \u2286 CC, a more careful analysis showing the lexicographically first maximal matching problem and its variants are CC-complete under AC0 many-one reductions, and an explanation of the relation between the Gale--Shapley algorithm and Subramanian\u2019s algorithm for stable marriage.<\/jats:p>\n          <jats:p>This article continues the previous work of Cook et al. [2011], which focused on Cook-Nguyen style uniform proof complexity, answering several open questions raised in that article.<\/jats:p>","DOI":"10.1145\/2635822","type":"journal-article","created":{"date-parts":[[2014,8,21]],"date-time":"2014-08-21T12:19:12Z","timestamp":1408623552000},"page":"1-44","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":8,"title":["The complexity of the comparator circuit value problem"],"prefix":"10.1145","volume":"6","author":[{"given":"Stephen A.","family":"Cook","sequence":"first","affiliation":[{"name":"University of Toronto"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuval","family":"Filmus","sequence":"additional","affiliation":[{"name":"University of Toronto"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dai Tri Man","family":"L\u00ea","sequence":"additional","affiliation":[{"name":"University of Toronto"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2014,8]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"crossref","unstructured":"K.\n      Aehlig S.\n      Cook and \n      P.\n      Nguyen\n  . \n  2007\n  . Relativizing small complexity classes and their theories. In Computer Science Logic Jacques Duparc and Thomas Henzinger Eds. Lecture Notes in Computer Science vol. \n  4646 Springer Berlin\/Heidelberg 374--388.   K. Aehlig S. Cook and P. Nguyen. 2007. Relativizing small complexity classes and their theories. In Computer Science Logic Jacques Duparc and Thomas Henzinger Eds. Lecture Notes in Computer Science vol. 4646 Springer Berlin\/Heidelberg 374--388.","DOI":"10.1007\/978-3-540-74915-8_29"},{"key":"e_1_2_1_2_1","volume-title":"Parallel and Distributed Computing (Advances in Computing Research)","volume":"4","author":"Anderson R.","unstructured":"R. Anderson and E. Mayr . 1987. Parallelism and greedy algorithms . In Parallel and Distributed Computing (Advances in Computing Research) , vol. 4 , JAI Press, 17--38. R. Anderson and E. Mayr. 1987. Parallelism and greedy algorithms. In Parallel and Distributed Computing (Advances in Computing Research), vol. 4, JAI Press, 17--38."},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0000(90)90022-D"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1145\/1468075.1468121"},{"key":"e_1_2_1_5_1","doi-asserted-by":"crossref","unstructured":"S. Cook and P. Nguyen. 2010. Logical Foundations of Proof Complexity. Cambridge University Press.   S. Cook and P. Nguyen. 2010. Logical Foundations of Proof Complexity . Cambridge University Press.","DOI":"10.1017\/CBO9780511676277"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0019-9958(85)80041-3"},{"key":"e_1_2_1_7_1","unstructured":"S. A. Cook D. T. M. L\u00ea and Y. Ye. 2011. Complexity classes and theories for the comparator circuit value problem. arXiv abs\/1106.4142.  S. A. Cook D. T. M. L\u00ea and Y. Ye. 2011. Complexity classes and theories for the comparator circuit value problem. arXiv abs\/1106.4142."},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539792226278"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0000(92)90048-N"},{"volume-title":"Stable Networks and Product Graphs","author":"Feder T.","key":"e_1_2_1_10_1","unstructured":"T. Feder . 1995. Stable Networks and Product Graphs . American Mathematical Society , Boston, MA . T. Feder. 1995. Stable Networks and Product Graphs. American Mathematical Society, Boston, MA."},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.1080\/00029890.1962.11989827"},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1145\/1008354.1008356"},{"key":"e_1_2_1_13_1","doi-asserted-by":"publisher","DOI":"10.5555\/203244"},{"key":"e_1_2_1_14_1","doi-asserted-by":"publisher","DOI":"10.1002\/cplx.v14:2"},{"key":"e_1_2_1_15_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02579407"},{"key":"e_1_2_1_16_1","doi-asserted-by":"publisher","DOI":"10.1145\/4221.4226"},{"key":"e_1_2_1_17_1","volume-title":"25th International Workshop\/20th Annual Conference of the EACSL (Leibniz International Proceedings in Informatics (LIPIcs)), Marc Bezem Ed.","volume":"12","author":"L\u00ea D. T. M.","unstructured":"D. T. M. L\u00ea , S. A. Cook , and Y. Ye . 2011. A formal theory for the complexity class associated with the stable marriage problem. In Computer Science Logic (CSL\u201911) - 25th International Workshop\/20th Annual Conference of the EACSL (Leibniz International Proceedings in Informatics (LIPIcs)), Marc Bezem Ed. , vol. 12 , Dagstuhl, Germany, 381--395. D. T. M. L\u00ea, S. A. Cook, and Y. Ye. 2011. A formal theory for the complexity class associated with the stable marriage problem. 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