{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T04:33:33Z","timestamp":1750221213929,"version":"3.41.0"},"reference-count":23,"publisher":"Association for Computing Machinery (ACM)","issue":"2","license":[{"start":{"date-parts":[[2018,12,7]],"date-time":"2018-12-07T00:00:00Z","timestamp":1544140800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Algorithms"],"published-print":{"date-parts":[[2019,4,30]]},"abstract":"<jats:p>\n            The main result of this article is a generalization of the classical blossom algorithm for finding perfect matchings. Our algorithm can efficiently solve Boolean CSPs where each variable appears in exactly two constraints (we call it edge CSP) and all constraints are\n            <jats:italic>even \u0394-matroid<\/jats:italic>\n            relations (represented by lists of tuples). As a consequence of this, we settle the complexity classification of planar Boolean CSPs started by Dvo\u0159\u00e1k and Kupec.\n          <\/jats:p>\n          <jats:p>\n            Using a reduction to even \u0394-matroids, we then extend the tractability result to larger classes of \u0394-matroids that we call\n            <jats:italic>efficiently coverable<\/jats:italic>\n            . It properly includes classes that were known to be tractable before, namely,\n            <jats:italic>co-independent<\/jats:italic>\n            ,\n            <jats:italic>compact<\/jats:italic>\n            ,\n            <jats:italic>local<\/jats:italic>\n            ,\n            <jats:italic>linear<\/jats:italic>\n            , and\n            <jats:italic>binary<\/jats:italic>\n            , with the following caveat: We represent \u0394-matroids by lists of tuples, while the last two use a representation by matrices. Since an\n            <jats:italic>n<\/jats:italic>\n            \u00d7\n            <jats:italic>n<\/jats:italic>\n            matrix can represent exponentially many tuples, our tractability result is not strictly stronger than the known algorithm for linear and binary \u0394-matroids.\n          <\/jats:p>","DOI":"10.1145\/3230649","type":"journal-article","created":{"date-parts":[[2018,12,7]],"date-time":"2018-12-07T13:17:29Z","timestamp":1544188649000},"page":"1-33","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":1,"title":["Even Delta-Matroids and the Complexity of Planar Boolean CSPs"],"prefix":"10.1145","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7338-037X","authenticated-orcid":false,"given":"Alexandr","family":"Kazda","sequence":"first","affiliation":[{"name":"IST Austria and Charles University, Praha, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vladimir","family":"Kolmogorov","sequence":"additional","affiliation":[{"name":"IST Austria, Klosterneuburg, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michal","family":"Rol\u00ednek","sequence":"additional","affiliation":[{"name":"IST Austria and Max Planck Institute for Intelligent Systems, T\u00fcbingen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2018,12,7]]},"reference":[{"volume-title":"The Theory of Graphs and Its Applications","author":"Berge Claude","key":"e_1_2_1_1_1"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218X(92)90272-C"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0895480191222926"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1145\/1970398.1970400"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1145\/1120582.1120584"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2017.37"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1137\/100814585"},{"volume-title":"Proceedings of the 28th International Symposium on Mathematical Foundations of Computer Science (MFCS\u201903)","year":"2003","author":"Dalmau Victor","key":"e_1_2_1_8_1"},{"volume-title":"Proceedings of the International Colloquium on Automata, Languages and Programming (ICALP\u201915)","series-title":"Lecture Notes in Computer Science","author":"Dvo\u0159\u00e1k Zden\u011bk","key":"e_1_2_1_9_1"},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1965-045-4"},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(99)00288-1"},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0895480104445009"},{"key":"e_1_2_1_13_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539794266766"},{"key":"e_1_2_1_14_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0095-8956(03)00039-X"},{"key":"e_1_2_1_15_1","doi-asserted-by":"publisher","DOI":"10.1145\/321850.321852"},{"key":"e_1_2_1_17_1","doi-asserted-by":"publisher","DOI":"10.1137\/0211014"},{"key":"e_1_2_1_18_1","doi-asserted-by":"publisher","DOI":"10.5555\/3039686.3039706"},{"volume-title":"Proceedings of the Conference on Algebraic Methods in Graph Theory. 495--517","year":"1978","author":"Lov\u00e1sz L.","key":"e_1_2_1_19_1"},{"key":"e_1_2_1_20_1","doi-asserted-by":"publisher","DOI":"10.1145\/49097.49099"},{"key":"e_1_2_1_21_1","doi-asserted-by":"publisher","DOI":"10.1145\/800133.804350"},{"volume-title":"Foundations of Constraint Satisfaction","author":"Tsang Edward P. 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