{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T02:50:33Z","timestamp":1774320633736,"version":"3.50.1"},"reference-count":77,"publisher":"Association for Computing Machinery (ACM)","issue":"2","license":[{"start":{"date-parts":[[2018,1,31]],"date-time":"2018-01-31T00:00:00Z","timestamp":1517356800000},"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":["J. ACM"],"published-print":{"date-parts":[[2018,4,30]]},"abstract":"<jats:p>\n            Two of the most widely used approaches to obtain polynomial-time approximation schemes (PTASs) on planar graphs are the Lipton-Tarjan separator-based approach and Baker\u2019s approach. In 2005, Demaine and Hajiaghayi strengthened both approaches using bidimensionality and obtained efficient polynomial-time approximation schemes (EPTASs) for several problems, including C\n            <jats:sc>onnected<\/jats:sc>\n            D\n            <jats:sc>ominating<\/jats:sc>\n            S\n            <jats:sc>et<\/jats:sc>\n            and F\n            <jats:sc>eedback<\/jats:sc>\n            V\n            <jats:sc>ertex<\/jats:sc>\n            S\n            <jats:sc>et<\/jats:sc>\n            . In this work, we unify the two strengthened approaches to combine the best of both worlds. We develop a framework allowing the design of EPTAS on classes of graphs with the subquadratic grid minor (SQGM) property. Roughly speaking, a class of graphs has the SQGM property if, for every graph\n            <jats:italic>G<\/jats:italic>\n            from the class, the fact that\n            <jats:italic>G<\/jats:italic>\n            contains no\n            <jats:italic>t<\/jats:italic>\n            \u00d7\n            <jats:italic>t<\/jats:italic>\n            grid as a minor guarantees that the treewidth of\n            <jats:italic>G<\/jats:italic>\n            is subquadratic in\n            <jats:italic>t<\/jats:italic>\n            . For example, the class of planar graphs and, more generally, classes of graphs excluding some fixed graph as a minor, have the SQGM property. At the heart of our framework is a decomposition lemma stating that for \u201cmost\u201d bidimensional problems on a graph class G with the SQGM property, there is a polynomial-time algorithm that, given a graph G \u03f5 G as input and an \u03f5 &gt; 0, outputs a vertex set\n            <jats:italic>X<\/jats:italic>\n            of size \u03f5 \u010b OPT such that the treewidth of\n            <jats:italic>G<\/jats:italic>\n            -\n            <jats:italic>X<\/jats:italic>\n            is\n            <jats:italic>f<\/jats:italic>\n            (\u03f5). Here, OPT is the objective function value of the problem in question and\n            <jats:italic>f<\/jats:italic>\n            is a function depending only on \u03f5. This allows us to obtain EPTASs on (apex)-minor-free graphs for all problems covered by the previous framework as well as for a wide range of packing problems, partial covering problems and problems that are neither closed under taking minors nor contractions. To the best of our knowledge, for many of these problems\u2014including C\n            <jats:sc>ycle<\/jats:sc>\n            P\n            <jats:sc>acking<\/jats:sc>\n            ,\n            <jats:italic>F<\/jats:italic>\n            -P\n            <jats:sc>acking<\/jats:sc>\n            ,\n            <jats:italic>F<\/jats:italic>\n            -D\n            <jats:sc>eletion<\/jats:sc>\n            , M\n            <jats:sc>ax<\/jats:sc>\n            L\n            <jats:sc>eaf<\/jats:sc>\n            S\n            <jats:sc>panning<\/jats:sc>\n            T\n            <jats:sc>ree<\/jats:sc>\n            , or P\n            <jats:sc>artial<\/jats:sc>\n            <jats:italic>r<\/jats:italic>\n            -D\n            <jats:sc>ominating<\/jats:sc>\n            S\n            <jats:sc>et<\/jats:sc>\n            \u2014no EPTASs, even on planar graphs, were previously known.\n          <\/jats:p>\n          <jats:p>We also prove novel excluded grid theorems in unit disk and map graphs without large cliques. Using these theorems, we show that these classes of graphs have the SQGM property. Based on the developed framework, we design EPTASs and subexponential time parameterized algorithms for various classes of problems on unit disk and map graphs.<\/jats:p>","DOI":"10.1145\/3154833","type":"journal-article","created":{"date-parts":[[2018,2,1]],"date-time":"2018-02-01T13:10:52Z","timestamp":1517490652000},"page":"1-44","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":14,"title":["Excluded Grid Minors and Efficient Polynomial-Time Approximation Schemes"],"prefix":"10.1145","volume":"65","author":[{"given":"Fedorr V.","family":"Fomin","sequence":"first","affiliation":[{"name":"University of Bergen, Bergen, Norway"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Daniel","family":"Lokshtanov","sequence":"additional","affiliation":[{"name":"University of Bergen, Bergen, Norway"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Saket","family":"Saurabh","sequence":"additional","affiliation":[{"name":"University of Bergen and the Institute of Mathematical Sciences, Chennai, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2018,1,31]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00453-001-0116-5"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1145\/990308.990309"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(98)00158-3"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1016\/0196-6774(91)90006-K"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1145\/1502793.1502794"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1145\/174644.174650"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539796305109"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1145\/2973749"},{"key":"e_1_2_1_9_1","volume-title":"Tovey","author":"Borie Richard B.","year":"1992","unstructured":"Richard B. 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