{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,28]],"date-time":"2025-10-28T03:18:27Z","timestamp":1761621507292,"version":"3.41.0"},"reference-count":44,"publisher":"Association for Computing Machinery (ACM)","issue":"2","license":[{"start":{"date-parts":[[2020,3,11]],"date-time":"2020-03-11T00:00:00Z","timestamp":1583884800000},"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":[[2020,4,30]]},"abstract":"<jats:p>\n            In this work, we focus on several completion problems for subclasses of chordal graphs: M\n            <jats:sc>INIMUM<\/jats:sc>\n            F\n            <jats:sc>ILL<\/jats:sc>\n            -I\n            <jats:sc>N<\/jats:sc>\n            , I\n            <jats:sc>NTERVAL<\/jats:sc>\n            C\n            <jats:sc>OMPLETION<\/jats:sc>\n            , P\n            <jats:sc>ROPER<\/jats:sc>\n            I\n            <jats:sc>NTERVAL<\/jats:sc>\n            C\n            <jats:sc>OMPLETION<\/jats:sc>\n            , T\n            <jats:sc>RIVIALLY<\/jats:sc>\n            P\n            <jats:sc>ERFECT<\/jats:sc>\n            C\n            <jats:sc>OMPLETION<\/jats:sc>\n            , and T\n            <jats:sc>HRESHOLD<\/jats:sc>\n            C\n            <jats:sc>OMPLETION<\/jats:sc>\n            . In these problems, the task is to add at most\n            <jats:italic>k<\/jats:italic>\n            edges to a given graph to obtain a chordal, interval, proper interval, threshold, or trivially perfect graph, respectively. We prove the following lower bounds for all these problems, as well as for the related C\n            <jats:sc>HAIN<\/jats:sc>\n            C\n            <jats:sc>OMPLETION<\/jats:sc>\n            problem:\n          <\/jats:p>\n          <jats:p>\n            \u2022 Assuming the Exponential Time Hypothesis, none of these problems can be solved in time 2\n            <jats:sup>O<\/jats:sup>\n            (\n            <jats:italic>n<\/jats:italic>\n            <jats:sup>1\/2<\/jats:sup>\n            \/log\n            <jats:sup>\n              <jats:italic>c<\/jats:italic>\n            <\/jats:sup>\n            <jats:italic>n<\/jats:italic>\n            ) or 2\n            <jats:sup>\n              <jats:italic>O<\/jats:italic>\n            <\/jats:sup>\n            (\n            <jats:italic>k<\/jats:italic>\n            <jats:sup>1\/4<\/jats:sup>\n            \/log\n            <jats:sup>\n              <jats:italic>c<\/jats:italic>\n            <\/jats:sup>\n            <jats:italic>k<\/jats:italic>\n            )\u00b7\n            <jats:italic>n<\/jats:italic>\n            <jats:sup>\n              <jats:italic>O<\/jats:italic>\n              (1)\n            <\/jats:sup>\n            , for some integer\n            <jats:italic>c<\/jats:italic>\n            .\n          <\/jats:p>\n          <jats:p>\n            \u2022 Assuming the non-existence of a subexponential-time approximation scheme for M\n            <jats:sc>IN<\/jats:sc>\n            B\n            <jats:sc>ISECTION<\/jats:sc>\n            on\n            <jats:italic>d<\/jats:italic>\n            -regular graphs, for some constant\n            <jats:italic>d<\/jats:italic>\n            , none of these problems can be solved in time 2\n            <jats:sup>\n              <jats:italic>o<\/jats:italic>\n              (\n              <jats:italic>n<\/jats:italic>\n              )\n            <\/jats:sup>\n            or 2\n            <jats:sup>\n              <jats:italic>o<\/jats:italic>\n              \u221ak)\n            <\/jats:sup>\n            }\u00b7\n            <jats:italic>n<\/jats:italic>\n            <jats:sup>\n              <jats:italic>O<\/jats:italic>\n              (1)\n            <\/jats:sup>\n            .\n          <\/jats:p>\n          <jats:p>\n            For all the aforementioned completion problems, apart from P\n            <jats:sc>ROPER<\/jats:sc>\n            I\n            <jats:sc>NTERVAL<\/jats:sc>\n            C\n            <jats:sc>OMPLETION<\/jats:sc>\n            , FPT algorithms with running time of the form 2\n            <jats:sup>\n              <jats:italic>O<\/jats:italic>\n              (\u221a\n              <jats:italic>k<\/jats:italic>\n              log\n              <jats:italic>k<\/jats:italic>\n              )\n            <\/jats:sup>\n            \u00b7\n            <jats:italic>n<\/jats:italic>\n            <jats:sup>\n              <jats:italic>O<\/jats:italic>\n              (1)\n            <\/jats:sup>\n            are known. Thus, the second result proves that a significant improvement of any of these algorithms would lead to a surprising breakthrough in the design of approximation algorithms for M\n            <jats:sc>IN<\/jats:sc>\n            B\n            <jats:sc>ISECTION<\/jats:sc>\n            .\n          <\/jats:p>\n          <jats:p>\n            To prove our results, we use a reduction methodology based on combining the classic approach of starting with a sparse instance of 3-S\n            <jats:sc>AT<\/jats:sc>\n            , prepared using the Sparsification Lemma, with the existence of almost linear-size Probabilistically Checkable Proofs. Apart from our main results, we also obtain lower bounds excluding the existence of subexponential algorithms for the O\n            <jats:sc>PTIMUM<\/jats:sc>\n            L\n            <jats:sc>INEAR<\/jats:sc>\n            A\n            <jats:sc>RRANGEMENT<\/jats:sc>\n            problem, as well as improved, yet still not tight, lower bounds for F\n            <jats:sc>EEDBACK<\/jats:sc>\n            A\n            <jats:sc>RC<\/jats:sc>\n            S\n            <jats:sc>ET<\/jats:sc>\n            <jats:sc>IN<\/jats:sc>\n            T\n            <jats:sc>OURNAMENTS<\/jats:sc>\n            .\n          <\/jats:p>","DOI":"10.1145\/3381426","type":"journal-article","created":{"date-parts":[[2020,3,11]],"date-time":"2020-03-11T10:44:36Z","timestamp":1583923476000},"page":"1-31","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":4,"title":["Lower Bounds for the Parameterized Complexity of Minimum Fill-in and Other Completion Problems"],"prefix":"10.1145","volume":"16","author":[{"given":"Ivan","family":"Bliznets","sequence":"first","affiliation":[{"name":"National Research University Higher School of Economics, St. Petersburg, Russia"}]},{"given":"Marek","family":"Cygan","sequence":"additional","affiliation":[{"name":"University of Warsaw, Warsaw, Poland"}]},{"given":"Pawe\u0142","family":"Komosa","sequence":"additional","affiliation":[{"name":"University of Warsaw, Warsaw, Poland"}]},{"given":"Micha\u0142","family":"Pilipczuk","sequence":"additional","affiliation":[{"name":"University of Warsaw, Warsaw, Poland"}]},{"given":"Luk\u00e1\u0161","family":"Mach","sequence":"additional","affiliation":[{"name":"University of Warwick, Coventry, England"}]}],"member":"320","published-online":{"date-parts":[[2020,3,11]]},"reference":[{"doi-asserted-by":"publisher","key":"e_1_2_1_1_1","DOI":"10.1145\/1411509.1411513"},{"doi-asserted-by":"publisher","key":"e_1_2_1_2_1","DOI":"10.1137\/050623905"},{"key":"e_1_2_1_3_1","volume-title":"Fast FAST. 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