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Comput. Theory"],"published-print":{"date-parts":[[2021,12,31]]},"abstract":"\n A graph operation that\n contracts edges<\/jats:italic>\n is one of the fundamental operations in the theory of graph minors. Parameterized Complexity of editing to a family of graphs by contracting\n k<\/jats:italic>\n edges has recently gained substantial scientific attention, and several new results have been obtained. Some important families of graphs, namely, the subfamilies of chordal graphs, in the context of edge contractions, have proven to be significantly difficult than one might expect. In this article, we study the\n \n F<\/jats:italic>\n -Contraction\n <\/jats:sc>\n problem, where\n F<\/jats:italic>\n is a subfamily of chordal graphs, in the realm of parameterized approximation. Formally, given a graph\n G<\/jats:italic>\n and an integer\n k<\/jats:italic>\n ,\n \n F<\/jats:italic>\n -Contraction\n <\/jats:sc>\n asks whether there exists\n X<\/jats:italic>\n \u2286\n E(G)<\/jats:italic>\n such that\n G\/X<\/jats:italic>\n \u2208\n F<\/jats:italic>\n and |\n X<\/jats:italic>\n | \u2264\n k<\/jats:italic>\n . Here,\n G\/X<\/jats:italic>\n is the graph obtained from\n G<\/jats:italic>\n by contracting edges in\n X<\/jats:italic>\n . We obtain the following results for the\n F<\/jats:italic>\n -\n Contraction<\/jats:sc>\n problem:\n \n \n \u2022<\/jats:label>\n \n Clique Contraction<\/jats:sc>\n is known to be\n FPT<\/jats:sans-serif>\n . However, unless NP\u2286 coNP\/\n poly<\/jats:italic>\n , it does not admit a polynomial kernel. We show that it admits a polynomial-size approximate kernelization scheme (\n PSAKS<\/jats:sans-serif>\n ). That is, it admits a (1 + \u03b5)-approximate kernel with\n O<\/jats:italic>\n (\n \n k\n f(\u03b5))<\/jats:sup>\n <\/jats:italic>\n vertices for every \u03b5 > 0.\n <\/jats:p>\n <\/jats:list-item>\n \n \u2022<\/jats:label>\n \n Split Contraction<\/jats:sc>\n is known to be\n W[1]-Hard<\/jats:sans-serif>\n . We deconstruct this intractability result in two ways. First, we give a (2+\u03b5)-approximate polynomial kernel for\n Split Contraction<\/jats:sc>\n (which also implies a factor (2+\u03b5)-\n FPT<\/jats:sans-serif>\n -approximation algorithm for\n Split Contraction<\/jats:sc>\n ). Furthermore, we show that, assuming\n Gap-ETH<\/jats:sans-serif>\n , there is no (5\/4-\u03b4)-\n FPT<\/jats:sans-serif>\n -approximation algorithm for\n Split Contraction<\/jats:sc>\n . Here, \u03b5, \u03b4 > 0 are fixed constants.\n <\/jats:p>\n <\/jats:list-item>\n \n \u2022<\/jats:label>\n \n Chordal Contraction<\/jats:sc>\n is known to be\n W[2]-Hard<\/jats:sans-serif>\n . We complement this result by observing that the existing\n W[2]-hardness<\/jats:sans-serif>\n reduction can be adapted to show that, assuming\n FPT<\/jats:sans-serif>\n \u2260\n W[1]<\/jats:sans-serif>\n , there is no\n F(k)<\/jats:italic>\n -\n FPT<\/jats:sans-serif>\n -approximation algorithm for\n Chordal Contraction<\/jats:sc>\n . Here,\n F(k)<\/jats:italic>\n is an arbitrary function depending on\n k<\/jats:italic>\n alone.\n <\/jats:p>\n <\/jats:list-item>\n <\/jats:list>\n We say that an algorithm is an\n h(k)<\/jats:italic>\n -\n FPT<\/jats:sans-serif>\n -approximation algorithm for the\n \n F<\/jats:italic>\n -Contraction\n <\/jats:sc>\n problem, if it runs in\n FPT<\/jats:sans-serif>\n time, and on any input\n (G, k)<\/jats:italic>\n such that there exists\n X<\/jats:italic>\n \u2286\n E(G)<\/jats:italic>\n satisfying\n G\/X<\/jats:italic>\n \u2208\n F<\/jats:italic>\n and |\n X<\/jats:italic>\n | \u2264\n k<\/jats:italic>\n , it outputs an edge set\n Y<\/jats:italic>\n of size at most\n h(k)<\/jats:italic>\n \u010b\n k<\/jats:italic>\n for which\n G\/Y<\/jats:italic>\n is in\n F<\/jats:italic>\n .\n <\/jats:p>","DOI":"10.1145\/3470869","type":"journal-article","created":{"date-parts":[[2021,9,1]],"date-time":"2021-09-01T19:28:46Z","timestamp":1630524526000},"page":"1-40","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":2,"title":["On the Parameterized Approximability of Contraction to Classes of Chordal Graphs"],"prefix":"10.1145","volume":"13","author":[{"given":"Spoorthy","family":"Gunda","sequence":"first","affiliation":[{"name":"Simon Fraser University, Burnaby, Canada"}]},{"given":"Pallavi","family":"Jain","sequence":"additional","affiliation":[{"name":"Indian Institute of Technology Jodhpur, Jodhpur, India"}]},{"given":"Daniel","family":"Lokshtanov","sequence":"additional","affiliation":[{"name":"University of California, Santa Barbara, California, USA"}]},{"given":"Saket","family":"Saurabh","sequence":"additional","affiliation":[{"name":"The Institute of Mathematical Sciences, HBNI, Chennai, India and University of Bergen, Bergen, Norway"}]},{"given":"Prafullkumar","family":"Tale","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbr\u00fccken, Germany"}]}],"member":"320","published-online":{"date-parts":[[2021,9]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00224-018-9892-z"},{"key":"e_1_2_1_2_1","first-page":"1","article-title":"Split contraction: The untold story","volume":"5","author":"Agrawal Akanksha","year":"2017","unstructured":"Akanksha Agrawal , Daniel Lokshtanov , Saket Saurabh , and Meirav Zehavi . 2017 . 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