{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,24]],"date-time":"2026-09-24T12:15:59Z","timestamp":1790252159556,"version":"4.1.0"},"reference-count":67,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/100010663","name":"H2020 European Research Council","doi-asserted-by":"publisher","award":["715744"],"award-info":[{"award-number":["715744"]}],"id":[{"id":"10.13039\/100010663","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100010663","name":"H2020 European Research Council","doi-asserted-by":"publisher","award":["280152"],"award-info":[{"award-number":["280152"]}],"id":[{"id":"10.13039\/100010663","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100010663","name":"H2020 European Research Council","doi-asserted-by":"publisher","award":["755978"],"award-info":[{"award-number":["755978"]}],"id":[{"id":"10.13039\/100010663","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100010663","name":"H2020 European Research Council","doi-asserted-by":"publisher","award":["306992"],"award-info":[{"award-number":["306992"]}],"id":[{"id":"10.13039\/100010663","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>A central problem in parameterized algorithms is to obtain algorithms with running time $f(k)\\cdot n^{O(1)}$ such that $f$ is as slow growing a function of the parameter $k$ as possible. In particular, a large number of basic parameterized problems admit parameterized algorithms where $f(k)$ is single-exponential, that is, $c^k$ for some constant $c$, which makes aiming for such a running time a natural goal for other problems as well. However, there are still plenty of problems where the $f(k)$ appearing in the best-known running time is worse than single-exponential and it remained \u201cslightly superexponential\u201d even after serious attempts to bring it down. A natural question to ask is whether the $f(k)$ appearing in the running time of the best-known algorithms is optimal for any of these problems. In this paper, we examine parameterized problems where $f(k)$ is $k^{O(k)}=2^{O(k\\log k)}$ in the best-known running time, and for a number of such problems we show that the dependence on $k$ in the running time cannot be improved to single-exponential. More precisely we prove the following tight lower bounds, for four natural problems, arising from three different domains: (1) In the Closest String problem, given strings $s_1$, $\\dots$, $s_t$ over an alphabet $\\Sigma$ of length $L$ each, and an integer $d$, the question is whether there exists a string $s$ over $\\Sigma$ of length $L$, such that its hamming distance from each of the strings $s_i$, $1\\leq i \\leq t$, is at most $d$. The pattern matching problem Closest String is known to be solvable in times $2^{O(d\\log d)}\\cdot n^{O(1)}$ and $2^{O(d\\log |\\Sigma|)}\\cdot n^{O(1)}$. We show that there are no $2^{o(d\\log d)}\\cdot n^{O(1)}$ or $2^{o(d\\log |\\Sigma|)}\\cdot n^{O(1)}$ time algorithms, unless the Exponential Time Hypothesis (ETH) fails. (2) The graph embedding problem Distortion, that is, deciding whether a graph $G$ has a metric embedding into the integers with distortion at most $d$ can be solved in time $2^{O(d\\log d)}\\cdot n^{O(1)}$. We show that there is no $2^{o(d\\log d)}\\cdot n^{O(1)}$ time algorithm, unless the ETH fails. (3) The Disjoint Paths problem can be solved in time $2^{O(w\\log w)}\\cdot n^{O(1)}$ on graphs of treewidth at most $w$. We show that there is no $2^{o(w\\log w)}\\cdot n^{O(1)}$ time algorithm, unless the ETH fails. (4) The Chromatic Number problem can be solved in time $2^{O(w\\log w)}\\cdot n^{O(1)}$ on graphs of treewidth at most $w$. We show that there is no $2^{o(w\\log w)}\\cdot n^{O(1)}$ time algorithm, unless the ETH fails. To obtain our results, we first prove the lower bound for variants of basic problems: finding cliques, independent sets, and hitting sets. These artificially constrained variants form a good starting point for proving lower bounds on natural problems without any technical restrictions and could be of independent interest. Several follow-up works have already obtained tight lower bounds by using our framework, and we believe it will prove useful in obtaining even more lower bounds in the future.<\/jats:p>","DOI":"10.1137\/16m1104834","type":"journal-article","created":{"date-parts":[[2018,5,23]],"date-time":"2018-05-23T09:26:28Z","timestamp":1527067588000},"page":"675-702","source":"Crossref","is-referenced-by-count":44,"title":["Slightly Superexponential Parameterized Problems"],"prefix":"10.1137","volume":"47","author":[{"given":"Daniel","family":"Lokshtanov","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"D\u00e1niel","family":"Marx","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Saket","family":"Saurabh","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,5,22]]},"reference":[{"key":"atypb1","first-page":"49","author":"Alon N.","year":"2009","journal-title":"Berlin"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1145\/210332.210337"},{"key":"atypb3","first-page":"225","author":"B\u0103doiu M.","year":"2005","journal-title":"New York"},{"key":"atypb4","first-page":"119","author":"B\u0103doiu M.","year":"2005","journal-title":"Philadelphia"},{"key":"atypb5","first-page":"512","author":"B\u0103doiu M.","year":"2007","journal-title":"Philadelphia"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1613\/jair.638"},{"key":"atypb7","first-page":"33","author":"Bienstock D.","year":"1991","journal-title":"RI"},{"key":"atypb8","first-page":"1116","author":"Bliznets I.","year":"2016","journal-title":"Philadelphia"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1016\/j.ic.2014.12.008"},{"key":"atypb10","first-page":"233","author":"Bonnet \u00c9.","year":"2016","journal-title":"Berlin"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2013.03.008"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-0000(03)00074-6"},{"key":"atypb13","first-page":"1096","author":"Cao Y.","year":"2016","journal-title":"Philadelphia"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2010.06.026"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1145\/1411509.1411511"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcss.2011.01.003"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1287\/ijoc.15.3.233.16078"},{"key":"atypb18","doi-asserted-by":"crossref","unstructured":"M. Cygan, F. V. Fomin, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh,\n                      Parameterized Algorithms\n                      , Springer, 2015,https:\/\/doi.org\/10.1007\/978-3-319-21275-3.","DOI":"10.1007\/978-3-319-21275-3"},{"key":"atypb19","doi-asserted-by":"crossref","unstructured":"M. Cygan, J. Nederlof, M. Pilipczuk, M. Pilipczuk, J. M. M. van Rooij, and J. O. 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Indyk,\n                      Algorithmic applications of low-distortion geometric embeddings\n                      , in Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science (FOCS), IEEE Computer Soc., Los Alamitos, CA, 2001, pp. 10-33.","DOI":"10.1109\/SFCS.2001.959878"},{"key":"atypb46","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-218X(96)00085-6"},{"key":"atypb47","doi-asserted-by":"publisher","DOI":"10.1002\/net.1975.5.1.45"},{"key":"atypb48","first-page":"272","author":"Kenyon C.","year":"2004","journal-title":"New York"},{"key":"atypb49","doi-asserted-by":"publisher","DOI":"10.1016\/S0890-5401(03)00057-9"},{"key":"atypb50","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-004-1108-4"},{"key":"atypb51","doi-asserted-by":"publisher","DOI":"10.1145\/506147.506150"},{"key":"atypb52","first-page":"573","author":"Linial N.","year":"2002","journal-title":"Beijing"},{"key":"atypb53","first-page":"760","author":"Lokshtanov D.","year":"2011","journal-title":"Philadelphia"},{"key":"atypb54","doi-asserted-by":"publisher","DOI":"10.1145\/1061425.1061430"},{"key":"atypb55","doi-asserted-by":"publisher","DOI":"10.1137\/080739069"},{"key":"atypb56","doi-asserted-by":"publisher","DOI":"10.1137\/060673898"},{"key":"atypb57","first-page":"239","author":"Monien B.","year":"1985","journal-title":"Amsterdam"},{"key":"atypb58","unstructured":"H. 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