{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,5]],"date-time":"2025-10-05T04:27:49Z","timestamp":1759638469937,"version":"3.41.0"},"reference-count":9,"publisher":"Association for Computing Machinery (ACM)","issue":"3","license":[{"start":{"date-parts":[[2017,9,18]],"date-time":"2017-09-18T00:00:00Z","timestamp":1505692800000},"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. Comput. Theory"],"published-print":{"date-parts":[[2017,9,30]]},"abstract":"\n The C\n losest<\/jats:sc>\n S\n ubstring<\/jats:sc>\n P\n roblem<\/jats:sc>\n is to decide, for given strings s\n 1<\/jats:sub>\n , \u2026 , s\n k<\/jats:sub>\n of length at most \u2113 and numbers\n m<\/jats:italic>\n and\n d<\/jats:italic>\n , whether there is a length-\n m<\/jats:italic>\n string\n s<\/jats:italic>\n and length-\n m<\/jats:italic>\n substrings\n s<\/jats:italic>\n \u2032<\/jats:sup>\n i<\/jats:sub>\n of\n s<\/jats:italic>\n i<\/jats:sub>\n , such that\n s<\/jats:italic>\n has a Hamming distance of at most\n d<\/jats:italic>\n from each\n s<\/jats:italic>\n \u2032<\/jats:sup>\n i<\/jats:sub>\n . If we instead require the sum of all the Hamming distances between\n s<\/jats:italic>\n and each\n s<\/jats:italic>\n \u2032<\/jats:sup>\n i<\/jats:sub>\n to be bounded by\n d<\/jats:italic>\n , then it is called the C\n onsensus<\/jats:sc>\n P\n atterns<\/jats:sc>\n P\n roblem<\/jats:sc>\n . We contribute to the parameterised complexity analysis of these classical NP-hard string problems by investigating the parameter (\u2113 \u2212\n m<\/jats:italic>\n ), i.e., the length difference between input and solution strings. For most combinations of (\u2113 \u2212\n m<\/jats:italic>\n ) and one of the classical parameters (\n m<\/jats:italic>\n , \u2113,\n k<\/jats:italic>\n , or\n d<\/jats:italic>\n ), we obtain fixed-parameter tractability. However, even for constant (\u2113 \u2212\n m<\/jats:italic>\n ) and constant alphabet size, both problems remain NP-hard. While this follows from known results with respect to the C\n losest<\/jats:sc>\n S\n ubstring<\/jats:sc>\n , we need a new reduction in the case of the C\n onsensus<\/jats:sc>\n P\n atterns<\/jats:sc>\n . As a by-product of this reduction, we obtain an exact exponential-time algorithm for both problems, which is based on an alphabet reduction.\n <\/jats:p>","DOI":"10.1145\/3110290","type":"journal-article","created":{"date-parts":[[2017,9,18]],"date-time":"2017-09-18T12:20:54Z","timestamp":1505737254000},"page":"1-18","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":4,"title":["Finding Consensus Strings with Small Length Difference between Input and Solution Strings"],"prefix":"10.1145","volume":"9","author":[{"given":"Markus L.","family":"Schmid","sequence":"first","affiliation":[{"name":"Universit\u00e4t Trier, Trier"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2017,9,18]]},"reference":[{"key":"e_1_2_1_1_1","first-page":"31","article-title":"Multivariate algorithmics for NP-hard string problems","volume":"114","author":"Bulteau L.","year":"2014","journal-title":"Bulletin of the EATCS"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(03)00320-7"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-006-0011-4"},{"key":"e_1_2_1_4_1","unstructured":"J. Flum and M. Grohe. 2006. Parameterized Complexity Theory. Springer-Verlag New York Inc. J. Flum and M. Grohe. 2006. Parameterized Complexity Theory. Springer-Verlag New York Inc."},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02679443"},{"key":"e_1_2_1_6_1","unstructured":"M. R. Garey and D. S. Johnson. 1979. Computers and Intractability. W. H. Freeman and Company. M. R. Garey and D. S. Johnson. 1979. Computers and Intractability. W. H. Freeman and Company."},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00453-003-1028-3"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1137\/060673898"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-48054-0_45"}],"container-title":["ACM Transactions on Computation Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3110290","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3110290","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T04:38:44Z","timestamp":1750221524000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3110290"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,9,18]]},"references-count":9,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2017,9,30]]}},"alternative-id":["10.1145\/3110290"],"URL":"https:\/\/doi.org\/10.1145\/3110290","relation":{},"ISSN":["1942-3454","1942-3462"],"issn-type":[{"type":"print","value":"1942-3454"},{"type":"electronic","value":"1942-3462"}],"subject":[],"published":{"date-parts":[[2017,9,18]]},"assertion":[{"value":"2016-12-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2017-06-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2017-09-18","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}