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<jats:inline-formula><jats:alternatives><jats:tex-math>$$G = (V,E)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>G<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>V<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>E<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$$A \\subseteq V$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>A<\/mml:mi>\n                    <mml:mo>\u2286<\/mml:mo>\n                    <mml:mi>V<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and integers <jats:italic>k<\/jats:italic> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, the <jats:inline-formula><jats:alternatives><jats:tex-math>$$(A,\\ell )$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>A<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>\u2113<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula><jats:sc>-Path Packing<\/jats:sc> problem asks to find <jats:italic>k<\/jats:italic> vertex-disjoint paths of length exactly <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> that have endpoints in <jats:italic>A<\/jats:italic> and internal points in <jats:inline-formula><jats:alternatives><jats:tex-math>$$V{\\setminus }A$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>V<\/mml:mi>\n                    <mml:mo>\\<\/mml:mo>\n                    <mml:mi>A<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We study the parameterized complexity of this problem with parameters |<jats:italic>A<\/jats:italic>|, <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, <jats:italic>k<\/jats:italic>, treewidth, pathwidth, and their combinations. We present sharp complexity contrasts with respect to these parameters. Among other results, we show that the problem is polynomial-time solvable when <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell \\le 3$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u2113<\/mml:mi>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mn>3<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, while it is NP-complete for constant <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell \\ge 4$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u2113<\/mml:mi>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:mn>4<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We also show that the problem is W[1]-hard parameterized by pathwidth<jats:inline-formula><jats:alternatives><jats:tex-math>$${}+|A|$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow\/>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mo>|<\/mml:mo>\n                    <mml:mi>A<\/mml:mi>\n                    <mml:mo>|<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, while it is fixed-parameter tractable parameterized by treewidth<jats:inline-formula><jats:alternatives><jats:tex-math>$${}+\\ell $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow\/>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mi>\u2113<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Additionally, we study a variant called <jats:sc>Short <\/jats:sc><jats:italic>A<\/jats:italic><jats:sc>-Path Packing<\/jats:sc> that asks to find <jats:italic>k<\/jats:italic> vertex-disjoint paths of length <jats:italic>at most<\/jats:italic><jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We show that all our positive results on the exact-length version can be translated to this version and show the hardness of the cases where |<jats:italic>A<\/jats:italic>| or <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is a constant.<\/jats:p>","DOI":"10.1007\/s00453-021-00875-y","type":"journal-article","created":{"date-parts":[[2021,10,15]],"date-time":"2021-10-15T16:40:41Z","timestamp":1634316041000},"page":"871-895","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Parameterized Complexity of $$(A,\\ell )$$-Path Packing"],"prefix":"10.1007","volume":"84","author":[{"given":"R\u00e9my","family":"Belmonte","sequence":"first","affiliation":[]},{"given":"Tesshu","family":"Hanaka","sequence":"additional","affiliation":[]},{"given":"Masaaki","family":"Kanzaki","sequence":"additional","affiliation":[]},{"given":"Masashi","family":"Kiyomi","sequence":"additional","affiliation":[]},{"given":"Yasuaki","family":"Kobayashi","sequence":"additional","affiliation":[]},{"given":"Yusuke","family":"Kobayashi","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Lampis","sequence":"additional","affiliation":[]},{"given":"Hirotaka","family":"Ono","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0087-853X","authenticated-orcid":false,"given":"Yota","family":"Otachi","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,10,15]]},"reference":[{"issue":"4","key":"875_CR1","doi-asserted-by":"publisher","first-page":"844","DOI":"10.1145\/210332.210337","volume":"42","author":"N Alon","year":"1995","unstructured":"Alon, N., Yuster, R., Zwick, U.: Color-coding. 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