Parameterized Complexity of (A,ℓ)-Path Packing

Given a graph G = ( V , E ) , A ⊆ V , and integers k and ℓ , the ( A , ℓ ) -Path Packing problem asks to find k vertex-disjoint paths of length exactly ℓ that have endpoints in A and internal points in V \ A . We study the parameterized complexity of this problem with parameters | A |, ℓ , k , treew...

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Published in	Algorithmica Vol. 84; no. 4; pp. 871 - 895
Main Authors	Belmonte, Rémy, Hanaka, Tesshu, Kanzaki, Masaaki, Kiyomi, Masashi, Kobayashi, Yasuaki, Kobayashi, Yusuke, Lampis, Michael, Ono, Hirotaka, Otachi, Yota
Format	Journal Article
Language	English
Published	New York Springer US 01.04.2022 Springer Nature B.V Springer Verlag
Subjects	Algorithm Analysis and Problem Complexity Algorithms Complexity Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Mathematics of Computing Parameterization Parameters Polynomials Special Issue on Combinatorial Algorithms (IWOCA 2020) Theory of Computation path packing Fixed-parameter tractability Treewidth
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Abstract	Given a graph G = ( V , E ) , A ⊆ V , and integers k and ℓ , the ( A , ℓ ) -Path Packing problem asks to find k vertex-disjoint paths of length exactly ℓ that have endpoints in A and internal points in V \ A . We study the parameterized complexity of this problem with parameters \| A \|, ℓ , k , 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 ℓ ≤ 3 , while it is NP-complete for constant ℓ ≥ 4 . We also show that the problem is W[1]-hard parameterized by pathwidth + \| A \| , while it is fixed-parameter tractable parameterized by treewidth + ℓ . Additionally, we study a variant called Short A -Path Packing that asks to find k vertex-disjoint paths of length at most ℓ . 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 \| A \| or ℓ is a constant.
AbstractList	Given a graph G = ( V , E ) , A ⊆ V , and integers k and ℓ , the ( A , ℓ ) -Path Packing problem asks to find k vertex-disjoint paths of length exactly ℓ that have endpoints in A and internal points in V \ A . We study the parameterized complexity of this problem with parameters \| A \|, ℓ , k , 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 ℓ ≤ 3 , while it is NP-complete for constant ℓ ≥ 4 . We also show that the problem is W[1]-hard parameterized by pathwidth + \| A \| , while it is fixed-parameter tractable parameterized by treewidth + ℓ . Additionally, we study a variant called Short A -Path Packing that asks to find k vertex-disjoint paths of length at most ℓ . 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 \| A \| or ℓ is a constant. Abstract Given a graph $$G = (V,E)$$ G = ( V , E ) , $$A \subseteq V$$ A ⊆ V , and integers k and $$\ell $$ ℓ , the $$(A,\ell )$$ ( A , ℓ ) -Path Packing problem asks to find k vertex-disjoint paths of length exactly $$\ell $$ ℓ that have endpoints in A and internal points in $$V{\setminus }A$$ V \ A . We study the parameterized complexity of this problem with parameters \| A \|, $$\ell $$ ℓ , k , 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 $$\ell \le 3$$ ℓ ≤ 3 , while it is NP-complete for constant $$\ell \ge 4$$ ℓ ≥ 4 . We also show that the problem is W[1]-hard parameterized by pathwidth $+\|A\|$$ + \| A \| , while it is fixed-parameter tractable parameterized by treewidth $+\ell $$ + ℓ . Additionally, we study a variant called Short A -Path Packing that asks to find k vertex-disjoint paths of length at most $$\ell $$ ℓ . 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 \| A \| or $$\ell $$ ℓ is a constant. Given a graph G=(V,E), A⊆V, and integers k and ℓ, the (A,ℓ)-Path Packing problem asks to find k vertex-disjoint paths of length exactly ℓ that have endpoints in A and internal points in V\A. We study the parameterized complexity of this problem with parameters \|A\|, ℓ, k, 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 ℓ≤3, while it is NP-complete for constant ℓ≥4. We also show that the problem is W[1]-hard parameterized by pathwidth+\|A\|, while it is fixed-parameter tractable parameterized by treewidth+ℓ. Additionally, we study a variant called Short A-Path Packing that asks to find k vertex-disjoint paths of length at mostℓ. 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 \|A\| or ℓ is a constant. Abstract Given a graph $$G = (V,E)$$ G = ( V , E ) , $$A \subseteq V$$ A ⊆ V , and integers k and $$\ell $$ ℓ , the $$(A,\ell )$$ ( A , ℓ ) -Path Packing problem asks to find k vertex-disjoint paths of length exactly $$\ell $$ ℓ that have endpoints in A and internal points in $$V{\setminus }A$$ V \ A . We study the parameterized complexity of this problem with parameters \| A \|, $$\ell $$ ℓ , k , 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 $$\ell \le 3$$ ℓ ≤ 3 , while it is NP-complete for constant $$\ell \ge 4$$ ℓ ≥ 4 . We also show that the problem is W[1]-hard parameterized by pathwidth $+\|A\|$$ + \| A \| , while it is fixed-parameter tractable parameterized by treewidth $+\ell $$ + ℓ . Additionally, we study a variant called Short A -Path Packing that asks to find k vertex-disjoint paths of length at most $$\ell $$ ℓ . 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 \| A \| or $$\ell $$ ℓ is a constant.
Author	Lampis, Michael Belmonte, Rémy Otachi, Yota Kobayashi, Yasuaki Kiyomi, Masashi Kobayashi, Yusuke Ono, Hirotaka Hanaka, Tesshu Kanzaki, Masaaki
Author_xml	– sequence: 1 givenname: Rémy surname: Belmonte fullname: Belmonte, Rémy organization: The University of Electro-Communications, PSL University, CNRS, LAMSADE, Université Paris-Dauphine – sequence: 2 givenname: Tesshu surname: Hanaka fullname: Hanaka, Tesshu organization: Nagoya University – sequence: 3 givenname: Masaaki surname: Kanzaki fullname: Kanzaki, Masaaki organization: Japan Advanced Institute of Science and Technology – sequence: 4 givenname: Masashi surname: Kiyomi fullname: Kiyomi, Masashi organization: Seikei University – sequence: 5 givenname: Yasuaki surname: Kobayashi fullname: Kobayashi, Yasuaki organization: Kyoto University – sequence: 6 givenname: Yusuke surname: Kobayashi fullname: Kobayashi, Yusuke organization: Kyoto University – sequence: 7 givenname: Michael surname: Lampis fullname: Lampis, Michael organization: PSL University, CNRS, LAMSADE, Université Paris-Dauphine – sequence: 8 givenname: Hirotaka surname: Ono fullname: Ono, Hirotaka organization: Nagoya University – sequence: 9 givenname: Yota orcidid: 0000-0002-0087-853X surname: Otachi fullname: Otachi, Yota email: otachi@nagoya-u.jp organization: Nagoya University
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Cites_doi	10.1007/BF01190507 10.1007/BF02066678 10.1051/ita/1992260302571 10.1093/comjnl/bxm052 10.1016/j.disc.2007.07.073 10.1145/210332.210337 10.1016/S0304-3975(02)00577-7 10.1016/S0304-3975(97)00228-4 10.1007/s00493-008-2157-8 10.1016/0012-365X(93)90223-G 10.1016/j.tcs.2008.09.065 10.1137/0211056 10.1016/0012-365X(85)90046-9 10.1007/978-3-642-27875-4 10.1016/j.orl.2006.12.004 10.1007/BF01226465 10.1007/s00493-006-0030-1 10.1016/j.tcs.2009.11.003 10.1016/S0166-218X(97)00012-7 10.1137/140981265 10.1137/0212040 10.1016/S0166-218X(97)00076-0 10.1007/978-3-319-21275-3 10.1109/TC.1981.1675758 10.1137/130947374 10.1137/S0097539793251219 10.1006/inco.1994.1064 10.1016/S0166-218X(97)00121-2 10.1007/s00493-007-0056-z 10.1007/BFb0045375 10.1109/SFCS.1980.12 10.1016/0196-6774(91)90006-K 10.1145/800157.805047
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Snippet	Given a graph G = ( V , E ) , A ⊆ V , and integers k and ℓ , the ( A , ℓ ) -Path Packing problem asks to find k vertex-disjoint paths of length exactly ℓ that... Abstract Given a graph $$G = (V,E)$$ G = ( V , E ) , $$A \subseteq V$$ A ⊆ V , and integers k and $$\ell $$ ℓ , the $$(A,\ell )$$ ( A , ℓ ) -Path Packing... Given a graph G=(V,E), A⊆V, and integers k and ℓ, the (A,ℓ)-Path Packing problem asks to find k vertex-disjoint paths of length exactly ℓ that have endpoints... Abstract Given a graph $$G = (V,E)$$ G = ( V , E ) , $$A \subseteq V$$ A ⊆ V , and integers k and $$\ell $$ ℓ , the $$(A,\ell )$$ ( A , ℓ ) -Path Packing...
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Complexity Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Mathematics of Computing Parameterization Parameters Polynomials Special Issue on Combinatorial Algorithms (IWOCA 2020) Theory of Computation
Title	Parameterized Complexity of (A,ℓ)-Path Packing
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