TY - JOUR

T1 - Parameterized Complexity of (A, ℓ) -Path Packing

AU - Belmonte, Rémy

AU - Hanaka, Tesshu

AU - Kanzaki, Masaaki

AU - Kiyomi, Masashi

AU - Kobayashi, Yasuaki

AU - Kobayashi, Yusuke

AU - Lampis, Michael

AU - Ono, Hirotaka

AU - Otachi, Yota

N1 - Funding Information:
Partially supported by PRC CNRS JSPS Project PARAGA, and by JSPS KAKENHI Grant Numbers JP17H01698, JP18H04091, JP18H05291, JP18K11157, JP18K11168, JP18K11169, JP19K21537, JP20K11692, JP20K19742. A preliminary version appeared in the proceedings of the 31st International Workshop on Combinatorial Algorithms (IWOCA 2020), Lecture Notes in Computer Science 12126 (2020) 43–55.
Publisher Copyright:
© 2021, The Author(s).

PY - 2022/4

Y1 - 2022/4

N2 - 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.

AB - 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.

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U2 - 10.1007/s00453-021-00875-y

DO - 10.1007/s00453-021-00875-y

M3 - Article

AN - SCOPUS:85117121393

VL - 84

SP - 871

EP - 895

JO - Algorithmica

JF - Algorithmica

SN - 0178-4617

IS - 4

ER -