Parameterized Complexity of (A,ℓ)-Path Packing

• 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
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-021-00875-y

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.