Masaaki Kanzaki’s research while affiliated with Japan Advanced Institute of Science and Technology and other places

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.

In combinatorial reconfiguration, the reconfiguration problems on a vertex subset (e.g., an independent set) are well investigated. In these problems, some tokens are placed on a subset of vertices of the graph, and there are three natural reconfiguration rules called “token sliding,” “token jumping,” and “token addition and removal”. In the context of computational complexity of puzzles, the sliding block puzzles play an important role. Depending on the rules and set of pieces, the sliding block puzzles characterize the computational complexity classes including P, NP, and PSPACE. The sliding block puzzles correspond to the token sliding model in the context of combinatorial reconfiguration. On the other hand, a relatively new notion of jumping block puzzles is proposed in puzzle society. This is the counterpart to the token jumping model of the combinatorial reconfiguration problems in the context of block puzzles. We investigate several variants of jumping block puzzles and determine their computational complexities.

In combinatorial reconfiguration, the reconfiguration problems on a vertex subset (e.g., an independent set) are well investigated. In these problems, some tokens are placed on a subset of vertices of the graph, and there are three natural reconfiguration rules called ``token sliding,'' ``token jumping,'' and ``token addition and removal''. In the context of computational complexity of puzzles, the sliding block puzzles play an important role. Depending on the rules and set of pieces, the sliding block puzzles characterize the computational complexity classes including P, NP, and PSPACE. The sliding block puzzles correspond to the token sliding model in the context of combinatorial reconfiguration. On the other hand, a relatively new notion of jumping block puzzles is proposed in puzzle society. This is the counterpart to the token jumping model of the combinatorial reconfiguration problems in the context of block puzzles. We investigate several variants of jumping block puzzles and determine their computational complexities.

Given a graph $G = (V,E)$, $A \subseteq V$, and integers k and $\ell$, the \textsc{$(A,\ell)$-Path Packing} problem asks to find k vertex-disjoint paths of length $\ell$ that have endpoints in A and internal points in $V \setminus A$. We study the parameterized complexity of this problem with parameters $|A|$, $\ell$, 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 $\ell \le 3$, while it is NP-complete for constant $\ell \ge 4$. We also show that the problem is W[1]-hard parameterized by pathwidth${}+|A|$, while it is fixed-parameter tractable parameterized by treewidth${}+\ell$.

Given a graph $G = (V,E)$, $A \subseteq V$, and integers k and $\ell$, the $(A,\ell )$ -Path Packing problem asks to find k vertex-disjoint paths of length $\ell$ that have endpoints in A and internal points in $V \setminus A$. We study the parameterized complexity of this problem with parameters |A|, $\ell$, 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 $\ell \le 3$, while it is NP-complete for constant $\ell \ge 4$. We also show that the problem is W[1]-hard parameterized by pathwidth${}+|A|$, while it is fixed-parameter tractable parameterized by treewidth${}+\ell$.