Token Sliding Reconfiguration on DAGs

TL;DR

The paper studies Independent Set Reconfiguration under Token Sliding (ISR-DTS) on DAGs, uncovering a sharp depth-based complexity landscape: ISR-DTS is polynomial for DAGs of depth $2$, NP-complete for depth $3$, and $W[1]$-hard for depth $4$ when parameterized by the number of tokens $k$. It achieves an FPT result when parameterized by the combined parameter of treewidth and $k$, by adapting the galactic reconfiguration framework to directed graphs and performing dynamic programming over tree decompositions. The authors also extend insights to undirected graphs under a token-visit restriction and discuss implications for the broader open problem of ISR-TS on bounded-treewidth graphs. Overall, the work provides tight hardness boundaries, a novel directed-graph DP framework, and practical parameterized strategies for reconfiguration problems on restricted graph classes.

Abstract

Given a graph $G$ and two independent sets of same size, the Independent Set Reconfiguration Problem under token sliding ask whether one can, in a step by step manner, transform the first independent set into the second one. In each step we must preserve the condition of independence. Further, referring to solution vertices as tokens, we are only permitted to slide a token along an edge. Until the recent work of Ito et al. [Ito et al. MFCS 2022] this problem was only considered on undirected graphs. In this work, we study reconfiguration under token sliding focusing on DAGs. We present a complete dichotomy of intractability in regard to the depth of the DAG, by proving that this problem is NP-complete for DAGs of depth 3 and $\textrm{W}[1]$-hard for depth 4 when parameterized by the number of tokens $k$, and that these bounds are tight. Further, we prove that it is fixed parameter tractable on DAGs parameterized by the combination of treewidth and $k$. We show that this result applies to undirected graphs, when the number of times a token can visit a vertex is restricted.

Figures (4)

• Figure 1: Schematic of the graph resulting from a 3-SAT formula, where the clause $c$ contains the positive literal $x$.
• Figure 2: Conceptual view on the graph resulting from the reduction. Arrows represent directed edges between vertices in the two blocs they connected. Unless labeled otherwise the connected is complete. Dashed arrows have no different meaning and are only stylized for clarity.
• Figure 3: Collapsing different graphs into each other.
• Figure 4: A graph, its tree-decomposition (left) and reconfiguration sequences with one token (red) over collapsed graphs (right), where $y_2$ is the right child of $x$. The reconfiguration over $\mathsf{hull}(x)$ and $\mathsf{significant}(y_2)$ can be glued together using \ref{['lem:gluing']} as they collapse to the same sequence over $\mathsf{warp}(y_2)$. The reconfiguration sequences are presented with repeating configuration for clarity.

Theorems & Definitions (29)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Definition 6
• Theorem 6
• Lemma 7
• Lemma 8
• Lemma 10