Token Sliding Reconfiguration on DAGs

2025-04-14

AI Summary

This paper investigates the reconfigurability of independent sets on directed acyclic graphs (DAGs) under the token sliding rule: given two independent sets of equal cardinality, can one be transformed into the other via a sequence of token slides along directed edges while preserving independence at every step? This is the first work to extend token sliding reconfiguration to directed graphs. Our contributions are threefold: (1) We establish a tight complexity dichotomy parameterized by DAG depth—NP-complete for depth 3 and W[1]-hard for depth 4; (2) We design an FPT algorithm parameterized by treewidth plus solution size *k*, overcoming limitations of prior results restricted to undirected graphs; (3) We introduce a novel analytical framework integrating directed path constraints, dynamic programming, and structural reduction. Our results also apply to undirected graphs with bounded token visit counts.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 $ extrm{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.

Problem

• Study token sliding reconfiguration on DAGs
• Determine NP-completeness for DAGs of depth 3
• Prove fixed parameter tractability with treewidth and k

Innovation

• Token sliding on DAGs for reconfiguration
• NP-complete for DAGs of depth 3
• FPT for treewidth and token count