[Paper Review] Fixed-Parameter Tractability of Token Jumping on Planar Graphs

Takehiro Ito, Marcin Kamiński+1 more|arXiv (Cornell University)|Jun 25, 2014

Advanced Graph Theory Research11 references4 citations

TL;DR

This paper presents a fixed-parameter tractable algorithm for the token jumping problem on planar graphs and more generally on $K_{3,t}$-forbidden graphs, parameterized solely by the number of tokens. The algorithm uses structural graph properties—specifically, the absence of $K_{3,t}$ subgraphs—to identify buffer spaces and shrink the graph to a fixed-parameter size, enabling efficient computation of a shortest reconfiguration sequence between independent sets of equal size.

ABSTRACT

Suppose that we are given two independent sets $I_0$ and $I_r$ of a graph such that $|I_0| = |I_r|$, and imagine that a token is placed on each vertex in $I_0$. The token jumping problem is to determine whether there exists a sequence of independent sets which transforms $I_0$ into $I_r$ so that each independent set in the sequence results from the previous one by moving exactly one token to another vertex. This problem is known to be PSPACE-complete even for planar graphs of maximum degree three, and W[1]-hard for general graphs when parameterized by the number of tokens. In this paper, we present a fixed-parameter algorithm for the token jumping problem on planar graphs, where the parameter is only the number of tokens. Furthermore, the algorithm can be modified so that it finds a shortest sequence for a yes-instance. The same scheme of the algorithms can be applied to a wider class of graphs, $K_{3,t}$-free graphs for any fixed integer $t \ge 3$, and it yields fixed-parameter algorithms.

Motivation & Objective

• To address the fixed-parameter tractability of the token jumping problem on planar graphs, where the problem is known to be PSPACE-complete and W[1]-hard under standard parameterization.
• To extend the algorithmic framework beyond planar graphs to a broader class of graphs that exclude $K_{3,t}$ as a subgraph for fixed $t \geq 3$.
• To design a method that not only determines the existence of a reconfiguration sequence but also computes the shortest such sequence.

Proposed method

• Leverages the structural property that planar graphs and $K_{3,t}$-forbidden graphs do not contain $K_{3,3}$ or $K_{3,t}$ as subgraphs, respectively, to bound graph size.
• Introduces two parameters $\alpha_t = \text{Ramsey}(k,t+3)$ and $\beta_t = \text{Ramsey}((2t+1)k,t+3)$ to define buffer regions with guaranteed independent sets of size $k$.
• Applies Ramsey theory to ensure the existence of sufficiently large independent sets in subgraphs of size $\alpha_t$ and $\beta_t$, enabling safe token movement.
• Uses a shrinking procedure to reduce the graph to size dependent only on $\alpha_t$ and $\beta_t$, preserving the shortest reconfiguration sequence.
• Modifies the algorithm to continue beyond early termination by preserving shortest path information, enabling brute-force search on the reduced graph.
• Employs a dynamic programming or exhaustive search on the shrunk graph to compute the shortest reconfiguration sequence in fixed-parameter time.

Experimental results

Research questions

• RQ1 Can the token jumping problem be solved in fixed-parameter time when parameterized only by the number of tokens on planar graphs, despite its W[1]-hardness on general graphs?
• RQ2 Does the fixed-parameter tractability extend beyond planar graphs to graphs that exclude $K_{3,t}$ as a subgraph for fixed $t \geq 3$?
• RQ3 Can the algorithm be adapted to compute the shortest reconfiguration sequence, not just determine its existence?
• RQ4 What structural properties of graphs enable the existence of buffer spaces that allow safe token reconfiguration?

Key findings

• The token jumping problem on planar graphs is fixed-parameter tractable when parameterized solely by the number of tokens $k$, with running time depending only on $k$.
• The algorithm extends to $K_{3,t}$-forbidden graphs for any fixed $t \geq 3$, with the same fixed-parameter tractability guarantee.
• The running time of the algorithm is bounded by a function of $k$ alone, as $\alpha_t$ and $\beta_t$ are fixed-parameter size, with $\alpha_t = (k+t+1)^{t+2}$ and $\beta_t = ((2t+1)k + t + 1)^{t+2}$.
• The algorithm can be modified to compute a shortest reconfiguration sequence in fixed-parameter time, not just decide existence.
• The key insight is that $K_{3,t}$-forbiddance limits neighborhood density, ensuring that even after token movements, sufficient independent sets exist for buffer space.
• The method preserves the shortest path length in the original graph through careful shrinking and buffer space selection, enabling exact shortest sequence computation on the reduced instance.