[Paper Review] Polynomial-Time Algorithms for Sliding Tokens on Cactus Graphs and Block Graphs

Duc A. Hoang, Ryuhei Uehara|arXiv (Cornell University)|May 1, 2017

Advanced Graph Theory Research3 citations

TL;DR

This paper presents polynomial-time algorithms for the Sliding Token problem on cactus graphs and block graphs, leveraging structural characterizations to identify forbidden configurations that block transformations. It proves that if no such configurations exist, a valid transformation sequence of polynomial length always exists between any two independent sets.

ABSTRACT

Given two independent sets $I, J$ of a graph $G$, and imagine that a token (coin) is placed at each vertex of $I$. The Sliding Token problem asks if one could transform $I$ to $J$ via a sequence of elementary steps, where each step requires sliding a token from one vertex to one of its neighbors so that the resulting set of vertices where tokens are placed remains independent. This problem is $\mathsf{PSPACE}$-complete even for planar graphs of maximum degree $3$ and bounded-treewidth. In this paper, we show that Sliding Token can be solved efficiently for cactus graphs and block graphs, and give upper bounds on the length of a transformation sequence between any two independent sets of these graph classes. Our algorithms are designed based on two main observations. First, all structures that forbid the existence of a sequence of token slidings between $I$ and $J$, if exist, can be found in polynomial time. A sufficient condition for determining no-instances can be easily derived using this characterization. Second, without such forbidden structures, a sequence of token slidings between $I$ and $J$ does exist. In this case, one can indeed transform $I$ to $J$ (and vice versa) using a polynomial number of token-slides.

Motivation & Objective

• To determine the computational complexity of the Sliding Token problem on cactus graphs and block graphs.
• To identify structural obstructions that prevent transformation between independent sets.
• To develop efficient algorithms that either confirm the existence of a transformation or detect impossibility.
• To establish upper bounds on the length of transformation sequences in these graph classes.

Proposed method

• Identifies all structures that forbid token sliding sequences between independent sets using polynomial-time characterization.
• Uses a sufficient condition based on forbidden structures to quickly determine no-instances.
• Employs a constructive approach that generates transformation sequences when no forbidden structures exist.
• Leverages the tree-like structure of cactus and block graphs to ensure polynomial-time solvability.
• Applies dynamic decomposition techniques tailored to the hierarchical structure of block graphs and cactus graphs.
• Proves that transformation sequences exist and are bounded by a polynomial in the number of vertices when no obstructions are present.

Experimental results

Research questions

• RQ1 Can the Sliding Token problem be solved in polynomial time on cactus graphs?
• RQ2 Are there efficient structural characterizations of obstructions that prevent token sliding between independent sets in block graphs and cactus graphs?
• RQ3 What is the maximum length of a valid token sliding sequence between any two independent sets in these graph classes?
• RQ4 Under what conditions does a transformation sequence exist between two independent sets in cactus and block graphs?
• RQ5 Can the existence of a transformation be decided efficiently using structural invariants?

Key findings

• The Sliding Token problem is solvable in polynomial time for cactus graphs and block graphs.
• Forbidden structures that block transformations can be detected in polynomial time.
• If no such forbidden structures exist, a transformation sequence between any two independent sets is guaranteed to exist.
• The length of the transformation sequence is bounded by a polynomial in the number of vertices.
• The existence of a transformation can be decided efficiently using a structural characterization.
• The proposed algorithms are constructive and generate valid transformation sequences when possible.