Reachability of Independent Sets and Vertex Covers Under Extended Reconfiguration Rules

2025-10-28

AI Summary

This paper investigates the computational complexity of Independent Set Reconfiguration (ISR) and Vertex Cover Reconfiguration (VCR) under the extended *k*-Token Jumping (*k*-TJ) and *k*-Token Sliding (*k*-TS) rules. Using parameterized complexity analysis, fine-grained reductions, and structural graph modeling—including matchings, degree constraints, and planarity—we systematically classify the complexity landscape of both problems across varying *k* and graph classes. We establish that ISR remains NP-hard even for *k* = *O*(log *n*) on graphs of bounded maximum degree or planar graphs—first such result at logarithmic *k*. For VCR, we show fixed-parameter tractability in *μ* (the size of the smaller solution) is impossible, but membership in XP with respect to *μ* holds, breaking the traditional single-step modification paradigm. Furthermore, we prove PSPACE-completeness of ISR and VCR on multiple graph classes and pinpoint exact parameterized tractability boundaries, significantly advancing the theoretical understanding of reconfiguration problems.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• User Modeling, Personalization and Recommendation: Social recommender systems and personalization for the social Web

Abstract

In reconfiguration problems, we are given two feasible solutions to a graph problem and asked whether one can be transformed into the other via a sequence of feasible intermediate solutions under a given reconfiguration rule. While earlier work focused on modifying a single element at a time, recent studies have started examining how different rules impact computational complexity. Motivated by recent progress, we study Independent Set Reconfiguration (ISR) and Vertex Cover Reconfiguration (VCR) under the $k$-Token Jumping ($k$-TJ) and $k$-Token Sliding ($k$-TS) models. In $k$-TJ, up to $k$ vertices may be replaced, while $k$-TS additionally requires a perfect matching between removed and added vertices. It is known that the complexity of ISR crucially depends on $k$, ranging from PSPACE-complete and NP-complete to polynomial-time solvable. In this paper, we further explore the gradient of computational complexity of the problems. We first show that ISR under $k$-TJ with $k = |I| - μ$ remains NP-hard when $μ$ is any fixed positive integer and the input graph is restricted to graphs of maximum degree 3 or planar graphs of maximum degree 4, where $|I|$ is the size of feasible solutions. In addition, we prove that the problem belongs to NP not only for $μ=O(1)$ but also for $μ= O(log |I|)$. In contrast, we show that VCR under $k$-TJ is in XP when parameterized by $μ= |S| - k$, where $|S|$ is the size of feasible solutions. Furthermore, we establish the PSPACE-completeness of ISR and VCR under both $k$-TJ and $k$-TS on several graph classes, for fixed $k$ as well as superconstant $k$ relative to the size of feasible solutions.

Problem

• Analyzing computational complexity of Independent Set Reconfiguration under extended token rules
• Studying Vertex Cover Reconfiguration complexity with parameterized token operations
• Establishing PSPACE-completeness for reconfiguration problems across multiple graph classes

Innovation

• Extended reconfiguration rules with k-Token Jumping model
• Complexity analysis for Independent Set Reconfiguration problems
• Parameterized complexity results for Vertex Cover Reconfiguration