How to Reconfigure Your Alliances

2025-09-10

AI Summary

This paper studies the reconfiguration problem for alliances in graphs: given two alliances of equal size, does there exist a reconfiguration sequence of length at most ℓ, where each step modifies the alliance by at most k vertices via sliding, jumping, or vertex addition/deletion? We introduce the novel notion of *reconfiguration monotonicity* to uniformly model both vertex additions/deletions and jumps. Using graph-theoretic modeling, PSPACE-completeness analysis, and parameterized algorithm design, we develop multiple fixed-parameter tractable (FPT) algorithms parameterized by neighborhood diversity, k + ℓ, and other structural parameters. We prove the problem is generally PSPACE-complete, yet solvable in LOGSPACE for certain restricted cases. Our main contributions are: (i) establishing the first unified framework for alliance reconfiguration; (ii) precisely characterizing its computational complexity landscape; and (iii) achieving fixed-parameter tractability for nontrivial parameter combinations—marking the first such result for alliance reconfiguration.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Different variations of alliances in graphs have been introduced into the graph-theoretic literature about twenty years ago. More broadly speaking, they can be interpreted as groups that collaborate to achieve a common goal, for instance, defending themselves against possible attacks from outside. In this paper, we initiate the study of reconfiguring alliances. This means that, with the understanding of having an interconnection map given by a graph, we look at two alliances of the same size~$k$ and investigate if there is a reconfiguration sequence (of length at most~$ell$) formed by alliances of size (at most)~$k$ that transfers one alliance into the other one. Here, we consider different (now classical) movements of tokens: sliding, jumping, addition/removal. We link the latter two regimes by introducing the concept of reconfiguration monotonicity. Concerning classical complexity, most of these reconfiguration problems are extsf{PSPACE}-complete, although some are solvable in extsf{Log-SPACE}. We also consider these reconfiguration questions through the lense of parameterized algorithms and prove various extsf{FPT}-results, in particular concerning the combined parameter $k+ell$ or neighborhood diversity together with $k$ or neighborhood diversity together with $k$.

Problem

• Reconfiguring graph alliances using token movements
• Studying complexity of PSPACE-complete reconfiguration problems
• Developing parameterized algorithms for alliance reconfiguration

Innovation

• Introduces alliance reconfiguration with sliding and jumping tokens
• Links token movements via reconfiguration monotonicity concept
• Provides FPT results for parameter k plus l