A Note on the Parameterised Complexity of Coverability in Vector Addition Systems

2025-11-24

AI Summary

This study systematically characterizes the parameterized complexity of the coverability problem for Vector Addition Systems (VAS). Given a set $V subseteq mathbb{Z}^d$ of vectors, an initial configuration $s in mathbb{N}^d$, and a target $t in mathbb{N}^d$, the problem asks whether some nonnegative path reaches a configuration that dominates $t$ coordinate-wise. We focus on two natural parameters—the dimension $d$ and the cardinality $|V|$—and provide fine-grained classifications under varying numerical encodings. Methodologically, we employ PL-reductions augmented with bit-length analysis and reachability reasoning. Our key contributions are: (i) the first proof that coverability is XNL-complete when parameterized by dimension $d$ under unary encoding, thereby pinpointing its exact location in the parameterized complexity landscape; (ii) tight complexity characterizations across multiple parameter settings; and (iii) identification of the tractability of coverability with fixed $|V|$ as an open problem.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Applications

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

We investigate the parameterised complexity of the classic coverability problem for vector addition systems (VAS): given a finite set of vectors $V subseteqmathbb{Z}^d$, an initial configuration $sinmathbb{N}^d$, and a target configuration $tinmathbb{N}^d$, decide whether starting from $s$, one can iteratively add vectors from $V$ to ultimately arrive at a configuration that is larger than or equal to $t$ on every coordinate, while not observing any negative value on any coordinate along the way. We consider two natural parameters for the problem: the dimension $d$ and the size of $V$, defined as the total bitsize of its encoding. We present several results charting the complexity of those two parameterisations, among which the highlight is that coverability for VAS parameterised by the dimension and with all the numbers in the input encoded in unary is complete for the class XNL under PL-reductions. We also discuss open problems in the topic, most notably the question about fixed-parameter tractability for the parameterisation by the size of $V$.

Problem

• Analyzing parameterized complexity of VAS coverability problem
• Studying dimension and vector size as key complexity parameters
• Determining completeness for complexity class XNL with unary encoding

Innovation

• Parameterised complexity analysis of coverability problem
• Dimension and vector size as natural parameters
• XNL completeness for unary encoded VAS