Strong Admissibility, a Tractable Algorithmic Approach (proofs)

TL;DR

This paper introduces two polynomial-time algorithms for constructing small strongly admissible labellings in argumentation frameworks, providing efficient explanations for an argument's membership in the grounded extension, with practical applications.

Contribution

The paper presents novel polynomial algorithms for constructing small strongly admissible labellings, improving efficiency over existing exponential-time methods.

Findings

Algorithms are polynomial-time and produce near-minimal labellings.

Runtime is significantly faster than previous approaches.

Results are practically useful for explanation in argumentation frameworks.

Abstract

Much like admissibility is the key concept underlying preferred semantics, strong admissibility is the key concept underlying grounded semantics, as membership of a strongly admissible set is sufficient to show membership of the grounded extension. As such, strongly admissible sets and labellings can be used as an explanation of membership of the grounded extension, as is for instance done in some of the proof procedures for grounded semantics. In the current paper, we present two polynomial algorithms for constructing relatively small strongly admissible labellings, with associated min-max numberings, for a particular argument. These labellings can be used as relatively small explanations for the argument's membership of the grounded extension. Although our algorithms are not guaranteed to yield an absolute minimal strongly admissible labelling for the argument (as doing do would have implied an exponential complexity), our best performing algorithm yields results that are only marginally bigger. Moreover, the runtime of this algorithm is an order of magnitude smaller than that of the existing approach for computing an absolute minimal strongly admissible labelling for a particular argument. As such, we believe that our algorithms can be of practical value in situations where the aim is to construct a minimal or near-minimal strongly admissible labelling in a time-efficient way.