On testing substitutability

TL;DR

This paper introduces a more efficient algorithm for testing substitutability of choice functions induced by preference lists, reducing the computational complexity from cubic to quadratic in key parameters.

Contribution

It presents a novel algorithm with a running time of O(|U|^2·N^2), improving significantly over previous algorithms.

Findings

New algorithm with quadratic time complexity

Applicable even when preference list length N is exponential in universe size

Faster testing of substitutability in choice functions

Abstract

The papers~\cite{hatfimmokomi11} and~\cite{azizbrilharr13} propose algorithms for testing whether the choice function induced by a (strict) preference list of length $N$ over a universe $U$ is substitutable. The running time of these algorithms is $O(|U|^3\cdot N^3)$, respectively $O(|U|^2\cdot N^3)$. In this note we present an algorithm with running time $O(|U|^2\cdot N^2)$. Note that $N$ may be exponential in the size $|U|$ of the universe.