Computational Complexity of Alignments

TL;DR

This paper analyzes the computational complexity of alignment algorithms in process mining, revealing hardness results for various Petri net classes and identifying conditions under which alignment is efficiently solvable.

Contribution

It provides a comprehensive complexity classification of the alignment problem across multiple Petri net subclasses, highlighting the impact of system properties like safeness and liveness.

Findings

Alignment is PSPACE-complete for safe Petri nets.

Optimal alignments exist in polynomial length for certain live, bounded, free-choice systems.

Alignment is solvable in polynomial time for live, safe S-systems.

Abstract

In process mining, alignments quantify the degree of deviation between an observed event trace and a business process model and constitute the most important conformance checking technique. We study the algorithmic complexity of computing alignments over important classes of Petri nets. First, we show that the alignment problem is PSPACE-complete on the class of safe Petri nets and also on the class of safe and sound workflow nets. For live, bounded, free-choice systems, we prove the existence of optimal alignments of polynomial length which positions the alignment problem in NP for this class. We further show that computing alignments is NP-complete even on basic subclasses such as process trees and T-systems. We establish NP-completeness on several related classes as well, including acyclic systems. Finally, we demonstrate that on live, safe S-systems the alignment problem is solvable in P and that both assumptions (liveness and safeness) are crucial for this result.