[論文レビュー] Computational Complexity of Alignments

Christopher T. Schwanen, Wied Pakusa+1 more|arXiv (Cornell University)|Mar 5, 2026

Business Process Modeling and Analysis被引用数 0

ひとこと要約

The paper analyzes the computational complexity of computing alignments between traces and process models across important Petri net classes, identifying PSPACE- and NP-hard cases and providing tractable subclasses.

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.

研究の動機と目的

• Identify the computational complexity of the alignment problem across key Petri net classes (safe nets, workflow nets, LBFC-systems, process trees, T-systems, acyclic systems).
• Determine which structural restrictions (safety, liveness, boundedness, free-choice) lead to tractable versus intractable alignment computation.
• Show existence of optimal alignments with polynomial-length for live, bounded, free-choice systems and derive NP membership for this class.
• Establish NP-hardness lower bounds for alignments on process trees, T-systems, and acyclic systems.
• Provide a comprehensive overview of complexity results and their implications for conformance checking in process mining.]
• method: ["Model the alignment problem as a search over synchronous product constructions between trace systems and process models.","Prove PSPACE-completeness of alignment for safe Petri nets and for safe and sound workflow nets.","Show polynomial-length optimal alignments and NP-membership for live, bounded, free-choice (LBFC) systems using structural results like the Shortest Sequence Theorem.","Demonstrate NP-completeness of alignment on basic subclasses such as process trees, T-systems, and acyclic systems; establish NP-hardness for these classes via membership reductions.","Analyze the role of liveness and safeness by proving polynomial-time solvability on live, safe S-systems and NP-hardness when either assumption is dropped.","Use the synchronous product construction to relate alignments to reachable markings and to connect traces with model behavior."]
• research_questions':['What is the computational complexity of computing alignments for various Petri net classes (safe nets, workflow nets, LBFC-systems, process trees, T-systems, acyclic systems)?','Do certain structural restrictions (safety, liveness, boundedness, free-choice) yield tractable alignment computation or render it intractable?','Can optimal alignments be of polynomial length for LBFC-systems, and does this place alignment in NP for this class?','Is alignment NP-complete for process trees, T-systems, and acyclic systems, and what are the corresponding lower/upper bounds?','Under what conditions do alignments become polynomial-time solvable (e.g., live, safe S-systems) and why are liveness and safeness crucial?'],
• key_findings':['Alignment is PSPACE-complete for safe Petri nets and for safe and sound workflow nets.', 'For live, bounded, free-choice systems, optimal alignments have polynomial length and the problem lies in NP.', 'Alignment is NP-complete on basic subclasses including process trees, T-systems, and acyclic systems.', 'NP-hardness results extend to several related classes beyond the above, including various acyclic configurations.', 'Live, safe S-systems yield polynomial-time solvability of alignment, and dropping liveness or safeness makes the problem NP-complete.', 'The paper provides a comprehensive table (Table 3) summarizing the complexity across model classes.'],
• table_headers: [],
• table_rows: []

提案手法

• Model the alignment problem as a search over synchronous product constructions between trace systems and process models.
• Prove PSPACE-completeness of alignment for safe Petri nets and for safe and sound workflow nets.
• Show polynomial-length optimal alignments and NP-membership for live, bounded, free-choice (LBFC) systems using structural results like the Shortest Sequence Theorem.
• Demonstrate NP-completeness of alignment on basic subclasses such as process trees, T-systems, and acyclic systems; establish NP-hardness for these classes via membership reductions.
• Analyze the role of liveness and safeness by proving polynomial-time solvability on live, safe S-systems and NP-hardness when either assumption is dropped.
• Use the synchronous product construction to relate alignments to reachable markings and to connect traces with model behavior.

実験結果

リサーチクエスチョン

• RQ1 What is the computational complexity of computing alignments for various Petri net classes (safe nets, workflow nets, LBFC-systems, process trees, T-systems, acyclic systems)?
• RQ2 Do certain structural restrictions (safety, liveness, boundedness, free-choice) yield tractable alignment computation or render it intractable?
• RQ3 Can optimal alignments be of polynomial length for LBFC-systems, and does this place alignment in NP for this class?
• RQ4 Is alignment NP-complete for process trees, T-systems, and acyclic systems, and what are the corresponding lower/upper bounds?
• RQ5 Under what conditions do alignments become polynomial-time solvable (e.g., live, safe S-systems) and why are liveness and safeness crucial?

主な発見

• Alignment is PSPACE-complete for safe Petri nets and for safe and sound workflow nets.
• For live, bounded, free-choice systems, optimal alignments have polynomial length and the problem lies in NP.
• Alignment is NP-complete on basic subclasses including process trees, T-systems, and acyclic systems.
• NP-hardness results extend to several related classes beyond the above, including various acyclic configurations.
• Live, safe S-systems yield polynomial-time solvability of alignment, and dropping liveness or safeness makes the problem NP-complete.
• The paper provides a comprehensive table (Table 3) summarizing the complexity across model classes.