[Paper Review] On the Quest for an Acyclic Graph

Mikoláš Janota, Radu Grigore+1 more|arXiv (Cornell University)|Aug 5, 2017

Formal Methods in Verification40 references3 citations

TL;DR

This paper proposes and evaluates multiple CNF encodings for enforcing graph acyclicity in SAT-based reasoning, comparing techniques like transitive closure, Warshall's algorithm, and binary labeling. The binary labeling encoding achieves the smallest formula size and best practical performance for graphs with n ≥ 100, outperforming cubic and super-cubic alternatives despite theoretical asymptotic advantages of others.

ABSTRACT

The paper aims at finding acyclic graphs under a given set of constraints. More specifically, given a propositional formula ϕ over edges of a fixed-size graph, the objective is to find a model of ϕ that corresponds to a graph that is acyclic. The paper proposes several encodings of the problem and compares them in an experimental evaluation using stateof-the-art SAT solvers.

Motivation & Objective

• To address the challenge of encoding graph acyclicity as a CNF formula for use in SAT solvers.
• To compare the efficiency and scalability of various encodings, including transitive closure, Warshall’s algorithm, and binary labeling.
• To identify encodings that balance theoretical efficiency with practical performance in real-world SAT solving.
• To evaluate the impact of SAT solver choice on encoding performance, especially for hard instances.
• To explore the feasibility of scalable encodings for large graphs (n > 1000) and propose future directions.

Proposed method

• The paper formulates acyclicity as a CNF constraint (the 'acyclicity checker') that, when conjoined with a base formula φₙ, ensures only acyclic graphs are modeled.
• It implements and evaluates five main encodings: transitive closure (tc1, tc2), Warshall’s algorithm (fw), unary labeling (unr), and binary labeling (bin).
• The binary labeling method uses log n bits per vertex to encode reachability, enabling compact CNF representations.
• The paper uses the Supervisor Problem—a generalization of the GT family—as a well-defined benchmark to evaluate encodings.
• Performance is measured via formula size, SAT solving time, and solver behavior across multiple solvers (e.g., Lingeling, MiniSAT).
• Resolution proof complexity is analyzed to assess the theoretical hardness of encodings, particularly regarding resolution refutation length.

Experimental results

Research questions

• RQ1 Which CNF encoding for graph acyclicity offers the best trade-off between formula size and SAT solver performance?
• RQ2 Why does the transitive closure encoding (tc1) perform significantly better than its alternative (tc2) across all solvers?
• RQ3 Can asymptotically superior encodings (e.g., Strassen-based or binary matrix multiplication) outperform cubic encodings in practice?
• RQ4 To what extent do unit propagation and resolution properties differ across encodings, particularly in detecting cycles early?
• RQ5 Is there a practical path to scaling acyclicity encodings beyond n = 1000 vertices using lazy or incremental SAT solving?

Key findings

• The binary labeling encoding (bin) produces the smallest CNF formulas for n ≥ 100, outperforming cubic and super-cubic alternatives in size and practical performance.
• The transitive closure encoding (tc1) is highly efficient with Lingeling, while its alternative (tc2) performs poorly across all solvers, suggesting structural differences in resolution complexity.
• The Strassen-based encoding (ss) has a formula size of O(n²·⁸¹ log³n), but its large constants make it impractical even for n = 10, with no crossover point expected before n ~ 10³².
• Boolean matrix multiplication (mm) is already worse than cubic encodings at n = 3 and degrades rapidly, making it unsuitable for practical use.
• The unary labeling (unr) and Warshall (fw) encodings are within a factor of 3 of each other in formula size, but bin is significantly smaller for large n.
• Unit propagation fails to detect cycles in unary and binary labeling encodings, indicating they are less amenable to early conflict detection than transitive closure or matrix multiplication methods.