[Paper Review] SAT Encodings for Bandwidth Coloring: A Systematic Design Study

Duc Trung Kim Nguyen, Tuyen Van Kieu+1 more|arXiv (Cornell University)|Feb 9, 2026

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TL;DR

The paper systematically studies SAT encodings for the Bandwidth Coloring Problem (BCP), proposing six encoding methods within a unified framework and evaluating their interaction with solver configurations to achieve state-of-the-art results on GEOM and MS-CAP benchmarks. Block encodings, in particular, solve GEOM120b to proven optimality in about 1000 seconds.

ABSTRACT

The Bandwidth Coloring Problem (BCP) generalizes graph coloring by enforcing minimum separation constraints between adjacent vertices and arises in frequency assignment applications. While SAT-based approaches have shown promise for exact BCP solving, the encoding design space remains largely unexplored. This paper presents a systematic study of SAT encodings for the BCP, proposing a unified framework with six encoding methods across three categories: one-variable, two-variable, and block encodings. We evaluate the impact of key features including incremental solving and symmetry breaking. While symmetry breaking has been studied for graph coloring, it has not been systematically evaluated for SAT-based BCP solvers. Our analysis reveals significant interaction effects between encoding choices and solver configurations. The proposed framework achieves state-of-the-art performance on GEOM and MS-CAP benchmarks. Block encodings solve GEOM120b, the hardest instance, to proven optimality in approximately 1000 seconds, whereas previous methods could not solve it within a one-hour time limit.

Motivation & Objective

• Motivate and address gaps in SAT-based solving for Bandwidth Coloring by exploring encoding design space.
• Develop a unified framework of six SAT encoding methods for the BCP across one-variable, two-variable, and block encodings.
• Evaluate the impact of incremental solving, symmetry breaking, and block width strategies on performance.
• Demonstrate state-of-the-art results on GEOM and MS-CAP benchmarks, including solving GEOM120b to optimality.

Proposed method

• Define six SAT encoding methods: 1G (one-variable greater-than), 1L (one-variable less-than), 2G (two-variable greater-than), 2L (two-variable less-than), X (block encoding without auxiliary variables), Xa (block encoding with auxiliary variables).
• Provide variable definitions and constraint formulations for each encoding, including channeling constraints and distance constraints for |c(u)-c(v)| ≥ d(u,v).
• Introduce block encodings with fixed or varying block widths and auxiliary range variables R to represent color ranges efficiently.
• Describe symmetry breaking by fixing the highest-degree vertex into the lower half of the color range.
• Discuss incremental solving modes (assumptions) and non-incremental solving, and how they interact with the encoding types.
• Outline a solving procedure that starts from an upper bound on the span H and searches downward for the optimal span, using incremental assumptions when enabled.

Experimental results

Research questions

• RQ1 What are the effects of different SAT encoding representations (one-variable, two-variable, block encodings) on exact BCP solving performance?
• RQ2 How do incremental solving and symmetry breaking influence SAT-based BCP solvers across encoding types?
• RQ3 What is the impact of block width strategies on the efficiency of block encodings for BCP?
• RQ4 Can a unified SAT encoding framework achieve improved performance on GEOM and MS-CAP benchmarks, including hard instances like GEOM120b?

Key findings

• Block encodings solve GEOM120b to proven optimality in around 1000 seconds, outperforming prior methods within a one-hour limit.
• The framework identifies important interaction effects between encoding choices and solver configurations. Incremental solving benefits block encodings substantially but not one-variable encodings. Symmetry breaking effectiveness varies across encoding types.
• All order-based encodings have clause counts independent of average edge weight, offering asymptotic advantages when edge weights are large.
• Block encodings provide state-of-the-art performance on GEOM and MS-CAP benchmarks compared to previous SAT-based approaches.