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SAT Encoding of Partial Ordering Models for Graph Coloring Problems

Daniel Faber, Adalat Jabrayilov, Petra Mutzel

TL;DR

This work revisits partial-ordering based encodings for exact graph coloring, introducing new SAT and ILP formulations that generalize to the bandwidth coloring problem. The authors compare assignment-based, POP-based, and hybrid models, demonstrating that SAT implementations based on partial-ordering (POP-S and POPH-S) achieve superior or competitive performance, particularly on sparse graphs and across DIMACS benchmarks. For BCP, their POP-based encodings (POP-S-B and POPH-S-B) yield the best results, solving many instances to optimality and even opening previously unsolved cases. The study highlights a favorable formulation size for POP approaches, practical efficiency with modern SAT solvers, and strong potential for integration into SAT-based solving frameworks in graph coloring and related problems.

Abstract

In this paper, we suggest new SAT encodings of the partial-ordering based ILP model for the graph coloring problem (GCP) and the bandwidth coloring problem (BCP). The GCP asks for the minimum number of colors that can be assigned to the vertices of a given graph such that each two adjacent vertices get different colors. The BCP is a generalization, where each edge has a weight that enforces a minimal "distance" between the assigned colors, and the goal is to minimize the "largest" color used. For the widely studied GCP, we experimentally compare our new SAT encoding to the state-of-the-art approaches on the DIMACS benchmark set. Our evaluation confirms that this SAT encoding is effective for sparse graphs and even outperforms the state-of-the-art on some DIMACS instances. For the BCP, our theoretical analysis shows that the partial-ordering based SAT and ILP formulations have an asymptotically smaller size than that of the classical assignment-based model. Our practical evaluation confirms not only a dominance compared to the assignment-based encodings but also to the state-of-the-art approaches on a set of benchmark instances. Up to our knowledge, we have solved several open instances of the BCP from the literature for the first time.

SAT Encoding of Partial Ordering Models for Graph Coloring Problems

TL;DR

This work revisits partial-ordering based encodings for exact graph coloring, introducing new SAT and ILP formulations that generalize to the bandwidth coloring problem. The authors compare assignment-based, POP-based, and hybrid models, demonstrating that SAT implementations based on partial-ordering (POP-S and POPH-S) achieve superior or competitive performance, particularly on sparse graphs and across DIMACS benchmarks. For BCP, their POP-based encodings (POP-S-B and POPH-S-B) yield the best results, solving many instances to optimality and even opening previously unsolved cases. The study highlights a favorable formulation size for POP approaches, practical efficiency with modern SAT solvers, and strong potential for integration into SAT-based solving frameworks in graph coloring and related problems.

Abstract

In this paper, we suggest new SAT encodings of the partial-ordering based ILP model for the graph coloring problem (GCP) and the bandwidth coloring problem (BCP). The GCP asks for the minimum number of colors that can be assigned to the vertices of a given graph such that each two adjacent vertices get different colors. The BCP is a generalization, where each edge has a weight that enforces a minimal "distance" between the assigned colors, and the goal is to minimize the "largest" color used. For the widely studied GCP, we experimentally compare our new SAT encoding to the state-of-the-art approaches on the DIMACS benchmark set. Our evaluation confirms that this SAT encoding is effective for sparse graphs and even outperforms the state-of-the-art on some DIMACS instances. For the BCP, our theoretical analysis shows that the partial-ordering based SAT and ILP formulations have an asymptotically smaller size than that of the classical assignment-based model. Our practical evaluation confirms not only a dominance compared to the assignment-based encodings but also to the state-of-the-art approaches on a set of benchmark instances. Up to our knowledge, we have solved several open instances of the BCP from the literature for the first time.
Paper Structure (23 sections, 2 theorems, 27 equations, 2 figures, 6 tables)

This paper contains 23 sections, 2 theorems, 27 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. State-of-the-art encodings
  3. Integer programming formulations
  4. The assignment models (ASS-I) and (ASS-I-B)
  5. The partial-ordering based model (POP-I) for the GCP
  6. The hybrid partial-ordering based model (POPH-I)
  7. SAT encodings (ASS-S) and (ASS-S-B)
  8. Formulations based on the partial-ordering approach
  9. SAT encodings based on partial-ordering: (POP-S) and (POPH-S)
  10. Hybrid partial-ordering based SAT encoding for the GCP
  11. Partial-ordering based ILP models (POP-I-B) and (POPH-I-B) for the BCP
  12. Hybrid partial-ordering ILP model for the BCP
  13. SAT encodings (POP-S-B) and (POPH-S-B) based on partial-ordering for the BCP
  14. Hybrid partial-ordering SAT encoding for the BCP
  15. Experimental evaluation
  16. ...and 8 more sections

Key Result

Lemma 1

ASS-I-B contains $H\cdot|V|+1$ variables and $(H+1) \cdot |V| + H\cdot|E|(2\bar{d}-1) - \sum_{e \in E} (d(e)^2-d(e))$ constraints.

Figures (2)

  • Figure 1: Number of DIMACS instances solved within a given runtime for the GCP
  • Figure 2: Number of GEOM instances solved within a given runtime for the BCP

Theorems & Definitions (2)

  • Lemma 1
  • Lemma 2