Solving Cyclic Antibandwidth Problem by SAT

TL;DR

This paper introduces SAT-CAB, the first exact SAT-based method for solving the NP-hard Cyclic Antibandwidth Problem on general graphs, providing guaranteed optimal solutions and surpassing existing heuristics.

Contribution

The paper presents a novel SAT encoding and an exact approach for CABP on general graphs, enabling guaranteed optimal solutions and improving over heuristic methods.

Findings

Successfully solves practical CABP instances with optimal solutions.

Identifies several previously unknown optimal values for benchmark instances.

Outperforms or matches state-of-the-art heuristics and solvers on general graphs.

Abstract

The Cyclic Antibandwidth Problem (CABP), a variant of the Antibandwidth Problem, is an NP-hard graph labeling problem with numerous applications. Despite significant research efforts, existing state-of-the-art approaches for CABP are exclusively heuristic or metaheuristic in nature, and exact methods have been limited to restricted graph classes. In this paper, we present the first exact approach for the CABP on general graphs, based on SAT solving, called SAT-CAB. The proposed method is able to systematically explore the solution space and guarantee global optimality, overcoming the limitations of previously reported heuristic algorithms. This approach relies on a novel and efficient SAT encoding of CABP, in which the problem is transformed into a sequence of At-Most-One constraints. In particular, we introduce a compact representation of the At-Most-One constraints inherent to CABP, which significantly reduces the size of the resulting formulas and enables modern SAT solvers to effectively explore the solution space and to certify global optimality. Extensive computational experiments on standard benchmark instances show that the proposed method efficiently solves CABP instances of practical relevance, while identifying several previously unknown optimal solutions. Moreover, global optimal cyclic antibandwidth values are proven for a number of benchmark instances for the first time. Comparative results indicate that SAT-CAB consistently matches or surpasses the best-known solutions obtained by state-of-the-art heuristic algorithms such as MS-GVNS, HABC-CAB, and MACAB, as well as strong commercial Constraint Programming and Mixed Integer Programming solvers like CPLEX and Gurobi, particularly on general graphs, while also providing optimality guarantees. These results advance the state of the art for CABP and provide a new baseline for exact and hybrid methods on general graphs.