Solving Cyclic Antibandwidth Problem by SAT
This solves a long-standing exact solution gap for CABP on general graphs, advancing the state of the art for researchers and practitioners in combinatorial optimization.
They tackled the Cyclic Antibandwidth Problem (CABP), an NP-hard graph labeling problem, by developing the first exact approach using SAT solving, which guarantees global optimality and matches or surpasses state-of-the-art heuristic and commercial solvers on benchmark instances.
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.