Hieu Truong Xuan | Searcharxiv

arXiv2026

Solving Minimum Span Antibandwidth and Cyclic Antibandwidth Labeling Problems

The Antibandwidth and Cyclic Antibandwidth problems are NP-hard graph labeling problems that aim to maximize the minimum (cyclic) distance between labels assigned to adjacent vertices. Extensive research on these problems has resulted in a variety of mathematical formulations and computational approaches. However, their minimum span perspective, in which a prescribed minimum (cyclic) distance is fixed and the objective is to minimize the label span, has received comparatively little attention. In this paper, we consider this complementary perspective by introducing the Minimum Span Antibandwidth/Cyclic Antibandwidth Labeling (MSABL/MSCABL) problems and developing a unified Boolean Satisfiability (SAT)-based framework for solving them. The SAT-based framework formulates MSABL/MSCABL as a sequence of decision problems and exploits their monotonicity to accelerate the search process. We also consider two SAT solving strategies, parallel and incremental SAT solving: the former examines multiple candidate spans concurrently, while the latter reuses a single SAT instance while progressively restricting the label domain. The proposed approaches are evaluated on benchmark instances from the Harwell-Boeing Sparse Matrix Collection and compared with CPLEXCP, CPLEXMIP, and Gurobi. The results show that SAT-based approaches are highly competitive in solution quality, with the parallel approach performing best overall for MSCABL and the incremental approach for MSABL. With the no-hole constraint, they remain competitive with CPLEXCP and significantly outperform CPLEXMIP and Gurobi, particularly for MSCABL. These results demonstrate the effectiveness of SAT solving as an exact approach for MSABL and MSCABL.

cs.AI↗

arXiv2026

Solving Cyclic Antibandwidth Problem by SAT

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.

cs.AI↗