circuit layout engineers

Abstract: The Directed Feedback Vertex Set problem (DFVS) asks whether a digraph can be made acyclic by deleting at most $k$ vertices. Whether DFVS admits a polynomial kernel parameterized by $k$ is a major open problem in kernelization, even for planar digraphs. We resolve the planar case by giving a deterministic kernel with $O(k^{66}\log^2 k)$ vertices and arcs. Our algorithm proceeds in three stages. First, we apply structural reduction rules to the input digraph, bounding the number of directed faces and some special vertices. Second, we pass to the planar dual, where vertex deletion corresponds to adding groups of reverse arcs to make each weakly connected component strongly connected. The structural bounds in the first stage yield a small retained vertex set in the dual. We then compress the dual instance by identifying vertices with the same distance records from this retained vertex set. The main technical contribution is a directed-cut argument showing that this identification preserves feasibility. Finally, we transform the polynomial-size dual instance back into an instance of Planar Directed Feedback Vertex Set via a $3$-CNF encoding and a planar graph construction.

Abstract: 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.