Exact and parameterized algorithms for window width minimization in bipartite arrangement

Abstract

We study 2-layer Window Width Minimization in Bipartite arrangements, where given a bipartite arrangement G with bipartition A?B, and fixed and distinct integer positions for vertices in A, the goal is to determine distinct integer positions for vertices in B such that the window width parameter b is minimized. The window width b ensures that each vertex v ? B maintains a horizontal distance at most b from its neighbors in A and from its sibling vertices in B that share a common neighbor in A. In this paper, we design two exact exponential algorithms for Window Width Minimization: (1) an O*(4.36n�2p+n+b) time and polynomial space algorithm and (2) O*(4.899n) time and O*(2n) space algorithm, where |A|=p,|B|=n. We also initiate the study of the problem from the perspective of parameterized complexity and design a fixed-parameter tractable algorithm parameterized by the window size, which stands in stark contrast with the closely related Bandwidth Minimization problem. Specifically, our algorithm runs in O*(bO(b)) time and space, either finds an ordering of children with window width b, or correctly determines that no such ordering exists. � 2026 Elsevier B.V.