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. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2025.