Exact and Parameterized Algorithms for Window Width Minimization in Bipartite Arrangement

Shashank Chauhan, Tanmay Inamdar, Lawqueen Kanesh, Sanjukta Roy

Abstract: We study 2-layer Window Width Minimization in Bipartite arrangements, where given a bipartite arrangement G with bipartition \(A \uplus B\), 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 \(v \in B \) 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 \(\mathcal {O}^*(4.36^n \cdot 2^{p+n+b})\) time and polynomial space algorithm and (2) \(\mathcal {O}^*(4.899^n)\) time and \(\mathcal {O}^*(2^n)\) 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 \(\mathcal {O}^*(b^{\mathcal {O}(b)})\) time and space, either finds an ordering of children with window width b, or correctly determines that no such ordering exists.