Endpoint Covering of Axis-Parallel Segments:Bichromatic and Monochromatic One-Center

We study exact one-center optimization for axis-parallel segments using axis-parallel squares under endpoint-based coverage, where a segment is \emph{$1$-covered} if the square contains at least one of its endpoints. In the monochromatic problem, we seek a minimum-side-length square that $1$-covers all $n$ input segments. We obtain $O(n\log n)$-time algorithms for both unrestricted and segment-constrained centers. The unrestricted bound matches the known bound implied by the two-representative color-spanning-square problem, whereas the segment-constrained result is new. We also prove matching $\mathrm{\Omega }(n\log n)$ lower bounds for both center models in the fixed-order algebraic decision-tree model. In the bichromatic problem, an admissible square must fully contain all $m$ blue segments, minimize the number of red segments with an endpoint in its interior, and, subject to this minimum, maximize its side length within a prescribed bounding box. We give deterministic $O(m+n{\log }^{2}n)$-time and $O(m+mn\log n)$-time algorithms for unrestricted and blue-segment-constrained centers, respectively.