arXiv subjects

Luis H. Herrera

Publications and source records attributed to Luis H. Herrera.

3 recordsLinked to original sources

arXiv2026

Maximum-Weight Two Boxes Symmetric Difference Problem

Let $P$ be a set of $n$ weighted points in the plane. We study optimization with axis-aligned rectangles under objectives determined by the incidence pattern of each point with the selected rectangles. As a central case, we consider two possibly overlapping rectangles maximizing the total weight of the points in their symmetric difference. We give a direct algorithm based on a generalized Maximum Consecutive Subsequence tree running in $O(n^4\log n)$ time and $O(n)$ space, and improve the running time to $O(n^4)$ via a reduction to Weighted Depth in $\mathbb{R}^8$. The resulting parameter-space instance has the signed coefficient pattern $(w,w,-2w)$ induced by symmetric difference. When the two factor boxes are allowed to be arbitrary and the associated weights may be signed, this coefficient structure is linearly equivalent, up to an additive constant, to general Weighted Depth. More generally, for any fixed $k$ and fixed Boolean incidence objective $h:\{0,1\}^k\to\mathbb{R}$, a Möbius expansion over the Boolean lattice yields an $O(n^{2k})$-time algorithm through Weighted Depth in $\mathbb{R}^{4k}$. This strictly extends depth-based objectives by allowing the value of a point to depend on which rectangles cover it. Symmetric difference, union, intersection, exact-depth and threshold coverage, and overlap rewards or penalties are all special cases.

cs.CG↗

arXiv2026

A Geometric View of Combinatorial Fiedler Theory

Recently, Andrade and Dahl introduced combinatorial Fiedler theory by studying a parameter $b(G)$ defined as the $\ell_1$-analog of the Rayleigh quotient minimization characterization of the algebraic connectivity of a graph $G=(V,E)$. In this work, we study the corresponding maximization problem, which plays the role of the $\ell_1$-analog of the largest Laplacian eigenvalue. We show that the new parameter $B(G)$ associated with this maximization problem admits a simple exact description: it is the average of the two largest vertex degrees of $G$. A unified combinatorial treatment of the minimization and maximization problems is presented first. Later, both optimization problems are reinterpreted in a geometrical setting. The feasible set is identified with a $(n-2)$-dimensional cuboctahedron shell where $n=|V|$. Additional structure is presented for this polyhedron, including the fact that maximizing solutions arise at its vertices and minimizing solutions arise at the centers of its facets. Finally, we analyze the number of optimal vectors for $b(G)$ and $B(G)$ for several graph families. Although the value of $B(G)$ is determined by the two largest degrees, we prove that counting the vectors that attain this value is actually $\#\mathrm{P}$-complete.

cs.CG↗

arXiv2018

On the intersection graph of the disks with diameters the sides of a convex $n$-gon

Given a convex $n$-gon, we can draw $n$ disks (called side disks) where each disk has a different side of the polygon as diameter and the midpoint of the side as its center. The intersection graph of such disks is the undirected graph with vertices the $n$ disks and two disks are adjacent if and only if they have a point in common. Such a graph was introduced by Huemer and Pérez-Lantero in 2016, proved to be planar and Hamiltonian. In this paper we study further combinatorial properties of this graph. We prove that the treewidth is at most 3, by showing an $O(n)$-time algorithm that builds a tree decomposition of width at most 3, given the polygon as input. This implies that we can construct the intersection graph of the side disks in $O(n)$ time. We further study the independence number of this graph, which is the maximum number of pairwise disjoint disks. The planarity condition implies that for every convex $n$-gon we can select at least $\lceil n/4 \rceil$ pairwise disjoint disks, and we prove that for every $n\ge 3$ there exist convex $n$-gons in which we cannot select more than this number. Finally, we show that our class of graphs includes all outerplanar Hamiltonian graphs except the cycle of length four, and that it is a proper subclass of the planar Hamiltonian graphs.

cs.CG↗