Let P be a set of n points in ℝ^d, d ≥ 2, and let t ≥ 2 be an integer. Let H_t(P) denote the t-uniform hypergraph on P, whose hyperedges consist of all t-tuples T ⊂ P for which ‖p-q‖ ≤ 1, for any two points p,q ∈ T. A matching in H_t(P) is a collection of vertex-disjoint hyperedges. We present a PTAS for finding a maximum matching in H_t(P). In particular, we present the first PTAS for the well-studied problem known as maximum (vertex-disjoint) triangle packing in unit disk graphs. Our approach consists of a sparsification stage, which replaces P by a subset Q with favorable properties, followed by an implementation of a PTAS for a maximum matching in H_t(Q). The two stages follow the high-level machinery in [Édouard Bonnet et al., 2023] and [Rom Aschner et al., 2013], respectively, but are considerably more involved.

We present efficient algorithms for the bottleneck path problem in two geometric settings that arise naturally in applications: directional-antenna graphs in the plane with antenna angles bounded from below by a constant, and visibility graphs whose vertices lie on or above a 1.5-dimensional terrain, both with Euclidean distances as edge weights. We provide near-linear algorithms for the corresponding decision problems, namely, determining whether the subgraph obtained by retaining all edges with weight at most some threshold bn contains a path from s to t. We then use the decision procedures to obtain algorithms for the bottleneck path problem that run in O^*(n^{8/7}) randomized expected time, where n is the input size and the O^*(⋅) notation hides subpolynomial factors. Within the same performance bounds, we can also solve the bounded-hop version, in which we only consider s-t paths with at most k edges, for a given integer k < n.

Let ℬ be a collection of n arbitrary balls in ℝ³, and let G₀(ℬ) be their intersection graph. We provide an algorithm for performing BFS on G₀(ℬ), which runs in O^*(n^{4/3}) time, where the O^*(⋅) notation hides subpolynomial factors. For r ≥ 0, let G_r(ℬ) be the intersection graph of the set ℬ_r = {B+r ∣ B ∈ ℬ}, where B+r is the ball concentric with B whose radius is larger by r than the radius of B. We provide an efficient algorithm for the reverse shortest path (RSP) problem, where we are given two designated balls B_s, B_t of ℬ and a parameter 0 < λ < n, and seek the smallest value r^* for which G_{r^*}(ℬ) contains a path from B_s to B_t of at most λ edges. For the special case of congruent balls (equivalently, for points in ℝ³), the algorithm runs in O^*(n^{29/21}) ≈ O^*(n^{1.381}) time. For the general case, the algorithm runs in O^*(n^{56/39}) ≈ O^*(n^{1.436}) time. We also extend the technique to handle other measures of expansion and higher dimensions.

We present a general technique, based on parametric search with some twist, for solving a variety of optimization problems on a set of semi-algebraic geometric objects of constant complexity. The common feature of these problems is that they involve a `growth parameter' $r$ and a semi-algebraic predicate $Π(o,o';r)$ of constant complexity on pairs of input objects, which depends on $r$ and is monotone in $r$. One then defines a graph $G(r)$ whose edges are all the pairs $(o,o')$ for which $Π(o,o';r)$ is true, and seeks the smallest value of $r$ for which some monotone property holds for $G(r)$. Problems that fit into this context include (i) the reverse shortest path problem in unit-disk graphs, recently studied by Wang and Zhao, (ii) the same problem for weighted unit-disk graphs, with a decision procedure recently provided by Wang and Xue, (iii) extensions of these problems to three and higher dimensions, (iv) the discrete Fréchet distance with one-sided shortcuts in higher dimensions, extending the study by Ben Avraham et al., (v) perfect matchings in intersection graphs: given, e.g., a set of fat ellipses of roughly the same size, find the smallest value $r$ such that if we expand each of the ellipses by $r$, the resulting intersection graph contains a perfect matching, (vi) generalized distance selection problems: given, e.g., a set of disjoint segments, find the $k$'th smallest distance among the pairwise distances determined by the segments, for a given (sufficiently small but superlinear) parameter $k$, and (vii) the maximum-height independent towers problem, in which we want to erect vertical towers of maximum height over a 1.5-dimensional terrain so that no pair of tower tips are mutually visible. We obtain significantly improved solutions for problems (i), (ii) and (vi), and new efficient solutions to the other problems.