Saeed Odak

Ahmad Biniaz, Prosenjit Bose, Chaeyoon Chung et al.

Let $S$ be a set of $n$ points in $\mathbb{R}^d$, where $d \geq 2$ is a constant, and let $H_1,H_2,\ldots,H_{m+1}$ be a sequence of vertical hyperplanes that are sorted by their first coordinates, such that exactly $n/m$ points of $S$ are between any two successive hyperplanes. Let $A(S,m)$ be the set of different closest pairs in the ${{m+1} \choose 2}$ vertical slabs that are bounded by $H_i$ and $H_j$, over all $1 \leq i < j \leq m+1$. We prove tight bounds for the largest possible size of $A(S,m)$, over all point sets of size $n$, and for all values of $1 \leq m \leq n$. As a result of these bounds, we obtain, for any constant $ε>0$, a data structure of size $O(n)$, such that for any vertical query slab $Q$, the closest pair in the set $Q \cap S$ can be reported in $O(n^{1/2+ε})$ time. Prior to this work, no linear space data structure with sublinear query time was known.

Sándor Kisfaludi-Bak, Saeed Odak, Satyam Singh et al.

We give an approximation scheme for the TSP in $d$-dimensional hyperbolic space that has optimal dependence on $\varepsilon$ under Gap-ETH. For any fixed dimension $d\geq 2$ and for any $\varepsilon>0$ our randomized algorithm gives a $(1+\varepsilon)$-approximation in time $2^{O(1/\varepsilon^{d-1})}n^{1+o(1)}$. We also provide an algorithm for the hyperbolic Steiner tree problem with the same running time. Our algorithm is an Arora-style dynamic program based on a randomly shifted hierarchical decomposition. However, we introduce a new hierarchical decomposition called the hybrid hyperbolic quadtree to achieve the desired large-scale structure, which deviates significantly from the recently proposed hyperbolic quadtree of Kisfaludi-Bak and Van Wordragen (JoCG'25). Moreover, we have a new non-uniform portal placement, and our structure theorem employs a new weighted crossing analysis. We believe that these techniques could form the basis for further developments in geometric optimization in curved spaces.

Manuel Cáceres, Sándor Kisfaludi-Bak, Saeed Odak

The shifting technique of Hochbaum and Maass [J.ACM'85] produces PTASes with the fastest known running times $n^{O(1/\varepsilon^{d-1})}$ for several $d$-dimensional geometric problems. However, it is only known, due to Marx [FOCS'07], that these algorithms are indeed optimal for dimension $d=2$. We show that these running times are optimal under Gap-ETH for every constant dimension. More precisely, we develop a framework that enables us to prove the conditional optimality of the shifting algorithms for several problems on unit ball graphs, such as maximum independent set, maximum induced forest, and others, as well as for the problem of piercing unit balls. Our framework is built using the cube wiring theorem of De Berg et al. [SICOMP'20] and the reduction steps of Marx and Sidiropoulos [SoCG'14] to create a convenient maximization version of geometric CSP that can be used as a basis for reductions.