arXiv subjects

Anastasiia Tkachenko

Publications and source records attributed to Anastasiia Tkachenko.

4 recordsLinked to original sources

arXiv2026

Maximum Independent Sets in Disk Graphs with Disks in Convex Position

For a set $\mathcal{D}$ of disks in the plane, its disk graph $G(\mathcal{D})$ is the graph with vertex set $\mathcal{D}$, where two vertices are adjacent if and only if the corresponding disks intersect. Given a set $\mathcal{D}$ of $n$ weighted disks, computing a maximum independent set of $G(\mathcal{D})$ is NP-hard. In this paper, we present an $O(n^3\log n)$-time algorithm for this problem in a special setting in which the disks are in convex position, meaning that every disk appears on the convex hull of $\mathcal{D}$. This setting has been studied previously for disks of equal radius, for which an $O(n^{37/11})$-time algorithm was known. Our algorithm also works in the weighted case where disks have weights and the goal is to compute a maximum-weight independent set. As an application of our result, we obtain an $O(n^3\log^2 n)$-time algorithm for the dispersion problem on a set of $n$ disks in convex position: given an integer $k$, compute a subset of $k$ disks that maximizes the minimum pairwise distance among all disks in the subset.

cs.CG↗

arXiv2026

Computing Dominating Sets in Disk Graphs with Centers in Convex Position

Given a set $P$ of $n$ points in the plane and a collection of disks centered at these points, the disk graph $G(P)$ has vertex set $P$, with an edge between two vertices if their corresponding disks intersect. We study the dominating set problem in $G(P)$ under the special case where the points of $P$ are in convex position. The problem is NP-hard in general disk graphs. Under the convex position assumption, however, we present the first polynomial-time algorithm for the problem. Specifically, we design an $O(k^2 n \log^2 n)$-time algorithm, where $k$ denotes the size of a minimum dominating set. For the weighted version, in which each disk has an associated weight and the goal is to compute a dominating set of minimum total weight, we obtain an $O(n^5 \log^2 n)$-time algorithm.

cs.CG↗

arXiv2025

Computing Maximum Cliques in Unit Disk Graphs

Given a set $P$ of $n$ points in the plane, the unit-disk graph $G(P)$ is a graph with $P$ as its vertex set such that two points of $P$ have an edge if their Euclidean distance is at most $1$. We consider the problem of computing a maximum clique in $G(P)$. The previously best algorithm for the problem runs in $O(n^{7/3+o(1)})$ time. We show that the problem can be solved in $O(n \log n + n K^{4/3+o(1)})$ time, where $K$ is the maximum clique size. The algorithm is faster than the previous one when $K=o(n)$. In addition, if $P$ is in convex position, we give a randomized algorithm that runs in $O(n^{15/7+o(1)})= O(n^{2.143})$ worst-case time and the algorithm can compute a maximum clique with high probability. For points in convex position, one special case we solve is when a point in the maximum clique is given; we present an $O(n^2\log n)$ time (deterministic) algorithm for this special case.

cs.CG↗

arXiv2024

Dominating Set, Independent Set, Discrete $k$-Center, Dispersion, and Related Problems for Planar Points in Convex Position

Given a set $P$ of $n$ points in the plane, its unit-disk graph $G(P)$ is a graph with $P$ as its vertex set such that two points of $P$ are connected by an edge if their (Euclidean) distance is at most $1$. We consider several classical problems on $G(P)$ in a special setting when points of $P$ are in convex position. These problems are all NP-hard in the general case. We present efficient algorithms for these problems under the convex position assumption. The considered problems include the following: finding a minimum weight dominating set in $G(P)$, the discrete $k$-center problem for $P$, finding a maximum weight independent set in $G(P)$, the dispersion problem for $P$, and several of their variations. For some of these problems, our algorithms improve the previously best results, while for others, our results provide first-known solutions.

cs.CG↗