Let S and T be two sets of points in a metric space with a total of n points. Each point in S and T has an associated value that specifies an upper limit on how many points it can be matched with from the other set. A multimatching between S and T is a way of pairing points such that each point in S is matched with at least as many points in T as its assigned value, and vice versa for each point in T. The cost of a multimatching is defined as the sum of the distances between all matched pairs of points. The geometric multimatching problem seeks to find a multimatching that minimizes this cost. A special case where each point is matched to at most one other point is known as the geometric many-to-many matching problem.
We present two results for these problems when the underlying metric space has a bounded doubling dimension. Specifically, we provide the first near-linear-time approximation scheme for the geometric multimatching problem in terms of the output size. Additionally, we improve upon the best-known approximation algorithm for the geometric many-to-many matching problem, previously introduced by Bandyapadhyay and Xue (SoCG 2024), which won the best paper award at SoCG 2024.
Given an undirected planar graph $G$ with $n$ vertices and a set $T$ of $k$ pairs $(s_i,t_i)_{i=1}^k$ of vertices, the goal of the planar disjoint paths problem is to find a set $\mathcal P$ of $k$ pairwise vertex-disjoint paths connecting $s_i$ and $t_i$ for all indices $i\in\{1,\ldots,k\}$. This problem has been studied extensively due to its numerous applications such as VLSI layout and circuit routing. However, this problem is NP-complete even for grid graphs. This motivates the study of this problem from the viewpoint of parameterized algorithms.
In this talk, I will present a $2^{O(k^2)}n$-time algorithm for the planar disjoint paths problem. This improves the two previously best-known algorithms: $2^{2^{O(k)}}n$-time algorithm [Discrete Applied Mathematics 1995] and $2^{O(k^2)}n^6$-time algorithm [STOC 2020].
This is joint work with Kyungjin Cho and Seunghyeok Oh.
On October 5, 2021, Eunjin Oh (오은진) from POSTECH gave a talk at the Discrete Math Seminar on the parameterized complexity of the feedback vertex set problem on unit disk graphs at the Discrete Math Seminar. The title of her talk was “Feedback Vertex Set on Geometric Intersection Graphs“.
I am going to present an algorithm for computing a feedback vertex set of a unit disk graph of size k, if it exists, which runs in time $2^{O(\sqrt{k})}(n + m)$, where $n$ and $m$ denote the numbers of vertices and edges, respectively. This improves the $2^{O(\sqrt{k}\log k)}(n + m)$-time algorithm for this problem on unit disk graphs by Fomin et al. [ICALP 2017].