Fully Dynamic Geometric Vertex Cover and Matching
Sujoy Bhore, Timothy M. Chan
TL;DR
This work addresses dynamic vertex cover and maximum-cardinality matching in intersection graphs of geometric objects in fixed-dimensional space, introducing two general frameworks that yield sublinear update times. The MVC framework combines a dynamic intersection-detection structure with a near-linear static approximation, leveraging MWU-based LP approximation and Nemhauser–Trotter kernelization to achieve $(1+\varepsilon)$- or $(\tfrac{3}{2}+\varepsilon)$-approximation across disks, rectangles, and fat boxes, with polylogarithmic update times. The MCM framework provides a complementary approach, using a modified Hopcroft–Karp algorithm for bipartite graphs and a color-coding-based reduction for general graphs to obtain a $(1+\varepsilon)$-approximation in dynamic geometric settings, including disks and boxes, with polylog or near-linear-time updates. Together, these results deliver near-optimal dynamic algorithms for fundamental geometric optimization problems, exploiting dynamic intersection detection and problem-specific kernelization or path-length reductions; they also establish near-linear static subroutines that underlie the dynamic guarantees. The work lays groundwork for efficient dynamic geometric optimization in practical settings and highlights open questions in weighted variants and broader object families."
Abstract
In this work, we study two fundamental graph optimization problems, minimum vertex cover (MVC) and maximum-cardinality matching (MCM), for intersection graphs of geometric objects, e.g., disks, rectangles, hypercubes, etc., in $d$-dimensional Euclidean space. We consider the problems in fully dynamic settings, allowing insertions and deletions of objects. We develop a general framework for dynamic MVC in intersection graphs, achieving sublinear amortized update time for most natural families of geometric objects. In particular, we show that - - For a dynamic collection of disks in $\mathbb{R}^2$ or hypercubes in $\mathbb{R}^d$ (for constant $d$), it is possible to maintain a $(1+\varepsilon)$-approximate vertex cover in polylog amortized update time. These results also hold in the bipartite case. - For a dynamic collection of rectangles in $\mathbb{R}^2$, it is possible to maintain a $(\frac{3}{2}+\varepsilon)$-approximate vertex cover in polylog amortized update time. Along the way, we obtain the first near-linear time static algorithms for MVC in the above two cases with the same approximation factors. Next, we turn our attention to the MCM problem. Although our MVC algorithms automatically allow us to approximate the size of the MCM in bipartite geometric intersection graphs, they do not produce a matching. We give another general framework to maintain an approximate maximum matching, and further extend the approach to handle non-bipartite intersection graphs. In particular, we show that - - For a dynamic collection of (bichromatic or monochromatic) disks in $\mathbb{R}^2$ or hypercubes in $\mathbb{R}^d$ (for constant $d$), it is possible to maintain a $(1+\varepsilon)$-approximate matching in polylog amortized update time.