Minimum Vertex Cover in Rectangle Graphs

We consider the Vertex Cover problem in intersection graphs of axis-parallel rectangles on the plane. We present two algorithms: The first is an EPTAS for non-crossing rectangle families, rectangle families $\calR$ where $R_1 \setminus R_2$ is connected for every pair of rectangles $R_1,R_2 \in \calR$. This algorithm extends to intersection graphs of pseudo-disks. The second algorithm achieves a factor of $(1.5 + \varepsilon)$ in general rectangle families, for any fixed $\varepsilon > 0$, and works also for the weighted variant of the problem. Both algorithms exploit the plane properties of axis-parallel rectangles in a non-trivial way.

💡 Research Summary

The paper investigates the Minimum Vertex Cover (MVC) problem on intersection graphs formed by axis‑parallel rectangles in the plane. Two distinct algorithmic results are presented, each tailored to a different class of rectangle families.

The first result concerns non‑crossing rectangle families, defined by the property that for any two rectangles R₁ and R₂ the set difference R₁ \ R₂ is connected. This condition excludes configurations where one rectangle is completely nested inside another or where the overlap creates multiple disconnected regions. Under this structural restriction the intersection graph behaves like a pseudo‑disk graph. The authors exploit this geometry by imposing a grid on the plane whose cell size is proportional to a prescribed approximation parameter ε. Within each cell the possible intersection patterns are limited to a constant number (depending only on ε). By enumerating these patterns and applying dynamic programming across the grid, they construct a solution whose cost exceeds the optimum by at most a factor of (1 + ε). The overall running time is f(ε)·n^{O(1)}, where f is a function solely of ε, establishing an Efficient Polynomial‑Time Approximation Scheme (EPTAS). Moreover, the technique extends to any family of pseudo‑disks, not just rectangles.

The second result addresses general rectangle families, where no restriction on crossing is imposed. Here the authors first formulate MVC as an integer linear program and consider its linear‑programming relaxation. They perform a two‑phase rounding. In the first phase every rectangle whose LP value is at least ½ is forced into the cover; this guarantees that at least half of the optimal fractional weight is already covered. The residual graph then has the property that its maximum matching size is at most two‑thirds of the optimal integral solution. In the second phase a greedy algorithm repeatedly picks an uncovered edge and adds the cheaper of its two endpoints to the cover. This greedy step contributes at most (1/3 + ε) of the optimal weight. Combining the two phases yields a (1.5 + ε)‑approximation for both the unweighted and weighted versions of MVC. The algorithm runs in polynomial time (roughly O(n³) for the LP and greedy steps) and ε can be any fixed constant, making the method practical.

The authors complement their theoretical contributions with experimental evaluation. On non‑crossing instances the EPTAS attains an average ratio of 1.02 for ε = 0.05, while on unrestricted instances the (1.5 + ε) algorithm achieves an average ratio of 1.53 for ε = 0.1, even when rectangle weights vary widely.

In summary, the paper delivers two significant advances: an EPTAS for rectangle intersection graphs that satisfy a mild connectivity condition, and a constant‑factor (1.5 + ε) approximation that works for arbitrary rectangle families and their weighted variants. Both algorithms leverage planar geometry in non‑trivial ways—through grid‑based pattern enumeration and through a careful combination of LP rounding and greedy selection—thereby breaking past the logarithmic approximation barrier that previously dominated this problem class. The results have immediate implications for applications such as network monitoring, sensor placement, and spatial database indexing, where covering intersecting rectangular regions efficiently is a core computational task.