Summary
A new paper breaks the long-standing factor-2 approximation barrier for the geometric hitting set problem on axis-parallel segments in the plane. Given horizontal and vertical line segments together with a weighted set of points, the goal is to select a minimum-weight subset of points such that every segment contains at least one chosen point. Standard approaches achieve only a 2-approximation for this computationally hard problem. The authors—Rajiv Raman, Siddhartha Sarkar, and Jatin Yadav—use an LP rounding technique to obtain a (1 + 2/e)-approximation (approximately 1.736) for the weighted version and a (1 + 1/(e-1))-approximation (approximately 1.582) for the unweighted version. The key idea is a hybrid strategy: the horizontal segments are handled approximately via LP rounding, while the vertical segments, having a simpler structure, can be solved exactly. This combination of approximation and exact solving is what enables the improvement over the classical factor-2 barrier. The preprint is available at arXiv:2605.14499.
Breaking the Factor-2 Approximation Barrier with LP Rounding
Consider the following geometric hitting set problem: given a set of horizontal and vertical line segments in the plane, together with a set of weighted points, select a minimum-weight subset of points such that every segment contains at least one selected point.
This is a classic geometric hitting set problem. It is computationally hard, and standard methods only guarantee a 2-approximation.
The Result
This paper uses LP rounding to break the factor-2 barrier:
- Weighted version: a
(1 + 2/e)-approximation ≈ 1.736
- Unweighted version: a
(1 + 1/(e-1))-approximation ≈ 1.582 The Key Idea
The approach is a hybrid "approximate + exact" strategy:
- The horizontal segments are handled with LP rounding (approximate solution).
- The vertical segments, due to their simpler structure, can be solved exactly.
Combining the approximate solution for one part with the exact solution for the other is what allows the algorithm to surpass the classical factor-2 barrier.
Paper Information
- Title: Hitting Axis-Parallel Segments with Weighted Points
- Authors: Rajiv Raman, Siddhartha Sarkar, Jatin Yadav
- Preprint: arXiv:2605.14499 (cs.CG)
- Link: https://arxiv.org/abs/2605.14499
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