Summary
A forum post discusses a new algorithmic result for the geometric hitting set problem: given weighted points and axis-parallel (horizontal and vertical) line segments in the plane, choose a minimum-weight subset of points such that every segment contains at least one selected point. The classic 2-approximation barrier for this problem has now been improved using linear programming (LP) rounding. For the weighted version, the paper achieves a (1+2/e)-approximation (about 1.736); for the unweighted version, it achieves a (1+1/(e-1))-approximation (about 1.582). The key insight is a hybrid strategy: the horizontal segments are handled with LP rounding, while the vertical segments, owing to their simpler structure, can be solved exactly. This 'approximate + exact' combination is what breaks the long-standing factor-2 barrier. The work is titled 'Hitting Axis-Parallel Segments with Weighted Points' by Rajiv Raman, Siddhartha Sarkar, and Jatin Yadav, available as arXiv preprint 2605.14499 in cs.CG.
In the plane, given a set of horizontal and vertical line segments and a set of weighted points, the goal is to select the lightest set of points such that every segment contains at least one selected point. This is the geometric hitting set problem, which is computationally hard, and standard methods only guarantee a 2-approximation.
This paper breaks the 2-approximation barrier using LP rounding:
- Weighted version: (1 + 2/e)-approximation ≈ 1.736
- Unweighted version: (1 + 1/(e-1))-approximation ≈ 1.582
Key idea: The horizontal segments are handled with LP rounding, while the vertical segments — due to their simpler structure — can be solved exactly. This hybrid "approximation + exact" strategy breaks the classic 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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https://zhichai.net/topic/177620157