Tight UGC Thresholds for Geometric Stabbing Problems

2026-07-30

AI Summary

This work establishes tight approximation thresholds for several geometric piercing problems under the Unique Games Conjecture (UGC). Focusing on three specific settings—axis-aligned cubes, horizontal line segments, and separated $d$-intervals—the authors construct integrality gap instances for the corresponding covering linear programs. By integrating the strict-CSP framework, randomized rounding, and full-support perturbation techniques, together with a novel analysis based on continuous trace structures, they derive the first UGC-based hardness results that match the performance of the best-known approximation algorithms. Specifically, they show that the UGC-hardness thresholds are $d$ for piercing $d$-dimensional axis-aligned cubes, $e/(e-1)$ for horizontal line segments, and $d$ for hitting sets of separated $d$-intervals. This provides a sharp complexity characterization for these fundamental geometric covering problems.

Abstract

Many geometric stabbing problems admit natural covering LPs in which each constraint is a union of consecutive traces on ordered candidate sets. We prove a transfer theorem showing that every fixed finite, bounded-arity integrality-gap instance of this form yields a matching hardness ratio under the Unique Games Conjecture. Using the strict-CSP framework of Kumar, Manokaran, Tulsiani, and Vishnoi [SODA 2011], we construct the required connected local distributions by randomized rounding and a full-support perturbation. Given a fractional vector $x$ on a block, the rounding selects candidate $i$ with marginal probability $x_i$ and hits each consecutive trace $T$ with probability $\min\{1,x(T)\}$. We obtain three tight UGC thresholds. First, for every fixed $d\ge 2$, stabbing arbitrary-size axis-parallel $d$-cubes with coordinate hyperplanes has threshold $d$. For $d=2$, the hardness holds for arbitrary-size squares and establishes threshold $2$ for rectangle and square stabbing, matching the $2$-approximation of Gaur, Ibaraki, and Krishnamurti [ESA 2000]. Second, stabbing horizontal segments with horizontal and vertical lines has threshold $e/(e-1)$, matching the $e/(e-1)$-approximation of Kovaleva and Spieksma [ESA 2004]. Third, separated $d$-interval transversal has threshold $d$ for every fixed $d\ge 2$, closing under UGC the gap left by the $d$-approximation of Ben-David, Grant, Ma, and Sharpe [CCCG 2012].

Problem

• geometric stabbing
• UGC hardness
• integrality gap
• approximation threshold
• covering LP

Innovation

• Unique Games Conjecture
• geometric stabbing
• integrality gap
• randomized rounding
• approximation threshold