AI 中文总结
该研究在独特游戏猜想下，通过严格CSP框架构造分布，得到几何刺探类问题的三个紧UGC阈值，填补了部分问题近似比与困难度之间的差距。
AI 中文摘要
许多几何刺探问题存在自然的覆盖线性规划（LP），其中每个约束是有序候选集上连续迹的并集。我们证明了一个转移定理，表明这类形式的每个固定有限、有界元数的整性间隙实例，在独特游戏猜想（Unique Games Conjecture，UGC）下会产生匹配的困难度比。利用Kumar、Manokaran、Tulsiani和Vishnoi[SODA 2011]的严格约束满足问题（strict-CSP）框架，我们通过随机舍入和全支撑扰动构造了所需的连通局部分布。给定块上的分数向量x，舍入操作以边缘概率x_i选择候选i，以概率min{1,x(T)}命中每个连续迹T。我们得到三个紧UGC阈值：第一，对于每个固定d≥2，用坐标超平面刺探任意大小的轴对齐d立方体的阈值为d；对于d=2，该困难度适用于任意大小的正方形，为矩形和正方形刺探确立了阈值2，与Gaur、Ibaraki和Krishnamurti[ESA 2000]的2-近似匹配。第二，用水平和垂直线刺探水平线段的阈值为e/(e-1)，与Kovaleva和Spieksma[ESA 2004]的e/(e-1)-近似匹配。第三，分离d区间横截的阈值为d，对于每个固定d≥2，在UGC下缩小了Ben-David、Grant、Ma和Sharpe[CCCG 2012]的d-近似留下的差距。
英文摘要
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].
Comments35 pages
