关于韦格纳猜想的一个64矩形反例以及高达5/2的线性规划间隙
A 64-Rectangle Counterexample to Wegner's Conjecture and LP Gaps up to $5/2$
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中文总结 AI 辅助
本文给出更简单可手动检验的64矩形反例，通过八矩形小装置构建无三角形族，定义矩形族$P_r$，在$P_3$处得到有限间隙$73/32$，改进之前基准，还构造递归分数解和匹配穿透集，表明每个有理数$t\in[1,5/2)$都作为间隙和填充 - 穿透比出现。
中文摘要 AI 辅助
韦格纳猜想每个有限的轴平行矩形族$\mathcal{R}$满足$\tau(\mathcal{R})\leq 2\nu(\mathcal{R}) - 1$，其中$\nu$是填充数，$\tau$是穿透数。Ajwani等人最近通过构造一个$2196\cdot 8^9$个矩形的无三角形反例并使用计算机辅助的打包和端口递归，得到矩形最大独立集的标准线性规划间隙为$17891/8064$。本文给出一个更简单且可手动检验的64矩形反例。它由一个八矩形小装置构建，其独立集注入四个有序插槽；然后使用该小装置的四个水平和四个垂直副本形成一个无三角形族，其中$\nu = 16$且$\tau\geq 32$。通过相同的水平 - 垂直步骤定义矩形族$P_r$，其$\nu(P_r)=4^{2^r}$。对于标准团，即点松弛，在$P_3$处得到有限间隙$73/32$，改进了之前的基准$17891/8064$。还构造了递归分数解和匹配穿透集，表明$\lim_r \alpha^*(P_r)/\nu(P_r)=\lim_r \tau(P_r)/\nu(P_r)=5/2$。最后，通过与孤立矩形的不相交并集表明，每个有理数$t\in[1,5/2)$都作为标准线性规划间隙以及合适矩形族的填充 - 穿透比出现。
英文摘要
Wegner conjectured that every finite family $\mathcal R$ of axis-parallel rectangles satisfies $τ(\mathcal R)\le 2ν(\mathcal R)-1$, where $ν$ is the packing number and $τ$ is the piercing number. Ajwani, Gajjala, Raman, and Ray recently disproved this by constructing a triangle-free counterexample on $2196\cdot 8^9$ rectangles and, using a computer-assisted package-and-port recursion, obtained a standard LP gap of $17891/8064$ for Maximum Independent Set of Rectangles. We give a simpler and hand-checkable counterexample with $64$ rectangles. It is built from an eight-rectangle gadget whose independent sets inject into four ordered slots; we then use four horizontal and four vertical copies of this gadget to form a triangle-free family with $ν=16$ and $τ\ge 32$. We use the same horizontal-vertical step to define recursive families of rectangles $P_r$ with $ν(P_r)=4^{2^r}$. For the standard clique, equivalently point, relaxation we obtain a finite gap $73/32$ at $P_3$, improving the previous benchmark of $17891/8064$. We then construct recursive fractional solutions and matching piercing sets showing $\lim_r α^*(P_r)/ν(P_r)=\lim_r τ(P_r)/ν(P_r)=5/2$. Finally, by disjoint union with isolated rectangles, we show that every rational $t\in[1,5/2)$ occurs as a standard LP gap and also as a packing-piercing ratio for suitable rectangle families.
