A 64-Rectangle Counterexample to Wegner's Conjecture and LP Gaps up to $5/2$

Abstract

Wegner conjectured that every finite family $\mathcal R$ of axis-parallel rectangles satisfies $\tau(\mathcal R)\le 2\nu(\mathcal R)-1$, where $\nu$ is the packing number and $\tau$ 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 $\nu=16$ and $\tau\ge 32$. We use the same horizontal-vertical step to define recursive families of rectangles $P_r$ with $\nu(P_r)=4^{2^r}$. 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 $\lim_r \alpha^*(P_r)/\nu(P_r)=\lim_r \tau(P_r)/\nu(P_r)=5/2$. 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.