Determine the integrality gap of unit-square stabbing

Determine the exact integrality gap of the set-cover linear-programming relaxation for stabbing finite families of axis-parallel unit squares, in particular whether its value is the conjectured constant (35 − √5)/20 ≈ 1.638.

Background

The unit-square stabbing problem asks for the minimum number of horizontal and vertical lines whose union intersects every square in a finite family. The paper defines the integer stabbing number τ(S) and the fractional set-cover optimum τ*(S), whose ratio measures the integrality gap.

The cited results establish a lower bound of 3/2 and an upper bound strictly below 2, specifically at most 1.93. The paper reports that Elbassioni and Ray conjectured the exact value to be (35 − √5)/20 ≈ 1.638. The authors’ machine-guided search produces finite constructions with gaps 20/13 and, using a structured staircase formulation, 14/9, but does not resolve the conjectured asymptotic value.

References

For unit squares, Elbassioni and Ray gave the best known bounds via threshold rounding: the integrality gap is strictly below 2, at most 1.93, and at least 3/2, the lower bound coming from a four-square construction whose LP optimum is 2 but whose integer optimum is 3 [14]; they conjecture the true value is (35 − √5)/20 ≈1.638.

— Towards Learning-Guided Discovery of Extremal Geometric Constructions: A PatternBoost Study  (2610.19980 - Ajwani et al., 14 Oct 2026) in Section 3.2, p. 4