Superlinear Integrality Gaps for the Minimum Majority Problem

Philip M. Long

Abstract: The minimum majority problem is as follows: given a matrix $A \in \{-1, 1\}^{m \times n}$, minimize $\sum_{i=1}^n x_i$ subject to $A {x} \geq {1}$ and ${x} \in ({\mathbb Z}^+)^n$. An approximation algorithm that finds a solution with value $O({opt}^2 \log m)$ in ${poly}(m,n,{opt})$ time is known, which can be obtained by rounding a linear programming relaxation. We establish integrality gaps that limit the prospects for improving upon this guarantee through improved rounding and/or the application of Lovász--Schrijver (LS) or Sherali--Adams (SA) tightening of the relaxation. These gaps show that applying LS and SA relaxations cannot improve upon the $O({opt}^2 \log m)$ guarantee by more than a constant factor in polynomial time.