Covariance-Rank Rounding and a Sharp Integrality Gap for a Matroid Edge-Coverage Moment Lift

Abstract

Selecting vertices covers their incident edges, but a matroid can prevent selecting the combinations that cover the most weight. We study whether pair-selection information improves rounding from a semidefinite relaxation of this problem. A feasible point $(x,Y)$ records individual and paired selection values. Its relaxation objective is $U$, and the matrix $Y-x{x}^{\top }$ has rank $d$. We compare the feasible marginal vectors obtained by fixing one element as selected or excluded, then round only the best vector. For $d\ge 1$, the resulting basis has value at least $(3/4+1/(4d))U$. The procedure is deterministic and polynomial-time for a supplied exact rational point and an independence oracle. Weighting the covariance spectrum by the objective gives a stronger version of the bound. To show the rank factor is sharp, we construct a rank-$2m$ rational linear matroid and $m$ disjoint four-cycles with integral optimum $3m+1$, relaxation optimum $4m$, and an optimal covariance matrix of rank $m$. Thus the relaxation has limiting reciprocal integrality gap $4/3$, despite its conditional constraints. One valid pair-exclusion inequality makes it exact on this family, as an explicit semidefinite certificate shows. The result concerns rounding and the strength of this relaxation; it does not supply a low-rank optimal point or a fixed universal approximation improvement.