Rank Bounds for a Hierarchy of Lovász and Schrijver

Lovász and Schrijver [17] introduced several lift and project methods for 0-1 integer programs, now collectively known as Lovász-Schrijver (LS) hierarchies. Several lower bounds have since been proven for the rank of various linear programming relaxations in the LS and LS+ hierarchies. In this paper we investigate rank bounds in the more general LS∗ hierarchy, which allows lifts by any derived inequality as opposed to just x ≥ 0 and 1 − x ≥ 0 in the LS hierarchy. Rank lower bounds for LS∗ were obtained for the symmetric knapsack polytope by Grigoriev et al [14]. In this paper we show that LS∗ rank is incomparable to other hierarchies like LS+ and Sherali-Adams (SA) and show rank lower bounds for PHPn+1 n and integrality gaps for optimization problems like MAX-CUT in LS∗. The rank lower bounds for LS∗ follow from rank lower bounds for the SA∗ hierarchy which is a generalization of the SA hierarchy in the same vein as LS∗. We show that the LS∗ rank of PHPn+1 n is ∼ log2 n. We also extend the polynomial rank lower bounds and integrality gaps for MAX-CUT studied in Charikar et al. [5] for SA hierarchy to corresponding logarithmic rank lower bounds and integrality gaps in the LS∗ hierarchy. The proof translates various known SA rank lower bounds [5] to weaker SA∗ (and LS∗) rank lower bounds as long as the number of variables in the constraints of the initial linear program is small. In the reverse direction we give an example of a linear program with large number of variables in a constraint which has unit rank in SA∗ (and LS∗) hierarchies but linear rank in SA (and LS+) hierarchies. ∗Department of Computer Science, University of Chicago, Chicago IL 60637; email: [email protected].