# Rank bounds for a hierarchy of Lovász and Schrijver

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## Abstract

Lovász and Schrijver 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 \(\hbox {LS}_+\) hierarchies. We investigate rank bounds in the more general \(\hbox {LS}_*\) hierarchy, which allows lifts by any derived inequality as opposed to just \(x\ge 0\) and \(1-x\ge 0\) in the LS hierarchy. Rank lower bounds for \(\hbox {LS}_*\) were obtained for the symmetric knapsack polytope by Grigoriev et al. We reinitiate further investigation into such general lifts. We prove simple upper bounds on rank which show that under such general lifts one can potentially converge to the integer solution much faster than \(\hbox {LS}_+\) or Sherali–Adams (SA) hierarchy. This motivates our investigation of rank lower bounds and integrality gaps for \(\hbox {LS}_*\) and the \(\hbox {SA}_*\) hierarchy, the latter is a generalization of the SA hierarchy in the same vein as \(\hbox {LS}_*\). In particular, we show that the \(\hbox {LS}_*\) rank of \(PHP_n^{n+1}\) is \(\sim \log _2n\). We also extend the rank lower bounds and integrality gaps for SA hierarchy to the \(\hbox {LS}_*\) and \(\hbox {SA}_*\) hierarchies as long as the maximum number of variables in any constraint of the initial linear program is bounded by a constant.

## Keywords

Linear programming Integrality gaps Proof complexity## Notes

### Acknowledgments

The author thanks Yury Makarychev for reading several drafts of this paper and also for his help with proofs in Sect. 7. The author thanks Alexander Razborov for introducing him to the problem and the subject, and for comments on an earlier draft. The author thanks Madhur Tulsiani for comments and helpful discussions regarding the presentation of the final draft and Janos Simon for his encouragement.

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