# Rank bounds for a hierarchy of Lovász and Schrijver

- 114 Downloads
- 2 Citations

## 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.

## References

- Au YH, Tunçel L (2011) Complexity analyses of Bienstock-Zuckerberg and Lasserre relaxations on the matching and stable set polytopes. In: IPCO, pp 14–26Google Scholar
- Beame P, Huynh T, Pitassi T (2010) Hardness amplification in proof complexity. In: STOC, pp 87–96Google Scholar
- Benabbas S, Georgiou K, Magen A, Tulsiani M (2012) SDP gaps from pairwise independence. Theory Comput 8(1):269–289MathSciNetCrossRefGoogle Scholar
- Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Buresh-Oppenheim J, Galesi N, Hoory S, Magen A, Pitassi T (2003) Rank bounds and integrality gaps for cutting planes procedures. In: FOCS, pp 318–327Google Scholar
- Charikar M, Makarychev K, Makarychev Y (2009) Integrality gaps for Sherali–Adams relaxations. In: STOC, pp 283–292Google Scholar
- Cheung KKH (2007) Computation of the lasserre ranks of some polytopes. Math Oper Res 32:88–94MathSciNetCrossRefGoogle Scholar
- Chlamatac E, Tulsiani M (2010) Convex relaxations and integrality gaps. In: Handbook on semidefinite, cone and polynomial optimizationGoogle Scholar
- Cook W, Dash S (2001) On the matrix-cut rank of polyhedra. Math Oper Res 26:19–31MathSciNetCrossRefGoogle Scholar
- Dantchev SS, Martin B, Rhodes MNC (2009) Tight rank lower bounds for the Sherali–Adams proof system. Theor Comput Sci 410(21–23):2054–2063MathSciNetCrossRefGoogle Scholar
- Dash S (2001) On the matrix-cut of polyhedra and their use in integer programming. PhD thesis. Department of Computer Science, Rice UniversityGoogle Scholar
- de la Vega WF, Kenyon-Mathieu C (2007) Linear programming relaxations of MAX-CUT. In: SODA, pp 53–61Google Scholar
- Eisenbrand F (1999) On the membership problem for the elementary closure of a polyhedron. Combinatorica 19(2):297–300MathSciNetCrossRefGoogle Scholar
- Goemans MX, Tunçel L (2001) When does the positive semidefiniteness constraint help in lifting procedures? Math Oper Res 26(4):796–815MathSciNetCrossRefGoogle Scholar
- Grigoriev D, Hirsch E, Pasechnik D (2002) Complexity of semi-algebraic proofs. In: STACS, LNCS, vol 2285, pp 419–430Google Scholar
- Hong SP, Tunçel L (2008) Unification of lower-bound analyses of the lift-and-project rank of combinatorial optimization polyhedra. Discrete Appl Math 156:25–41MathSciNetCrossRefGoogle Scholar
- Krajíček J (1995) Bounded arithmetic, propositional logic, and complexity theory. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Laurent M (2003) A comparison of the Sherali–Adams, Lovász–Schrijver, and Lasserre relaxations for 0–1 programming. Math Oper Res 28(3):470–496MathSciNetCrossRefMATHGoogle Scholar
- Lovász L, Schrijver A (1991) Cones of matrices and set-functions and 0–1 optimization. SIAM J Optim 1:166–190MathSciNetCrossRefGoogle Scholar
- Mathieu C, Sinclair A (2009) Sherali–Adams relaxations of the matching polytope. In: STOC, pp 293–302Google Scholar
- Pitassi T, Segerlind N (2009) Exponential lower bounds and integrality gaps for tree-like Lovász–Schrijver procedures. In: SODA, pp 355–364Google Scholar
- Pudlak P (1999) On the complexity of propositional calculus. In: Sets and proofs, invited papers from Logic Colloquium’97, Cambridge University Press, Cambridge, pp 197–218Google Scholar
- Razborov AA (2001) Proof complexity of Pigeonhole principles. In: 5th Developments in language theory, LNCS, vol 2295, pp 100–116Google Scholar
- Rudich S, Wigderson A (2004) Computational complexity theory. AMS, IAS/Park City Mathematics SeriesGoogle Scholar
- Segerlind N (2007) The complexity of propositional proofs. Bull Symb Log 13(4):417–481MathSciNetCrossRefMATHGoogle Scholar
- Sherali HD, Adams WP (1990) A hierarchy of relaxations between the continuous and Convex hull representations for zero-one programming problems. SIAM J Disc Math 3:411–430MathSciNetCrossRefMATHGoogle Scholar
- Sherali HD, Adams WP, Driscoll PJ (1998) Exploiting special structures in constructing a hierarchy for relaxations for 0–1 mixed integer problems. Oper Res 46:396–405MathSciNetCrossRefGoogle Scholar
- Stephen T, Tunçel L (1999) On a representation of the matching polytope via semidefinite liftings. Math Oper Res 24:1–7MathSciNetCrossRefMATHGoogle Scholar
- Vazirani VV (2004) Approximation algorithms. Springer, New YorkGoogle Scholar