Linear Integrality Gaps and the Lasserre Hierarchy: Tight Bounds and Approximation Algorithms

1. Linear RoundIntegrality Gapsfor theLasserre Hierarchy Grant Schoenebeck

2. Max Cut IP Given graph G Partition vertices into two sets to Maximize # edges crossing partition

3. Max Cut IP Homogenized

4. Max Cut SDP [GW94] Integral Solution SDP Solution Integrality Gap = min Integrality Gap = ) – Approximation Algorithm Integrality Gap ¸ .878… (rounding)[GW] Integrality Gap · .878… (bad instance) [FS]

5. Max Cut SDP 0 v0 1 v1 4 v4 3 2 v3 v2

6. Max Cut SDP and ▲ inequality

7. Max Cut SDP and ▲ inequality • SDP value of 5-cycle = 4 • General Integrality Gap Remains 0.878… [KV05]

8. Max Cut IP r-juntas Homogenized

9. Max Cut Lasserre r-rounds

10. CSP Maximization IP

11. CSP Maximization Lasserre r-rounds SDP

12. CSP Satisfaction IP

13. CSP SatisfiablityLasserre r-rounds SDP

14. Lasserre Facts • Runs in time nr • Strength of Lasserre • Tighter than other hieracheis • Serali-Adams • Lavasz-Schrijver (LP and SDP) • r-rounds imply all valid constraints on r variables • tight after n rounds • Few rounds often work well • 1-round )Lovasz -function • 1-round )Goemans-Williamson • 3-rounds ) ARV sparsest cut • 2-rounds )MaxCut with ▲inequality • In general unknown and a great open question

15. Main Result Theorem: Random 3XOR instance not refuted by (n) rounds of Lasserre 3XOR:  =

16. Previous LS+ Results 3-SAT • 7/8+(n) LS+ rounds [AAT] Vertex Cover • 7/6-1 rounds [FO] • 7/6- (n) LS+ rounds [STT] • 2-(√log(n)/loglog(n)) LS+ rounds [GMPT]

17. LB for Random 3XOR Theorem: Random 3XOR instance not refuted by (n) rounds of Lasserre Proof: • Random 3XOR cannot be refuted by width-w resolutions for w = (n) [BW] • No width-w resolution )no w/4-Lasserre refutation

18. Width w-Resolution • Combine if result has · w variables

19. Width w-Resolution • Combine if result has · w variables

20. Idea / Proof • ) width-2r Res ) • F = linear functions “in”  • L(r) = linear function of r-variables • L1, L22 FÅ) L1Δ L22 • ξ=L(r)/F = {[Ø][L*2], [L*2], …} • Good-PA = Partial assignment that satisfies  •  ~ , • for every Good-PA:  =  • for every Good-PA: 

21. Idea / Proof • L(r) = linear function of r-variables • F = linear functions in C • ξ = L(r)/F = {[Ø][L*2], [L*2], …}

22. Multiplication Check ^

23. Corollaries Meta-Corollary: Reductions easy The (n) level of Lasserre: • Cannot refute K-SAT • IG of ½ +  for Max-k-XOR • IG of 1 – ½k +  for Max-k-SAT • IG of 7/6 +  for Vertex Cover • IG ½ +  for UniformHGVertexCover • IG any constant for UniformHGIndependentSet

24. Corollary I Random 3SAT instances not refuted by (n) rounds of Lasserre • Pick random 3SAT formula  • Pretend it is a 3XOR formula • Use vectors from 3XOR SDP to satisfy 3SAT SDP

25. Corollary II, III • Integrality gap of ½ + ε after (n) rounds of Lasserre forRandom 3XOR instance • Integrality gap of 7/8 + ε after (n) rounds of Lasserre forRandom 3SAT instance

26. Vertex Cover Corollary Integrality gap of 7/6 - ε after (n) rounds of Lasserre for Vertex Cover • FGLSS graphs from Random 3XOR formula (m = cn clauses) • (y1, …, yn) Lasr(VC)  (1-y1, …, 1-yn) Lasr(IS) • Transformation previously constructed vectors x3 + x4 + x5 = 0 x1 + x2 + x3 = 1 001 010 110 000 011 101 111 100

27. SDP Hierarchies from a Distance • Approximation Algorithms • Unconditional Lower Bounds • Proof Complexity • Local-Global Tradeoffs

28. Future Directions • Other Lasserre Integrality Gaps • Positive Results • Relationship to Resolution