Linear Round Integrality Gaps for the Lasserre Hierarchy Grant Schoenebeck

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

Max Cut IP Homogenized

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

Max Cut SDP v0v0 v1v1 v4v4 v2v2 v3v3

Max Cut SDP and ▲ inequality

SDP value of 5-cycle = 4 General Integrality Gap Remains 0.878… [KV05]

Max Cut IP r-juntas Homogenized

Max Cut Lasserre r-rounds

CSP Maximization IP

CSP Maximization Lasserre r-rounds SDP

CSP Satisfaction IP

CSP Satisfiablity Lasserre r-rounds SDP

Lasserre Facts Runs in time n r 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

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

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

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

Width w-Resolution Combine if result has · w variables

Width w-Resolution Combine if result has · w variables

Idea / Proof  ) width-2r Res )  F = linear functions “in”  L (r) = linear function of r-variables L 1, L 2 2 F Å  ) L 1 Δ L 2 2  ξ=L (r) /F = {[Ø][L * 2 ], [L * 2 ], …} Good-PA = Partial assignment that satisfies   ~ , for every Good-PA:  =  for every Good-PA:   

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

Multiplication Check ^

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

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

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

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) (y 1, …, y n )  Las r (VC)  (1-y 1, …, 1-y n )  Las r (IS) Transformation previously constructed vectors x 1 + x 2 + x 3 = x 3 + x 4 + x 5 =

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

Future Directions Other Lasserre Integrality Gaps Positive Results Relationship to Resolution