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Linear Round Integrality Gaps for the Lasserre Hierarchy Grant Schoenebeck

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

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Max Cut IP Homogenized

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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

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Max Cut SDP 0 1 4 23 v0v0 v1v1 v4v4 v2v2 v3v3

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Max Cut SDP and ▲ inequality

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

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Max Cut IP r-juntas Homogenized

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Max Cut Lasserre r-rounds

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CSP Maximization IP

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CSP Maximization Lasserre r-rounds SDP

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CSP Satisfaction IP

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CSP Satisfiablity Lasserre r-rounds SDP

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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

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

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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]

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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

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Width w-Resolution Combine if result has · w variables

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Width w-Resolution Combine if result has · w variables

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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:

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

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Multiplication Check ^

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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

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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

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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

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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 = 1 001 100111 010 x 3 + x 4 + x 5 = 0 101 110 011 000

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

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Future Directions Other Lasserre Integrality Gaps Positive Results Relationship to Resolution

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