Lower bounds on the size of semidefinite programs
Lower bounds on the size of semidefinite programs
James Lee
Univ of Washington.
Prasad Raghavendra U.C.Berkeley.
David Steurer Cornell University
Extended formula/ons of polytopes traveling salesman problem (TSP)
𝑑↓14 1
𝑑↓24
𝑑↓13 3
4
𝑑↓35 5 𝑑↓25
given 𝑛×𝑛 cost matrix 𝐷=(𝑑↓𝑖𝑗 ), find minimum cost 𝑛-‐tour traveling salesman polytope tsp↓𝑛 =convex−hull 𝟏↓𝐸(𝐶) ∈{0,1}↑𝑛↑2 𝐶 is 𝑛−𝑡𝑜𝑢𝑟
characteriza8on: solving TSP on 𝑛 ciJes same as opJmizing linear funcJons over tsp↓𝑛
tsp↓𝑛 has exponenJal number of facets à no “direct“ LP algorithms
2
Extended formula/ons of polytopes idea: reduce opJmizing linear funcJons over tsp↓𝑛 to opJmizing linear funcJons over polytope defined by few linear inequali8es Size-‐𝑅 extended LP formulaJon for 𝑇𝑆𝑃↓𝑛
size-‐𝑅 polytope 𝑃, defined by ≤𝑅 linear inequaliJes, such that tsp↓𝑛 is image of 𝑃 under some linear map ℓ𝓁
tsp↓𝑛 ℝ↑𝑛↑2
linear map ℓ𝓁
𝑃 ℝ↑𝑅 (for 𝑅≫𝑛↑2 )
size-‐𝑅 SDP algorithm for tsp↓𝑛 (aka size-‐𝑅 extended SDP formulaJon) size-‐𝑅 spectrahedron 𝑃, defined by 𝑅×𝑅 linear matrix inequality, such that tsp↓𝑛 is image of 𝑃 under some linear map ℓ𝓁
complicated polytopes can have simple li1s (here: complicated = many inequaliJes; simple = few inequaliJes) unit ℓ𝓁↓1 -‐ball
projecJon (█■Id&0 ) {█■∑𝑖↑▒|𝑥↓𝑖 | ≤1 𝑥∈ℝ↑𝑛 }
{█■█■−𝑦≤𝑥≤𝑦∑𝑖↑▒𝑦↓𝑖 ≤1 𝑥,𝑦∈ℝ
{±𝑥↓1 ±…±𝑥↓𝑛 ≤1}
comparison: 2↑𝑛 linear inequaliJes vs. 2𝑛+1 linear inequaliJes idea: introduce variables for absolute values |𝑥| other polytopes: spanning trees, Held–Karp TSP LP / SDP hierarchies introduce new variables systemaJcally
lower bounds on extended LP/SDP formula=ons minimum size of LP/SDP algorithms for tsp↓𝑛 symmetric LP: 𝟐↑𝛀(𝒏) [Yannakakis]
symmetric SDP: 𝟐↑𝛀(𝒏) [Lee-‐Raghavendra-‐S.-‐Tan Fawzi-‐Saunderson-‐Parrio]
general LP: 𝟐↑𝛀(𝒏) [Fiorini-‐Massar-‐Poku[a-‐ Tiwary-‐de Wolf Rothvoss]
general SDP: 𝟐↑𝒏↑𝟏/𝟏𝟑 [this talk] Similar lower bounds for the 𝐶𝑈𝑇↓𝑛 and 𝐶𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛↓𝑛 polytopes.
Uncondi=onal lower bounds for restricted but powerful model of computaJon.
GW SDP for MaxCut
1
Semidefinite Program for MaxCut:
[Goemans-‐Williamson 94]
Maximize 1/4 ∑(𝑖,𝑗)∈𝐸↑▒|𝑣↓𝑖↑ − 𝑣↓𝑗↑ |↑2 ↑ Subject to |𝑣↓𝑖 |↑2 =1
Depends only on 𝑛 not on graph 𝐺
Linear funcJon represenJng cut value on graph 𝐺
Spectrahedron: 𝑌 is a 𝑛× 𝑛-‐p.s.d matrix
𝑌↓𝑖𝑖 =1 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖∈[𝑛]
Maximize:
1/4 ∑(𝑖,𝑗)∈𝐸↑▒(𝑌↓𝑖𝑖 +𝑌↓𝑗𝑗 −2𝑌↓𝑖𝑗 )
-‐1
Generic SDP for MaxCut Maximize ⟨𝑤,𝑌⟩ subject to 𝑌∈Spectrahedron S 𝑆=𝑆↓𝑅↑+ ∩{𝑌|𝐴𝑌=𝑏}
𝑎𝑓𝑓𝑖𝑛𝑒 𝑠𝑝𝑎𝑐𝑒 𝐴𝑌=𝑏
𝑝.𝑠.𝑑 𝑐𝑜𝑛𝑒 𝑅×𝑅
𝑆↓𝑅↑+ =𝑐𝑜𝑛𝑒 𝑜𝑓 𝑅×𝑅 𝑝.𝑠.𝑑 𝑚𝑎𝑡𝑟𝑖𝑐𝑒𝑠 AfSine space AY = b (𝑌∈ 𝑆𝑖𝑧𝑒 𝑜𝑓 𝑆𝐷𝑃 = 𝑅
𝑹↑𝑅×𝑅 )
General SDP Relaxa=on for MaxCut Spectrahedron
𝑆=𝑆↓𝑅↑+ ∩{𝑌|𝐴𝑌=𝑏}
𝑄:{−1,1}↑𝑛 →𝑆
For all 𝑥∈ {−1,1}↑𝑛 , there is a corresponding solu8on 𝑄↓𝑥 ∈ 𝑆 𝑤: 𝐺𝑟𝑎𝑝ℎ𝑠→𝑹↑𝑅×𝑅
Every Graph 𝐺 has a `lineariza=on’ 𝑤↓𝐺 ∈ 𝑹↑𝑅×𝑅
(on 𝑛 verJces).
Maximize ⟨𝑤↓𝐺 ,𝑌⟩ subject to 𝑌∈Spectrahedron S 𝑆=𝑆↓𝑅↑+ ∩{𝑌|𝐴𝑌=𝑏}
Expressing objec/ve value: For all integer assignments 𝑥∈ {−1,1}
↑𝑛 and graph 𝐺, 𝐺(𝑥) =⟨𝑤↓𝐺 , 𝑄↓𝑥 ⟩ where 𝐺(𝑥) = value of cut 𝑥 on the graph 𝐺
Result (Informal Statement): For every Max-‐CSP (like MaxCut), The k-‐round Lasserre/low-‐degree SOS SDP relaxa8on achieves the best approxima8on among all SDP relaxa8ons of roughly the same size. Theorem: For every integer 𝑘∈𝑁, for every Max-‐CSP, A SDP relaxaJon of size 𝑛↑𝑘/𝐶 is no more powerful than a degree 𝑘-‐SoS SDP relaxaJon (C = absolute constant) Using known lower-‐bounds against low-‐degree SoS hierarchies, Corollary: For every integer 𝑘, for every Max-‐CSP, a SDP relaxaJon of size 𝑛↑𝑘 cannot yield the following approximaJons: • 7/8 +𝜖-‐approximaJon for Max-‐3-‐SAT for any constant 𝜖. • 1/2 +𝜖-‐approximaJon for Max-‐3-‐LIN for any constant 𝜖.
Prior Work (Linear Programming Extended FormulaJons) • For `symmetric LPs’, an exponenJal lower bound for exact TSP and exact non-‐biparJte matching. [Yannakakis 89] • For general LPs, an exponenJal lower bound for exact TSP. [Fiorini-‐ etal] • For general LPs, an exponenJal lower bound for exact Perfect Matching. [Rothvoss]
In a slightly-‐more restricJve model,
2↑𝑛↑𝜖
• A -‐lower bound for Clique. [Fiorini etal] • A
𝑛↑1/2 −𝜖 -‐approximaJon to Max-‐
2↑𝑛↑𝜖 -‐lower bound for 𝑛↑1−𝜖 -‐approximaJon to Max-‐Clique.
Prior Work (Linear Programming Extended FormulaJons) Theorem: [Chan-‐Lee-‐R-‐Steurer] For every integer 𝑘 𝑐 ⇔ 𝑠𝑜𝑠𝑑𝑒𝑔(𝑐−𝐺(𝑥))>𝑑
𝑐
𝑠𝑜𝑠𝑑𝑒𝑔(𝑐−𝐺(𝑥))=𝑑
𝑐−𝐺(𝑥)
〈 𝐷, 𝑓〉=𝐸↓𝑥 𝐷(𝑥)⋅𝑓(𝑥)
Yannakakis’ characteriza/on of extension complexity
SoS characterization of SDPs Maximize ⟨𝑤↓𝐺 ,𝑌⟩ subject to 𝑌∈Spectrahedron S 𝑆=𝑆↓𝑅↑+ ∩{𝑌|𝐴𝑌=𝑏}
Vector Space of funcJons ( {−1,1}↑𝑛 → 𝑅) 𝑉 = 𝑠𝑝𝑎𝑛{ 𝑣↓1↑1 , …,𝑣↓𝑅↑𝑅 } Minimize 𝑐 such that
𝑐−𝐺(𝑥)=∑𝑞∈𝑉↑▒𝑞↑2 (𝑥)
Theorem: For any graph G, SDP-‐OPT(𝐺)≤𝑐 if and only if 𝑐 −𝐺=𝑆𝑢𝑚 𝑜𝑓 𝑆𝑞𝑢𝑎𝑟𝑒𝑠 𝑜𝑓 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠 𝑖𝑛 𝑉
SoS characterization of SDPs Vector Space of funcJons ( Maximize ⟨𝑤↓𝐺 ,𝑌⟩ subject to 𝑌∈Spectrahedron S 𝑆=𝑆↓𝑅↑+ ∩{𝑌|𝐴𝑌=𝑏}
{−1,1}↑𝑛 → 𝑹) 𝑉 = 𝑠𝑝𝑎𝑛{ 𝑣↓1↑1 , …,𝑣↓𝑅↑𝑅 }
Minimize 𝑐 such that
𝑐−𝐺(𝑥)=∑𝑞∈𝑉↑▒𝑞↑2 (𝑥)
For d-‐round Lasserre SDP, V = { vector space of polynomials of degree ≤𝑑} Degree d SOS SDP for MaxCut Minimize 𝑐 such that
𝑐−∑(𝑖,𝑗)∈𝐸↑▒𝑤↓𝑖𝑗 (𝑥↓𝑖 −𝑥↓𝑗 )↑2 =∑deg(𝑞) ≤𝑑↑▒𝑞(𝑥)↑2 for x∈{−1,1}↑𝑛
PSD rank Defini/on: PSD rank(M) = smallest R so that ∃ factorizaJon: 𝑀=𝑃⋅𝑄↑⊤ where rows of P, Q are 𝑅× 𝑅 p.s.d matrices
M =
Theorem:
𝑄↑⊤
𝑃
[Yannakakis]
∃ size-‐𝑅 SDP with approx. raJo 𝛼 for 𝑛-‐variable MAXCUT ⇔ psd-‐rank(𝑀)≤𝑅 for
variable assignments 𝑥∈{0,1}↑𝑛
MAXCUT
instances 𝐺 in 𝑛 variables
𝑀(𝐺,𝑥)= 𝛼⋅max(𝐺) −𝐺(𝑥)
Theorem:
[Yannakakis]
∃ size-‐𝑅 SDP with approx. raJo 𝛼 for 𝑛-‐variable MAXCUT ⇔ psd-‐rank(𝑀)≤𝑅 for
variable assignments 𝑥∈{0,1}↑𝑛
MAXCUT
instances 𝐺 in 𝑛 variables
𝑀(𝐺,𝑥)= 𝛼⋅max(𝐺) −𝐺(𝑥)
Proof: Suppose following SDP gives an 𝛼-‐approximaJon. Maximize ⟨𝑤↓𝐺 ,𝑌⟩ subject to 𝑌∈Spectrahedron S 𝑆=𝑆↓𝑅↑+ ∩{𝑌|𝐴𝑌=𝑏}
For each assignment 𝑥∈{0,1}↑𝑛 → 𝑄↓𝑥 ∈𝑆↓𝑅↑+ , For all graphs 𝐺 𝐺(𝑥)=〈𝑤↓𝐺 ,𝑄↓𝑥 〉
nonneg.-‐rank and psd-‐rank factorizaJon: 𝑀=𝑃⋅𝑄↑⊤
𝑃
𝑄↑⊤
two ways to evaluate: 1. inner products of rows of 𝑃 and 𝑄 2. outer products of columns of 𝑃 and 𝑄 for nonneg.-‐rank: nonneg. rows ⇔ nonneg. columns à can use both ways to evaluate factorizaJon all known lower bounds work with outer-‐product view (nonnegaJve rectangles) for psd-‐rank psd rows ⇏ psd columns
Main Technical Result
main theorem main theorem: suppose 𝑓 has sos deg > d. then exists 𝐶↓𝑓 ≥1 such that psd-‐rank(𝑀↓𝑛↑𝑓 )≥𝐶↓𝑓 ⋅𝑛↑𝑑/5 for all 𝑛∈ℕ points 𝑥∈{0,1}↑𝑛 subsets 𝑆⊆[𝑛] with |𝑆|=𝑚
𝑀↓𝑛↑𝑓 (𝑆,𝑥)=𝑓(𝑥↓𝑆 )
evaluaJons of 𝑓 applied to subcubes
𝑥↓𝑆 =(𝑥↓𝑠↓1 ,…,𝑥↓ 𝑠↓𝑚 ) for 𝑆={𝑠↓1 ,…, 𝑠↓𝑚 }
main theorem: suppose 𝑓 has sos deg > d. then exists 𝐶↓𝑓 ≥1 such that psd-‐rank(𝑀↓𝑛↑𝑓 )≥𝐶↓𝑓 ⋅𝑛↑𝑑/5 for all 𝑛∈ℕ points 𝑥∈{0,1}↑𝑛 subsets 𝑆⊆[𝑛] with |𝑆|=𝑚
𝑀↓𝑛↑𝑓 (𝑆,𝑥)=𝑓(𝑥↓𝑆 )
Main theorem ⇒ Op=mality of Lasserre SDP
evaluaJons of 𝑓 applied to subcubes
𝑥↓𝑆 =(𝑥↓𝑠↓1 ,…,𝑥↓ 𝑠↓𝑚 ) for 𝑆={𝑠↓1 ,…, 𝑠↓𝑚 }
Proof:Suppose degree 𝑑 Lasserre has an 𝛼-‐integrality gap for MaxCut ⇔∃ some graph G such that deg↓𝑠𝑜𝑠 (𝛼⋅max(𝐺) −𝐺(𝑥)) >𝑑
⇔ For 𝑓=𝛼⋅max(𝐺) −𝐺(𝑥) 𝑝𝑠𝑑𝑟𝑎𝑛𝑘(𝑀↓𝑛↑𝑓 )≥𝐶↓𝑓 𝑛↑𝑑/5
⇔ No SDP of size 𝑜(𝑛↑𝑑/5 ) gets an 𝛼−𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛.
main theorem: suppose 𝑓 has sos deg > d. then exists 𝐶↓𝑓 ≥1 such that psd-‐rank(𝑀↓𝑛↑𝑓 )≥𝐶↓𝑓 ⋅𝑛↑𝑑/5 for all 𝑛∈ℕ points 𝑥∈{0,1}↑𝑛 subsets 𝑆⊆[𝑛] with |𝑆|=𝑚
𝑀↓𝑛↑𝑓 (𝑆,𝑥)=𝑓(𝑥↓𝑆 )
evaluaJons of 𝑓 applied to subcubes
𝑥↓𝑆 =(𝑥↓𝑠↓1 ,…,𝑥↓ 𝑠↓𝑚 ) for 𝑆={𝑠↓1 ,…, 𝑠↓𝑚 }
sanity check: 𝑀↓𝑛↑𝑓 does not have low-‐deg. factorizaJons (Tr 𝑃↓𝑆 𝑄↓𝑥 )↓𝑆,𝑥 suppose 𝑄↓𝑥 =𝑅(𝑥)↑2 and 𝑥↦𝑅(𝑥) has degree at most 𝑑/2 then, Tr 𝑃↓𝑆 𝑄↓𝑥 =‖𝑃↓𝑆↑1∕2 𝑅(𝑥)‖↓𝐹↑2 à sum of squares of degree-‐𝑑/2 polynomials for each 𝑆 à cannot be factorizaJon of 𝑀↓𝑛↑𝑓 (because 𝑓 has sos deg > 𝑑)
Separa=ng from low-‐degree factoriza=ons By duality, sos-‐deg(𝑓)>𝑑 if and only if ∃ deg.-‐𝑑 pseudo-‐distr. 𝐷 with 𝔼↓𝑥 𝐷(𝑥)𝑓(𝑥)𝑑/2 in 𝑆 ) =ℙ↓S (|𝑇∩𝑆|>𝑑/2)≤(|𝑆|⋅|𝑇|/𝑛 )↑𝑑/2 ≈ 𝑛↑−𝑑/2
small enough to counter effect of 𝑃↓𝑆
Thank You
James Lee
Univ of Washington.
Prasad Raghavendra U.C.Berkeley.
David Steurer Cornell University
Extended formula/ons of polytopes traveling salesman problem (TSP)
𝑑↓14 1
𝑑↓24
𝑑↓13 3
4
𝑑↓35 5 𝑑↓25
given 𝑛×𝑛 cost matrix 𝐷=(𝑑↓𝑖𝑗 ), find minimum cost 𝑛-‐tour traveling salesman polytope tsp↓𝑛 =convex−hull 𝟏↓𝐸(𝐶) ∈{0,1}↑𝑛↑2 𝐶 is 𝑛−𝑡𝑜𝑢𝑟
characteriza8on: solving TSP on 𝑛 ciJes same as opJmizing linear funcJons over tsp↓𝑛
tsp↓𝑛 has exponenJal number of facets à no “direct“ LP algorithms
2
Extended formula/ons of polytopes idea: reduce opJmizing linear funcJons over tsp↓𝑛 to opJmizing linear funcJons over polytope defined by few linear inequali8es Size-‐𝑅 extended LP formulaJon for 𝑇𝑆𝑃↓𝑛
size-‐𝑅 polytope 𝑃, defined by ≤𝑅 linear inequaliJes, such that tsp↓𝑛 is image of 𝑃 under some linear map ℓ𝓁
tsp↓𝑛 ℝ↑𝑛↑2
linear map ℓ𝓁
𝑃 ℝ↑𝑅 (for 𝑅≫𝑛↑2 )
size-‐𝑅 SDP algorithm for tsp↓𝑛 (aka size-‐𝑅 extended SDP formulaJon) size-‐𝑅 spectrahedron 𝑃, defined by 𝑅×𝑅 linear matrix inequality, such that tsp↓𝑛 is image of 𝑃 under some linear map ℓ𝓁
complicated polytopes can have simple li1s (here: complicated = many inequaliJes; simple = few inequaliJes) unit ℓ𝓁↓1 -‐ball
projecJon (█■Id&0 ) {█■∑𝑖↑▒|𝑥↓𝑖 | ≤1 𝑥∈ℝ↑𝑛 }
{█■█■−𝑦≤𝑥≤𝑦∑𝑖↑▒𝑦↓𝑖 ≤1 𝑥,𝑦∈ℝ
{±𝑥↓1 ±…±𝑥↓𝑛 ≤1}
comparison: 2↑𝑛 linear inequaliJes vs. 2𝑛+1 linear inequaliJes idea: introduce variables for absolute values |𝑥| other polytopes: spanning trees, Held–Karp TSP LP / SDP hierarchies introduce new variables systemaJcally
lower bounds on extended LP/SDP formula=ons minimum size of LP/SDP algorithms for tsp↓𝑛 symmetric LP: 𝟐↑𝛀(𝒏) [Yannakakis]
symmetric SDP: 𝟐↑𝛀(𝒏) [Lee-‐Raghavendra-‐S.-‐Tan Fawzi-‐Saunderson-‐Parrio]
general LP: 𝟐↑𝛀(𝒏) [Fiorini-‐Massar-‐Poku[a-‐ Tiwary-‐de Wolf Rothvoss]
general SDP: 𝟐↑𝒏↑𝟏/𝟏𝟑 [this talk] Similar lower bounds for the 𝐶𝑈𝑇↓𝑛 and 𝐶𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛↓𝑛 polytopes.
Uncondi=onal lower bounds for restricted but powerful model of computaJon.
GW SDP for MaxCut
1
Semidefinite Program for MaxCut:
[Goemans-‐Williamson 94]
Maximize 1/4 ∑(𝑖,𝑗)∈𝐸↑▒|𝑣↓𝑖↑ − 𝑣↓𝑗↑ |↑2 ↑ Subject to |𝑣↓𝑖 |↑2 =1
Depends only on 𝑛 not on graph 𝐺
Linear funcJon represenJng cut value on graph 𝐺
Spectrahedron: 𝑌 is a 𝑛× 𝑛-‐p.s.d matrix
𝑌↓𝑖𝑖 =1 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖∈[𝑛]
Maximize:
1/4 ∑(𝑖,𝑗)∈𝐸↑▒(𝑌↓𝑖𝑖 +𝑌↓𝑗𝑗 −2𝑌↓𝑖𝑗 )
-‐1
Generic SDP for MaxCut Maximize ⟨𝑤,𝑌⟩ subject to 𝑌∈Spectrahedron S 𝑆=𝑆↓𝑅↑+ ∩{𝑌|𝐴𝑌=𝑏}
𝑎𝑓𝑓𝑖𝑛𝑒 𝑠𝑝𝑎𝑐𝑒 𝐴𝑌=𝑏
𝑝.𝑠.𝑑 𝑐𝑜𝑛𝑒 𝑅×𝑅
𝑆↓𝑅↑+ =𝑐𝑜𝑛𝑒 𝑜𝑓 𝑅×𝑅 𝑝.𝑠.𝑑 𝑚𝑎𝑡𝑟𝑖𝑐𝑒𝑠 AfSine space AY = b (𝑌∈ 𝑆𝑖𝑧𝑒 𝑜𝑓 𝑆𝐷𝑃 = 𝑅
𝑹↑𝑅×𝑅 )
General SDP Relaxa=on for MaxCut Spectrahedron
𝑆=𝑆↓𝑅↑+ ∩{𝑌|𝐴𝑌=𝑏}
𝑄:{−1,1}↑𝑛 →𝑆
For all 𝑥∈ {−1,1}↑𝑛 , there is a corresponding solu8on 𝑄↓𝑥 ∈ 𝑆 𝑤: 𝐺𝑟𝑎𝑝ℎ𝑠→𝑹↑𝑅×𝑅
Every Graph 𝐺 has a `lineariza=on’ 𝑤↓𝐺 ∈ 𝑹↑𝑅×𝑅
(on 𝑛 verJces).
Maximize ⟨𝑤↓𝐺 ,𝑌⟩ subject to 𝑌∈Spectrahedron S 𝑆=𝑆↓𝑅↑+ ∩{𝑌|𝐴𝑌=𝑏}
Expressing objec/ve value: For all integer assignments 𝑥∈ {−1,1}
↑𝑛 and graph 𝐺, 𝐺(𝑥) =⟨𝑤↓𝐺 , 𝑄↓𝑥 ⟩ where 𝐺(𝑥) = value of cut 𝑥 on the graph 𝐺
Result (Informal Statement): For every Max-‐CSP (like MaxCut), The k-‐round Lasserre/low-‐degree SOS SDP relaxa8on achieves the best approxima8on among all SDP relaxa8ons of roughly the same size. Theorem: For every integer 𝑘∈𝑁, for every Max-‐CSP, A SDP relaxaJon of size 𝑛↑𝑘/𝐶 is no more powerful than a degree 𝑘-‐SoS SDP relaxaJon (C = absolute constant) Using known lower-‐bounds against low-‐degree SoS hierarchies, Corollary: For every integer 𝑘, for every Max-‐CSP, a SDP relaxaJon of size 𝑛↑𝑘 cannot yield the following approximaJons: • 7/8 +𝜖-‐approximaJon for Max-‐3-‐SAT for any constant 𝜖. • 1/2 +𝜖-‐approximaJon for Max-‐3-‐LIN for any constant 𝜖.
Prior Work (Linear Programming Extended FormulaJons) • For `symmetric LPs’, an exponenJal lower bound for exact TSP and exact non-‐biparJte matching. [Yannakakis 89] • For general LPs, an exponenJal lower bound for exact TSP. [Fiorini-‐ etal] • For general LPs, an exponenJal lower bound for exact Perfect Matching. [Rothvoss]
In a slightly-‐more restricJve model,
2↑𝑛↑𝜖
• A -‐lower bound for Clique. [Fiorini etal] • A
𝑛↑1/2 −𝜖 -‐approximaJon to Max-‐
2↑𝑛↑𝜖 -‐lower bound for 𝑛↑1−𝜖 -‐approximaJon to Max-‐Clique.
Prior Work (Linear Programming Extended FormulaJons) Theorem: [Chan-‐Lee-‐R-‐Steurer] For every integer 𝑘 𝑐 ⇔ 𝑠𝑜𝑠𝑑𝑒𝑔(𝑐−𝐺(𝑥))>𝑑
𝑐
𝑠𝑜𝑠𝑑𝑒𝑔(𝑐−𝐺(𝑥))=𝑑
𝑐−𝐺(𝑥)
〈 𝐷, 𝑓〉=𝐸↓𝑥 𝐷(𝑥)⋅𝑓(𝑥)
Yannakakis’ characteriza/on of extension complexity
SoS characterization of SDPs Maximize ⟨𝑤↓𝐺 ,𝑌⟩ subject to 𝑌∈Spectrahedron S 𝑆=𝑆↓𝑅↑+ ∩{𝑌|𝐴𝑌=𝑏}
Vector Space of funcJons ( {−1,1}↑𝑛 → 𝑅) 𝑉 = 𝑠𝑝𝑎𝑛{ 𝑣↓1↑1 , …,𝑣↓𝑅↑𝑅 } Minimize 𝑐 such that
𝑐−𝐺(𝑥)=∑𝑞∈𝑉↑▒𝑞↑2 (𝑥)
Theorem: For any graph G, SDP-‐OPT(𝐺)≤𝑐 if and only if 𝑐 −𝐺=𝑆𝑢𝑚 𝑜𝑓 𝑆𝑞𝑢𝑎𝑟𝑒𝑠 𝑜𝑓 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠 𝑖𝑛 𝑉
SoS characterization of SDPs Vector Space of funcJons ( Maximize ⟨𝑤↓𝐺 ,𝑌⟩ subject to 𝑌∈Spectrahedron S 𝑆=𝑆↓𝑅↑+ ∩{𝑌|𝐴𝑌=𝑏}
{−1,1}↑𝑛 → 𝑹) 𝑉 = 𝑠𝑝𝑎𝑛{ 𝑣↓1↑1 , …,𝑣↓𝑅↑𝑅 }
Minimize 𝑐 such that
𝑐−𝐺(𝑥)=∑𝑞∈𝑉↑▒𝑞↑2 (𝑥)
For d-‐round Lasserre SDP, V = { vector space of polynomials of degree ≤𝑑} Degree d SOS SDP for MaxCut Minimize 𝑐 such that
𝑐−∑(𝑖,𝑗)∈𝐸↑▒𝑤↓𝑖𝑗 (𝑥↓𝑖 −𝑥↓𝑗 )↑2 =∑deg(𝑞) ≤𝑑↑▒𝑞(𝑥)↑2 for x∈{−1,1}↑𝑛
PSD rank Defini/on: PSD rank(M) = smallest R so that ∃ factorizaJon: 𝑀=𝑃⋅𝑄↑⊤ where rows of P, Q are 𝑅× 𝑅 p.s.d matrices
M =
Theorem:
𝑄↑⊤
𝑃
[Yannakakis]
∃ size-‐𝑅 SDP with approx. raJo 𝛼 for 𝑛-‐variable MAXCUT ⇔ psd-‐rank(𝑀)≤𝑅 for
variable assignments 𝑥∈{0,1}↑𝑛
MAXCUT
instances 𝐺 in 𝑛 variables
𝑀(𝐺,𝑥)= 𝛼⋅max(𝐺) −𝐺(𝑥)
Theorem:
[Yannakakis]
∃ size-‐𝑅 SDP with approx. raJo 𝛼 for 𝑛-‐variable MAXCUT ⇔ psd-‐rank(𝑀)≤𝑅 for
variable assignments 𝑥∈{0,1}↑𝑛
MAXCUT
instances 𝐺 in 𝑛 variables
𝑀(𝐺,𝑥)= 𝛼⋅max(𝐺) −𝐺(𝑥)
Proof: Suppose following SDP gives an 𝛼-‐approximaJon. Maximize ⟨𝑤↓𝐺 ,𝑌⟩ subject to 𝑌∈Spectrahedron S 𝑆=𝑆↓𝑅↑+ ∩{𝑌|𝐴𝑌=𝑏}
For each assignment 𝑥∈{0,1}↑𝑛 → 𝑄↓𝑥 ∈𝑆↓𝑅↑+ , For all graphs 𝐺 𝐺(𝑥)=〈𝑤↓𝐺 ,𝑄↓𝑥 〉
nonneg.-‐rank and psd-‐rank factorizaJon: 𝑀=𝑃⋅𝑄↑⊤
𝑃
𝑄↑⊤
two ways to evaluate: 1. inner products of rows of 𝑃 and 𝑄 2. outer products of columns of 𝑃 and 𝑄 for nonneg.-‐rank: nonneg. rows ⇔ nonneg. columns à can use both ways to evaluate factorizaJon all known lower bounds work with outer-‐product view (nonnegaJve rectangles) for psd-‐rank psd rows ⇏ psd columns
Main Technical Result
main theorem main theorem: suppose 𝑓 has sos deg > d. then exists 𝐶↓𝑓 ≥1 such that psd-‐rank(𝑀↓𝑛↑𝑓 )≥𝐶↓𝑓 ⋅𝑛↑𝑑/5 for all 𝑛∈ℕ points 𝑥∈{0,1}↑𝑛 subsets 𝑆⊆[𝑛] with |𝑆|=𝑚
𝑀↓𝑛↑𝑓 (𝑆,𝑥)=𝑓(𝑥↓𝑆 )
evaluaJons of 𝑓 applied to subcubes
𝑥↓𝑆 =(𝑥↓𝑠↓1 ,…,𝑥↓ 𝑠↓𝑚 ) for 𝑆={𝑠↓1 ,…, 𝑠↓𝑚 }
main theorem: suppose 𝑓 has sos deg > d. then exists 𝐶↓𝑓 ≥1 such that psd-‐rank(𝑀↓𝑛↑𝑓 )≥𝐶↓𝑓 ⋅𝑛↑𝑑/5 for all 𝑛∈ℕ points 𝑥∈{0,1}↑𝑛 subsets 𝑆⊆[𝑛] with |𝑆|=𝑚
𝑀↓𝑛↑𝑓 (𝑆,𝑥)=𝑓(𝑥↓𝑆 )
Main theorem ⇒ Op=mality of Lasserre SDP
evaluaJons of 𝑓 applied to subcubes
𝑥↓𝑆 =(𝑥↓𝑠↓1 ,…,𝑥↓ 𝑠↓𝑚 ) for 𝑆={𝑠↓1 ,…, 𝑠↓𝑚 }
Proof:Suppose degree 𝑑 Lasserre has an 𝛼-‐integrality gap for MaxCut ⇔∃ some graph G such that deg↓𝑠𝑜𝑠 (𝛼⋅max(𝐺) −𝐺(𝑥)) >𝑑
⇔ For 𝑓=𝛼⋅max(𝐺) −𝐺(𝑥) 𝑝𝑠𝑑𝑟𝑎𝑛𝑘(𝑀↓𝑛↑𝑓 )≥𝐶↓𝑓 𝑛↑𝑑/5
⇔ No SDP of size 𝑜(𝑛↑𝑑/5 ) gets an 𝛼−𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛.
main theorem: suppose 𝑓 has sos deg > d. then exists 𝐶↓𝑓 ≥1 such that psd-‐rank(𝑀↓𝑛↑𝑓 )≥𝐶↓𝑓 ⋅𝑛↑𝑑/5 for all 𝑛∈ℕ points 𝑥∈{0,1}↑𝑛 subsets 𝑆⊆[𝑛] with |𝑆|=𝑚
𝑀↓𝑛↑𝑓 (𝑆,𝑥)=𝑓(𝑥↓𝑆 )
evaluaJons of 𝑓 applied to subcubes
𝑥↓𝑆 =(𝑥↓𝑠↓1 ,…,𝑥↓ 𝑠↓𝑚 ) for 𝑆={𝑠↓1 ,…, 𝑠↓𝑚 }
sanity check: 𝑀↓𝑛↑𝑓 does not have low-‐deg. factorizaJons (Tr 𝑃↓𝑆 𝑄↓𝑥 )↓𝑆,𝑥 suppose 𝑄↓𝑥 =𝑅(𝑥)↑2 and 𝑥↦𝑅(𝑥) has degree at most 𝑑/2 then, Tr 𝑃↓𝑆 𝑄↓𝑥 =‖𝑃↓𝑆↑1∕2 𝑅(𝑥)‖↓𝐹↑2 à sum of squares of degree-‐𝑑/2 polynomials for each 𝑆 à cannot be factorizaJon of 𝑀↓𝑛↑𝑓 (because 𝑓 has sos deg > 𝑑)
Separa=ng from low-‐degree factoriza=ons By duality, sos-‐deg(𝑓)>𝑑 if and only if ∃ deg.-‐𝑑 pseudo-‐distr. 𝐷 with 𝔼↓𝑥 𝐷(𝑥)𝑓(𝑥)𝑑/2 in 𝑆 ) =ℙ↓S (|𝑇∩𝑆|>𝑑/2)≤(|𝑆|⋅|𝑇|/𝑛 )↑𝑑/2 ≈ 𝑛↑−𝑑/2
small enough to counter effect of 𝑃↓𝑆
Thank You
