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Exponential Lower Bounds for Polytopes in Combinatorial Optimization

Ronald de Wolf Joint with Samuel Fiorini (ULB), Serge Massar (ULB), Sebastian Pokutta (Erlangen), Hans Raj Tiwary (ULB)

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 1/13

Background: solving NP by LP?

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 2/13

Background: solving NP by LP?

Famous P-problem: linear programming (Khachian’79)

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 2/13

Background: solving NP by LP?

Famous P-problem: linear programming (Khachian’79) Famous NP-hard problem: traveling salesman problem

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 2/13

Background: solving NP by LP?

Famous P-problem: linear programming (Khachian’79) Famous NP-hard problem: traveling salesman problem A polynomial-size LP for TSP would show P = NP

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 2/13

Background: solving NP by LP?

Famous P-problem: linear programming (Khachian’79) Famous NP-hard problem: traveling salesman problem A polynomial-size LP for TSP would show P = NP Swart’86–87 claimed to have found such LPs

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 2/13

Background: solving NP by LP?

Famous P-problem: linear programming (Khachian’79) Famous NP-hard problem: traveling salesman problem A polynomial-size LP for TSP would show P = NP Swart’86–87 claimed to have found such LPs Yannakakis’88: symmetric LPs for TSP are exponential

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 2/13

Background: solving NP by LP?

Famous P-problem: linear programming (Khachian’79) Famous NP-hard problem: traveling salesman problem A polynomial-size LP for TSP would show P = NP Swart’86–87 claimed to have found such LPs Yannakakis’88: symmetric LPs for TSP are exponential Swart’s LPs were symmetric, so they couldn’t work

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 2/13

Background: solving NP by LP?

Famous P-problem: linear programming (Khachian’79) Famous NP-hard problem: traveling salesman problem A polynomial-size LP for TSP would show P = NP Swart’86–87 claimed to have found such LPs Yannakakis’88: symmetric LPs for TSP are exponential Swart’s LPs were symmetric, so they couldn’t work 20-year open problem: what about non-symmetric LP?

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 2/13

Background: solving NP by LP?

Famous P-problem: linear programming (Khachian’79) Famous NP-hard problem: traveling salesman problem A polynomial-size LP for TSP would show P = NP Swart’86–87 claimed to have found such LPs Yannakakis’88: symmetric LPs for TSP are exponential Swart’s LPs were symmetric, so they couldn’t work 20-year open problem: what about non-symmetric LP? Sometimes non-symmetry helps a lot! (Kaibel et al’10)

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 2/13

Background: solving NP by LP?

Famous P-problem: linear programming (Khachian’79) Famous NP-hard problem: traveling salesman problem A polynomial-size LP for TSP would show P = NP Swart’86–87 claimed to have found such LPs Yannakakis’88: symmetric LPs for TSP are exponential Swart’s LPs were symmetric, so they couldn’t work 20-year open problem: what about non-symmetric LP? Sometimes non-symmetry helps a lot! (Kaibel et al’10) Yannakakis, May 2011: “I believe in fact that it should be

possible to prove that there is no polynomial-size formulation for the TSP polytope or any other NP-hard problem, although

• f course showing this remains a challenging task”

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 2/13

Basics of polytopes

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 3/13

Basics of polytopes

Polytope P: convex hull of finite set of points in Rd

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 3/13

Basics of polytopes

Polytope P: convex hull of finite set of points in Rd

⇔ bounded intersection of finitely many halfspaces

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 3/13

Basics of polytopes

Polytope P: convex hull of finite set of points in Rd

⇔ bounded intersection of finitely many halfspaces

Can be written as system of linear inequalities:

P = {x ∈ Rd | Ax ≤ b}

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 3/13

Basics of polytopes

Polytope P: convex hull of finite set of points in Rd

⇔ bounded intersection of finitely many halfspaces

Can be written as system of linear inequalities:

P = {x ∈ Rd | Ax ≤ b}

Different systems “Ax ≤ b” can define the same P

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 3/13

Basics of polytopes

Polytope P: convex hull of finite set of points in Rd

⇔ bounded intersection of finitely many halfspaces

Can be written as system of linear inequalities:

P = {x ∈ Rd | Ax ≤ b}

Different systems “Ax ≤ b” can define the same P The size of P is the minimal number of inequalities

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 3/13

Basics of polytopes

Polytope P: convex hull of finite set of points in Rd

⇔ bounded intersection of finitely many halfspaces

Can be written as system of linear inequalities:

P = {x ∈ Rd | Ax ≤ b}

Different systems “Ax ≤ b” can define the same P The size of P is the minimal number of inequalities TSP polytope: convex hull of Hamiltonian cycles in Kn

PTSP = conv{χF ∈ {0, 1}(

n 2) | F ⊆ En is a tour of Kn}

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 3/13

Basics of polytopes

Polytope P: convex hull of finite set of points in Rd

⇔ bounded intersection of finitely many halfspaces

Can be written as system of linear inequalities:

P = {x ∈ Rd | Ax ≤ b}

Different systems “Ax ≤ b” can define the same P The size of P is the minimal number of inequalities TSP polytope: convex hull of Hamiltonian cycles in Kn

PTSP = conv{χF ∈ {0, 1}(

n 2) | F ⊆ En is a tour of Kn}

Solving TSP w.r.t. weight function wij: minimize the linear function

i,j wijxij over x ∈ PTSP

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 3/13

Basics of polytopes

Polytope P: convex hull of finite set of points in Rd

⇔ bounded intersection of finitely many halfspaces

Can be written as system of linear inequalities:

P = {x ∈ Rd | Ax ≤ b}

Different systems “Ax ≤ b” can define the same P The size of P is the minimal number of inequalities TSP polytope: convex hull of Hamiltonian cycles in Kn

PTSP = conv{χF ∈ {0, 1}(

n 2) | F ⊆ En is a tour of Kn}

Solving TSP w.r.t. weight function wij: minimize the linear function

i,j wijxij over x ∈ PTSP

PTSP has exponential size, so corresponding LP is huge

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 3/13

Extended formulations of polytopes

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 4/13

Extended formulations of polytopes

Sometimes extra variables/dimensions can reduce size very much.

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 4/13

Extended formulations of polytopes

Sometimes extra variables/dimensions can reduce size very much. Regular n-gon in R2 has size n, but is the projection of polytope in higher dimension, of size O(log n)

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 4/13

Extended formulations of polytopes

Sometimes extra variables/dimensions can reduce size very much. Regular n-gon in R2 has size n, but is the projection of polytope in higher dimension, of size O(log n) Extended formulation of P: polytope Q ⊆ Rd+k s.t. P = {x | ∃ y s.t. (x, y) ∈ Q}

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 4/13

Extended formulations of polytopes

Sometimes extra variables/dimensions can reduce size very much. Regular n-gon in R2 has size n, but is the projection of polytope in higher dimension, of size O(log n) Extended formulation of P: polytope Q ⊆ Rd+k s.t. P = {x | ∃ y s.t. (x, y) ∈ Q} Optimizing over P reduces to optimizing over Q. If Q has small size, this can be done efficiently!

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 4/13

Extended formulations of polytopes

Sometimes extra variables/dimensions can reduce size very much. Regular n-gon in R2 has size n, but is the projection of polytope in higher dimension, of size O(log n) Extended formulation of P: polytope Q ⊆ Rd+k s.t. P = {x | ∃ y s.t. (x, y) ∈ Q} Optimizing over P reduces to optimizing over Q. If Q has small size, this can be done efficiently! How small can size(Q) be? Extension complexity:

xc(P) = min{size(Q) | Q is an EF of P}

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 4/13

Extended formulations of polytopes

Sometimes extra variables/dimensions can reduce size very much. Regular n-gon in R2 has size n, but is the projection of polytope in higher dimension, of size O(log n) Extended formulation of P: polytope Q ⊆ Rd+k s.t. P = {x | ∃ y s.t. (x, y) ∈ Q} Optimizing over P reduces to optimizing over Q. If Q has small size, this can be done efficiently! How small can size(Q) be? Extension complexity:

xc(P) = min{size(Q) | Q is an EF of P}

Our goal: strong lower bounds on xc(P) for interesting P

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 4/13

The TSP polytope: main result

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 5/13

The TSP polytope: main result

PTSP = conv{χF ∈ {0, 1}(n

2) | F ⊆ En is a tour of Kn}

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 5/13

The TSP polytope: main result

PTSP = conv{χF ∈ {0, 1}(n

2) | F ⊆ En is a tour of Kn}

Our main result: xc(PTSP) ≥ 2Ω(√n)

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 5/13

The TSP polytope: main result

PTSP = conv{χF ∈ {0, 1}(n

2) | F ⊆ En is a tour of Kn}

Our main result: xc(PTSP) ≥ 2Ω(√n) Hence every LP for TSP based on extended formulation

• f TSP-polytope needs exponential time

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 5/13

The TSP polytope: main result

PTSP = conv{χF ∈ {0, 1}(n

2) | F ⊆ En is a tour of Kn}

Our main result: xc(PTSP) ≥ 2Ω(√n) Hence every LP for TSP based on extended formulation

• f TSP-polytope needs exponential time

This rules out a lot of potential algorithms

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 5/13

The TSP polytope: main result

PTSP = conv{χF ∈ {0, 1}(n

2) | F ⊆ En is a tour of Kn}

Our main result: xc(PTSP) ≥ 2Ω(√n) Hence every LP for TSP based on extended formulation

• f TSP-polytope needs exponential time

This rules out a lot of potential algorithms Roadmap for the proof:

2n lower bound on xc of correlation polytope

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 5/13

The TSP polytope: main result

PTSP = conv{χF ∈ {0, 1}(n

2) | F ⊆ En is a tour of Kn}

Our main result: xc(PTSP) ≥ 2Ω(√n) Hence every LP for TSP based on extended formulation

• f TSP-polytope needs exponential time

This rules out a lot of potential algorithms Roadmap for the proof:

2n lower bound on xc of correlation polytope

[inspired by quantum communication complexity!]

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 5/13

The TSP polytope: main result

PTSP = conv{χF ∈ {0, 1}(n

2) | F ⊆ En is a tour of Kn}

Our main result: xc(PTSP) ≥ 2Ω(√n) Hence every LP for TSP based on extended formulation

• f TSP-polytope needs exponential time

This rules out a lot of potential algorithms Roadmap for the proof:

2n lower bound on xc of correlation polytope

[inspired by quantum communication complexity!]

⇓ gadget-based reduction 2

√n lower bound for TSP-polytope

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 5/13

How to lower bound extension compl?

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 6/13

How to lower bound extension compl?

Slack matrix S of a polytope P = conv(V ) with inequalities {Aix ≤ bi} and points V = {vj}:

Sij = bi − Aivj

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 6/13

How to lower bound extension compl?

Slack matrix S of a polytope P = conv(V ) with inequalities {Aix ≤ bi} and points V = {vj}:

Sij = bi − Aivj

NB: every entry is nonnegative; S is not unique

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 6/13

How to lower bound extension compl?

Slack matrix S of a polytope P = conv(V ) with inequalities {Aix ≤ bi} and points V = {vj}:

Sij = bi − Aivj

NB: every entry is nonnegative; S is not unique Positive factorization S =

r

• j=1

ajbT

j , vectors aj, bj ≥ 0

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 6/13

How to lower bound extension compl?

Slack matrix S of a polytope P = conv(V ) with inequalities {Aix ≤ bi} and points V = {vj}:

Sij = bi − Aivj

NB: every entry is nonnegative; S is not unique Positive factorization S =

r

• j=1

ajbT

j , vectors aj, bj ≥ 0

Nonnegative rank: rank+(S) = min such r

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 6/13

How to lower bound extension compl?

Slack matrix S of a polytope P = conv(V ) with inequalities {Aix ≤ bi} and points V = {vj}:

Sij = bi − Aivj

NB: every entry is nonnegative; S is not unique Positive factorization S =

r

• j=1

ajbT

j , vectors aj, bj ≥ 0

Nonnegative rank: rank+(S) = min such r Yannakakis’88: xc(P) = rank+(S)

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 6/13

How to lower bound extension compl?

Slack matrix S of a polytope P = conv(V ) with inequalities {Aix ≤ bi} and points V = {vj}:

Sij = bi − Aivj

NB: every entry is nonnegative; S is not unique Positive factorization S =

r

• j=1

ajbT

j , vectors aj, bj ≥ 0

Nonnegative rank: rank+(S) = min such r Yannakakis’88: xc(P) = rank+(S) Yannakakis’88: rank+(S) ≥ minimal # rectangles needed to cover all (and only) non-zero entries of S (rectangle = set of row-indices × set of column-indices)

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 6/13

Connection w communication compl

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 7/13

Connection w communication compl

de Wolf’00: exponential separation of quantum and classical nondeterminisic communication complexity, based on 2n × 2n matrix M, indexed by a, b ∈ {0, 1}n

Mab = (1 − aT b)2

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 7/13

Connection w communication compl

de Wolf’00: exponential separation of quantum and classical nondeterminisic communication complexity, based on 2n × 2n matrix M, indexed by a, b ∈ {0, 1}n

Mab = (1 − aT b)2

Claim: 2Ω(n) rectangles needed to cover support of M

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 7/13

Connection w communication compl

de Wolf’00: exponential separation of quantum and classical nondeterminisic communication complexity, based on 2n × 2n matrix M, indexed by a, b ∈ {0, 1}n

Mab = (1 − aT b)2

Claim: 2Ω(n) rectangles needed to cover support of M

Razborov: ∃ a measure µ with weight 1/2 on each of

A = {(a, b) : aT b = 0}, B = {(a, b) : aT b = 1}; constants α, δ, s.t. ∀ rectangles R: µ(B ∩ R) ≥ αµ(A ∩ R) − 2−δn.

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 7/13

Connection w communication compl

de Wolf’00: exponential separation of quantum and classical nondeterminisic communication complexity, based on 2n × 2n matrix M, indexed by a, b ∈ {0, 1}n

Mab = (1 − aT b)2

Claim: 2Ω(n) rectangles needed to cover support of M

Razborov: ∃ a measure µ with weight 1/2 on each of

A = {(a, b) : aT b = 0}, B = {(a, b) : aT b = 1}; constants α, δ, s.t. ∀ rectangles R: µ(B ∩ R) ≥ αµ(A ∩ R) − 2−δn.

Every monochrome R has B ∩ R = ∅, µ(A ∩ R) ≤ 2−Ω(n).

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 7/13

Connection w communication compl

de Wolf’00: exponential separation of quantum and classical nondeterminisic communication complexity, based on 2n × 2n matrix M, indexed by a, b ∈ {0, 1}n

Mab = (1 − aT b)2

Claim: 2Ω(n) rectangles needed to cover support of M

Razborov: ∃ a measure µ with weight 1/2 on each of

A = {(a, b) : aT b = 0}, B = {(a, b) : aT b = 1}; constants α, δ, s.t. ∀ rectangles R: µ(B ∩ R) ≥ αµ(A ∩ R) − 2−δn.

Every monochrome R has B ∩ R = ∅, µ(A ∩ R) ≤ 2−Ω(n). Hence 2Ω(n) rectangles are needed to cover A

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 7/13

Connection w communication compl

de Wolf’00: exponential separation of quantum and classical nondeterminisic communication complexity, based on 2n × 2n matrix M, indexed by a, b ∈ {0, 1}n

Mab = (1 − aT b)2

Claim: 2Ω(n) rectangles needed to cover support of M

Razborov: ∃ a measure µ with weight 1/2 on each of

A = {(a, b) : aT b = 0}, B = {(a, b) : aT b = 1}; constants α, δ, s.t. ∀ rectangles R: µ(B ∩ R) ≥ αµ(A ∩ R) − 2−δn.

Every monochrome R has B ∩ R = ∅, µ(A ∩ R) ≤ 2−Ω(n). Hence 2Ω(n) rectangles are needed to cover A By previous slide, rank+(M) ≥ 2Ω(n)

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 7/13

Lower bound for correlation polytope

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 8/13

Lower bound for correlation polytope

Correlation polytope: COR(n) = conv{bbT | b ∈ {0, 1}n}

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 8/13

Lower bound for correlation polytope

Correlation polytope: COR(n) = conv{bbT | b ∈ {0, 1}n} The following constraints hold (one for each a ∈ {0, 1}n):

∀x ∈ COR(n) : Tr

• (2diag(a) − aaT )x
• ≤ 1

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 8/13

Lower bound for correlation polytope

Correlation polytope: COR(n) = conv{bbT | b ∈ {0, 1}n} The following constraints hold (one for each a ∈ {0, 1}n):

∀x ∈ COR(n) : Tr

• (2diag(a) − aaT )x
• ≤ 1

Slack of this a-constraint w.r.t. vertex bbT:

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 8/13

Lower bound for correlation polytope

Correlation polytope: COR(n) = conv{bbT | b ∈ {0, 1}n} The following constraints hold (one for each a ∈ {0, 1}n):

∀x ∈ COR(n) : Tr

• (2diag(a) − aaT )x
• ≤ 1

Slack of this a-constraint w.r.t. vertex bbT:

Sab = 1 − Tr

• (2diag(a) − aaT )bbT

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 8/13

Lower bound for correlation polytope

Correlation polytope: COR(n) = conv{bbT | b ∈ {0, 1}n} The following constraints hold (one for each a ∈ {0, 1}n):

∀x ∈ COR(n) : Tr

• (2diag(a) − aaT )x
• ≤ 1

Slack of this a-constraint w.r.t. vertex bbT:

Sab = 1 − Tr

• (2diag(a) − aaT )bbT

= (1 − aT b)2

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 8/13

Lower bound for correlation polytope

Correlation polytope: COR(n) = conv{bbT | b ∈ {0, 1}n} The following constraints hold (one for each a ∈ {0, 1}n):

∀x ∈ COR(n) : Tr

• (2diag(a) − aaT )x
• ≤ 1

Slack of this a-constraint w.r.t. vertex bbT:

Sab = 1 − Tr

• (2diag(a) − aaT )bbT

= (1 − aT b)2 = Mab

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 8/13

Lower bound for correlation polytope

Correlation polytope: COR(n) = conv{bbT | b ∈ {0, 1}n} The following constraints hold (one for each a ∈ {0, 1}n):

∀x ∈ COR(n) : Tr

• (2diag(a) − aaT )x
• ≤ 1

Slack of this a-constraint w.r.t. vertex bbT:

Sab = 1 − Tr

• (2diag(a) − aaT )bbT

= (1 − aT b)2 = Mab

Take slack matrix S for COR, with 2n vertices bbT for columns,

2n a-constraints for first 2n rows,

remaining facets for other rows

S = 2 6 6 6 6 6 6 6 4 . . . · · · Mab · · · . . . . . . 3 7 7 7 7 7 7 7 5

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 8/13

Lower bound for correlation polytope

Correlation polytope: COR(n) = conv{bbT | b ∈ {0, 1}n} The following constraints hold (one for each a ∈ {0, 1}n):

∀x ∈ COR(n) : Tr

• (2diag(a) − aaT )x
• ≤ 1

Slack of this a-constraint w.r.t. vertex bbT:

Sab = 1 − Tr

• (2diag(a) − aaT )bbT

= (1 − aT b)2 = Mab

Take slack matrix S for COR, with 2n vertices bbT for columns,

2n a-constraints for first 2n rows,

remaining facets for other rows

S = 2 6 6 6 6 6 6 6 4 . . . · · · Mab · · · . . . . . . 3 7 7 7 7 7 7 7 5

xc(COR(n))

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 8/13

Lower bound for correlation polytope

Correlation polytope: COR(n) = conv{bbT | b ∈ {0, 1}n} The following constraints hold (one for each a ∈ {0, 1}n):

∀x ∈ COR(n) : Tr

• (2diag(a) − aaT )x
• ≤ 1

Slack of this a-constraint w.r.t. vertex bbT:

Sab = 1 − Tr

• (2diag(a) − aaT )bbT

= (1 − aT b)2 = Mab

Take slack matrix S for COR, with 2n vertices bbT for columns,

2n a-constraints for first 2n rows,

remaining facets for other rows

S = 2 6 6 6 6 6 6 6 4 . . . · · · Mab · · · . . . . . . 3 7 7 7 7 7 7 7 5

xc(COR(n)) = rank+(S)

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 8/13

Lower bound for correlation polytope

Correlation polytope: COR(n) = conv{bbT | b ∈ {0, 1}n} The following constraints hold (one for each a ∈ {0, 1}n):

∀x ∈ COR(n) : Tr

• (2diag(a) − aaT )x
• ≤ 1

Slack of this a-constraint w.r.t. vertex bbT:

Sab = 1 − Tr

• (2diag(a) − aaT )bbT

= (1 − aT b)2 = Mab

Take slack matrix S for COR, with 2n vertices bbT for columns,

2n a-constraints for first 2n rows,

remaining facets for other rows

S = 2 6 6 6 6 6 6 6 4 . . . · · · Mab · · · . . . . . . 3 7 7 7 7 7 7 7 5

xc(COR(n)) = rank+(S) ≥ rank+(M)

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 8/13

Lower bound for correlation polytope

Correlation polytope: COR(n) = conv{bbT | b ∈ {0, 1}n} The following constraints hold (one for each a ∈ {0, 1}n):

∀x ∈ COR(n) : Tr

• (2diag(a) − aaT )x
• ≤ 1

Slack of this a-constraint w.r.t. vertex bbT:

Sab = 1 − Tr

• (2diag(a) − aaT )bbT

= (1 − aT b)2 = Mab

Take slack matrix S for COR, with 2n vertices bbT for columns,

2n a-constraints for first 2n rows,

remaining facets for other rows

S = 2 6 6 6 6 6 6 6 4 . . . · · · Mab · · · . . . . . . 3 7 7 7 7 7 7 7 5

xc(COR(n)) = rank+(S) ≥ rank+(M) ≥ 2Ω(n)

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 8/13

Consequences for the TSP polytope

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 9/13

Consequences for the TSP polytope

We can “embed” COR(k) into PTSP with n = O(k2).

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 9/13

Consequences for the TSP polytope

We can “embed” COR(k) into PTSP with n = O(k2). Since COR(k) has extension complexity exp(k),

PTSP has extension complexity ≥ exp(k) = exp(√n)

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 9/13

Consequences for the TSP polytope

We can “embed” COR(k) into PTSP with n = O(k2). Since COR(k) has extension complexity exp(k),

PTSP has extension complexity ≥ exp(k) = exp(√n)

• 1. Define 3-SAT formula φ with k2 variables

s.t. vertices of COR(k) ⇔ satisfying assignments of φ

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 9/13

Consequences for the TSP polytope

We can “embed” COR(k) into PTSP with n = O(k2). Since COR(k) has extension complexity exp(k),

PTSP has extension complexity ≥ exp(k) = exp(√n)

• 1. Define 3-SAT formula φ with k2 variables

s.t. vertices of COR(k) ⇔ satisfying assignments of φ

• 2. Construct directed graph Dn with n = O(k2) vertices

s.t. satisfying assignments ⇔ directed tours in Dn

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 9/13

Consequences for the TSP polytope

We can “embed” COR(k) into PTSP with n = O(k2). Since COR(k) has extension complexity exp(k),

PTSP has extension complexity ≥ exp(k) = exp(√n)

• 1. Define 3-SAT formula φ with k2 variables

s.t. vertices of COR(k) ⇔ satisfying assignments of φ

• 2. Construct directed graph Dn with n = O(k2) vertices

s.t. satisfying assignments ⇔ directed tours in Dn

• 3. Convert Dk to undirected graph Gn

s.t. directed tours in Dn ⇔ undirected tours in Gn

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 9/13

Consequences for the TSP polytope

We can “embed” COR(k) into PTSP with n = O(k2). Since COR(k) has extension complexity exp(k),

PTSP has extension complexity ≥ exp(k) = exp(√n)

• 1. Define 3-SAT formula φ with k2 variables

s.t. vertices of COR(k) ⇔ satisfying assignments of φ

• 2. Construct directed graph Dn with n = O(k2) vertices

s.t. satisfying assignments ⇔ directed tours in Dn

• 3. Convert Dk to undirected graph Gn

s.t. directed tours in Dn ⇔ undirected tours in Gn Consider the face of PTSP obtained by extra contraints

xij = 0 for all (i, j) ∈ Gn

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 9/13

Consequences for the TSP polytope

We can “embed” COR(k) into PTSP with n = O(k2). Since COR(k) has extension complexity exp(k),

PTSP has extension complexity ≥ exp(k) = exp(√n)

• 1. Define 3-SAT formula φ with k2 variables

s.t. vertices of COR(k) ⇔ satisfying assignments of φ

• 2. Construct directed graph Dn with n = O(k2) vertices

s.t. satisfying assignments ⇔ directed tours in Dn

• 3. Convert Dk to undirected graph Gn

s.t. directed tours in Dn ⇔ undirected tours in Gn Consider the face of PTSP obtained by extra contraints

xij = 0 for all (i, j) ∈ Gn. This is an EF of COR(k)

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 9/13

Consequences for the TSP polytope

We can “embed” COR(k) into PTSP with n = O(k2). Since COR(k) has extension complexity exp(k),

PTSP has extension complexity ≥ exp(k) = exp(√n)

• 1. Define 3-SAT formula φ with k2 variables

s.t. vertices of COR(k) ⇔ satisfying assignments of φ

• 2. Construct directed graph Dn with n = O(k2) vertices

s.t. satisfying assignments ⇔ directed tours in Dn

• 3. Convert Dk to undirected graph Gn

s.t. directed tours in Dn ⇔ undirected tours in Gn Consider the face of PTSP obtained by extra contraints

xij = 0 for all (i, j) ∈ Gn. This is an EF of COR(k)

This refutes all P=NP “proofs” à la Swart

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 9/13

Cartoon by Pavel Pudlak

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 10/13

Summary

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 11/13

Summary

We studied the extension complexity of polytopes

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 11/13

Summary

We studied the extension complexity of polytopes Showed exponential lower bounds on the extension complexities of the correlation, cut, stable set, and TSP polytopes, even for non-symmetric extensions. This solves a 20-year old problem of Yannakakis

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 11/13

Summary

We studied the extension complexity of polytopes Showed exponential lower bounds on the extension complexities of the correlation, cut, stable set, and TSP polytopes, even for non-symmetric extensions. This solves a 20-year old problem of Yannakakis Further research: Lower bound for the matching polytope? (Yannakakis: exponential LB for symmetric)

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 11/13

Summary

We studied the extension complexity of polytopes Showed exponential lower bounds on the extension complexities of the correlation, cut, stable set, and TSP polytopes, even for non-symmetric extensions. This solves a 20-year old problem of Yannakakis Further research: Lower bound for the matching polytope? (Yannakakis: exponential LB for symmetric) Lower bounds on positive semidefinite extensions? [Not shown here: this is closely connected to

• ne-way quantum communication complexity]

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 11/13

Summary

We studied the extension complexity of polytopes Showed exponential lower bounds on the extension complexities of the correlation, cut, stable set, and TSP polytopes, even for non-symmetric extensions. This solves a 20-year old problem of Yannakakis Further research: Lower bound for the matching polytope? (Yannakakis: exponential LB for symmetric) Lower bounds on positive semidefinite extensions? [Not shown here: this is closely connected to

• ne-way quantum communication complexity]

Lower bounds for approximation? [BFPS’12,BM’12]

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 11/13

Extra slide 1: PSD extensions?

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 12/13

Extra slide 1: PSD extensions?

Small semidefinite programs for TSP?

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 12/13

Extra slide 1: PSD extensions?

Small semidefinite programs for TSP? Nonnegative factorization of a matrix S: nonnegative vectors ax, by ∈ Rr s.t. ax · by = Sxy

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 12/13

Extra slide 1: PSD extensions?

Small semidefinite programs for TSP? Nonnegative factorization of a matrix S: nonnegative vectors ax, by ∈ Rr s.t. ax · by = Sxy extension complexity of polytope P

= (Yannakakis’87)

nonnegative rank of its slack matrix S

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 12/13

Extra slide 1: PSD extensions?

Small semidefinite programs for TSP? Nonnegative factorization of a matrix S: nonnegative vectors ax, by ∈ Rr s.t. ax · by = Sxy extension complexity of polytope P

= (Yannakakis’87)

nonnegative rank of its slack matrix S

= (Faenza et al.’11)

exp(communication complexity of S “in expectation”)

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 12/13

Extra slide 1: PSD extensions?

Small semidefinite programs for TSP? Nonnegative factorization of a matrix S: nonnegative vectors ax, by ∈ Rr s.t. ax · by = Sxy extension complexity of polytope P

= (Yannakakis’87)

nonnegative rank of its slack matrix S

= (Faenza et al.’11)

exp(communication complexity of S “in expectation”) Positive semidefinite factorization of S: PSD matrices Ax, By ∈ Rr×r s.t. Tr(Ax · By) = Sxy

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 12/13

Extra slide 1: PSD extensions?

Small semidefinite programs for TSP? Nonnegative factorization of a matrix S: nonnegative vectors ax, by ∈ Rr s.t. ax · by = Sxy extension complexity of polytope P

= (Yannakakis’87)

nonnegative rank of its slack matrix S

= (Faenza et al.’11)

exp(communication complexity of S “in expectation”) Positive semidefinite factorization of S: PSD matrices Ax, By ∈ Rr×r s.t. Tr(Ax · By) = Sxy PSD extension complexity of polytope P

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 12/13

Extra slide 1: PSD extensions?

Small semidefinite programs for TSP? Nonnegative factorization of a matrix S: nonnegative vectors ax, by ∈ Rr s.t. ax · by = Sxy extension complexity of polytope P

= (Yannakakis’87)

nonnegative rank of its slack matrix S

= (Faenza et al.’11)

exp(communication complexity of S “in expectation”) Positive semidefinite factorization of S: PSD matrices Ax, By ∈ Rr×r s.t. Tr(Ax · By) = Sxy PSD extension complexity of polytope P

= PSD rank of its slack matrix S

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 12/13

Extra slide 1: PSD extensions?

Small semidefinite programs for TSP? Nonnegative factorization of a matrix S: nonnegative vectors ax, by ∈ Rr s.t. ax · by = Sxy extension complexity of polytope P

= (Yannakakis’87)

nonnegative rank of its slack matrix S

= (Faenza et al.’11)

exp(communication complexity of S “in expectation”) Positive semidefinite factorization of S: PSD matrices Ax, By ∈ Rr×r s.t. Tr(Ax · By) = Sxy PSD extension complexity of polytope P

= PSD rank of its slack matrix S = exp(quantum comm comp of S “in expectation”)

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 12/13

Extra slide 2: rank+ vs PSD-rank

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 13/13

Extra slide 2: rank+ vs PSD-rank

Recall our 2n × 2n matrix: Mxy = (1 − xTy)2

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 13/13

Extra slide 2: rank+ vs PSD-rank

Recall our 2n × 2n matrix: Mxy = (1 − xTy)2 We showed that rank+(M) ≥ 2Ω(n)

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 13/13

Extra slide 2: rank+ vs PSD-rank

Recall our 2n × 2n matrix: Mxy = (1 − xTy)2 We showed that rank+(M) ≥ 2Ω(n) But M has PSD-rank ≤ n + 1:

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 13/13

Extra slide 2: rank+ vs PSD-rank

Recall our 2n × 2n matrix: Mxy = (1 − xTy)2 We showed that rank+(M) ≥ 2Ω(n) But M has PSD-rank ≤ n + 1: set Ax to outer product of vector (x, 1) ∈ Rn+1 with itself, and By to outer product of (y, −1) with itself

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 13/13

Extra slide 2: rank+ vs PSD-rank

Recall our 2n × 2n matrix: Mxy = (1 − xTy)2 We showed that rank+(M) ≥ 2Ω(n) But M has PSD-rank ≤ n + 1: set Ax to outer product of vector (x, 1) ∈ Rn+1 with itself, and By to outer product of (y, −1) with itself, then Tr(Ax · By) = Mxy

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 13/13

Extra slide 2: rank+ vs PSD-rank

Recall our 2n × 2n matrix: Mxy = (1 − xTy)2 We showed that rank+(M) ≥ 2Ω(n) But M has PSD-rank ≤ n + 1: set Ax to outer product of vector (x, 1) ∈ Rn+1 with itself, and By to outer product of (y, −1) with itself, then Tr(Ax · By) = Mxy Exponential separation between rank+ and PSD rank

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 13/13

Extra slide 2: rank+ vs PSD-rank

Recall our 2n × 2n matrix: Mxy = (1 − xTy)2 We showed that rank+(M) ≥ 2Ω(n) But M has PSD-rank ≤ n + 1: set Ax to outer product of vector (x, 1) ∈ Rn+1 with itself, and By to outer product of (y, −1) with itself, then Tr(Ax · By) = Mxy Exponential separation between rank+ and PSD rank Also: log-rank conjecture is equivalent to efficient classical simulation of quantum communication in expectation, for all Boolean matrices

Exponential Lower Bounds for Polytopes in Combinatorial Optimization – p. 13/13