can linear programs solve np-hard problems? · can we solve np-hard problems by lp? famous np-hard...

Can linear programs solveNP-hard problems?

Ronald de Wolf

Can linear programs solve NP-hard problems? – p. 1/9

Linear programs

Can linear programs solve NP-hard problems? – p. 2/9

Linear programs

A factory produces 2 types of goods

Can linear programs solve NP-hard problems? – p. 2/9

Linear programs

A factory produces 2 types of goods- unit of type 1 gives profit e 1; type 2 gives profit e 4

Can linear programs solve NP-hard problems? – p. 2/9

Linear programs

A factory produces 2 types of goods- unit of type 1 gives profit e 1; type 2 gives profit e 4

It has 4 kinds of resources: 650 of R1, 100 of R2,. . .

Can linear programs solve NP-hard problems? – p. 2/9

Linear programs

A factory produces 2 types of goods- unit of type 1 gives profit e 1; type 2 gives profit e 4

It has 4 kinds of resources: 650 of R1, 100 of R2,. . .- type 1 uses 34 units of R1, type 2 uses 16 units of R1

Can linear programs solve NP-hard problems? – p. 2/9

Linear programs

A factory produces 2 types of goods- unit of type 1 gives profit e 1; type 2 gives profit e 4

It has 4 kinds of resources: 650 of R1, 100 of R2,. . .- type 1 uses 34 units of R1, type 2 uses 16 units of R1

Maximizing profit is linear program

Can linear programs solve NP-hard problems? – p. 2/9

Linear programs

A factory produces 2 types of goods- unit of type 1 gives profit e 1; type 2 gives profit e 4

It has 4 kinds of resources: 650 of R1, 100 of R2,. . .- type 1 uses 34 units of R1, type 2 uses 16 units of R1

Maximizing profit is linear program: max x1 + 4x2

s.t. 34x1 + 16x2 ≤ 650

. . . ≤ 100

. . .

x1, x2 ≥ 0

Can linear programs solve NP-hard problems? – p. 2/9

Linear programs

A factory produces 2 types of goods- unit of type 1 gives profit e 1; type 2 gives profit e 4

It has 4 kinds of resources: 650 of R1, 100 of R2,. . .- type 1 uses 34 units of R1, type 2 uses 16 units of R1

Maximizing profit is linear program: max x1 + 4x2

s.t. 34x1 + 16x2 ≤ 650

. . . ≤ 100

. . .

x1, x2 ≥ 0Feasible region is a polytope

Can linear programs solve NP-hard problems? – p. 2/9

A bit of history of LPs

Can linear programs solve NP-hard problems? – p. 3/9

A bit of history of LPs

Simplex algorithm: developed by Kantorovich andDantzig for use in WWII, published by Dantzig in 1947.

Can linear programs solve NP-hard problems? – p. 3/9

A bit of history of LPs

Simplex algorithm: developed by Kantorovich andDantzig for use in WWII, published by Dantzig in 1947.

Walks along vertices of polytope, optimizing objective

Can linear programs solve NP-hard problems? – p. 3/9

A bit of history of LPs

Simplex algorithm: developed by Kantorovich andDantzig for use in WWII, published by Dantzig in 1947.

Walks along vertices of polytope, optimizing objective

Usually efficient in practice, worst-case exponential time

Can linear programs solve NP-hard problems? – p. 3/9

A bit of history of LPs

Simplex algorithm: developed by Kantorovich andDantzig for use in WWII, published by Dantzig in 1947.

Walks along vertices of polytope, optimizing objective

Usually efficient in practice, worst-case exponential time

Ellipsoid method (Khachiyan’79): takes polynomial timein the worst case, but is not practical

Can linear programs solve NP-hard problems? – p. 3/9

A bit of history of LPs

Simplex algorithm: developed by Kantorovich andDantzig for use in WWII, published by Dantzig in 1947.

Walks along vertices of polytope, optimizing objective

Usually efficient in practice, worst-case exponential time

Ellipsoid method (Khachiyan’79): takes polynomial timein the worst case, but is not practical

Interior point method (Karmarkar’84): reasonablyefficient in practice

Can linear programs solve NP-hard problems? – p. 3/9

Can we solve NP-hard problems by LP?

Can linear programs solve NP-hard problems? – p. 4/9

Can we solve NP-hard problems by LP?

Famous NP-hard problem: traveling salesman problem

Find shortest cycle that goes through each of n citiesexactly once

Can linear programs solve NP-hard problems? – p. 4/9

Can we solve NP-hard problems by LP?

Famous NP-hard problem: traveling salesman problem

Find shortest cycle that goes through each of n citiesexactly once

A polynomial-size LP for TSP would show P = NP

Can linear programs solve NP-hard problems? – p. 4/9

Can we solve NP-hard problems by LP?

Famous NP-hard problem: traveling salesman problem

Find shortest cycle that goes through each of n citiesexactly once

A polynomial-size LP for TSP would show P = NP

Swart’86–87 claimed to have found such LPs

Can linear programs solve NP-hard problems? – p. 4/9

Can we solve NP-hard problems by LP?

Famous NP-hard problem: traveling salesman problem

Find shortest cycle that goes through each of n citiesexactly once

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

Can linear programs solve NP-hard problems? – p. 4/9

Can we solve NP-hard problems by LP?

Famous NP-hard problem: traveling salesman problem

Find shortest cycle that goes through each of n citiesexactly once

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

Can linear programs solve NP-hard problems? – p. 4/9

Can we solve NP-hard problems by LP?

Famous NP-hard problem: traveling salesman problem

Find shortest cycle that goes through each of n citiesexactly once

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?

Can linear programs solve NP-hard problems? – p. 4/9

Can we solve NP-hard problems by LP?

Famous NP-hard problem: traveling salesman problem

Find shortest cycle that goes through each of n citiesexactly once

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)

Can linear programs solve NP-hard problems? – p. 4/9

Can we solve NP-hard problems by LP?

Famous NP-hard problem: traveling salesman problem

Find shortest cycle that goes through each of n citiesexactly once

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)

Fiorini, Massar, Pokutta, Tiwary, dW (STOC’12):any LP for TSP needs exponential size

Can linear programs solve NP-hard problems? – p. 4/9

Polytopes and optimization problems

Can linear programs solve NP-hard problems? – p. 5/9

Polytopes and optimization problems

Polytope P : convex hull of

finite set of points in Rd

Can linear programs solve NP-hard problems? – p. 5/9

Polytopes and optimization problems

Polytope P : convex hull of

finite set of points in Rd

⇔ bounded intersection of finitely many halfspaces

Can linear programs solve NP-hard problems? – p. 5/9

Polytopes and optimization problems

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

Can linear programs solve NP-hard problems? – p. 5/9

Polytopes and optimization problems

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

Can linear programs solve NP-hard problems? – p. 5/9

Polytopes and optimization problems

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

Can linear programs solve NP-hard problems? – p. 5/9

Polytopes and optimization problems

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: each cycle in Kn is vector ∈ 0, 1(n

2)

Can linear programs solve NP-hard problems? – p. 5/9

Polytopes and optimization problems

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: each cycle in Kn is vector ∈ 0, 1(n

2).

TSP(n) is the convex hull of all those vectors

Can linear programs solve NP-hard problems? – p. 5/9

Polytopes and optimization problems

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: each cycle in Kn is vector ∈ 0, 1(n

2).

TSP(n) is the convex hull of all those vectors

Solving TSP w.r.t. weight function wij:

minimize the linear function∑

i,j wijxij over x ∈ TSP(n)

Can linear programs solve NP-hard problems? – p. 5/9

Polytopes and optimization problems

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: each cycle in Kn is vector ∈ 0, 1(n

2).

TSP(n) is the convex hull of all those vectors

Solving TSP w.r.t. weight function wij:

minimize the linear function∑

i,j wijxij over x ∈ TSP(n)

TSP(n) has exponential size

Can linear programs solve NP-hard problems? – p. 5/9

Extended formulations of polytopes

Can linear programs solve NP-hard problems? – p. 6/9

Extended formulations of polytopes

Sometimes extra variables/dimensionscan reduce size very much.

Can linear programs solve NP-hard problems? – p. 6/9

Extended formulations of polytopes

Sometimes extra variables/dimensionscan reduce size very much.

Regular n-gon in R2 has size n,

but is the projection of polytope Q

in higher dimension, of size O(logn)

Can linear programs solve NP-hard problems? – p. 6/9

Extended formulations of polytopes

Sometimes extra variables/dimensionscan reduce size very much.

Regular n-gon in R2 has size n,

but is the projection of polytope Q

in higher dimension, of size O(logn)

Optimizing over P reduces to optimizing over Q.If Q has small size, this can be done efficiently!

Can linear programs solve NP-hard problems? – p. 6/9

Extended formulations of polytopes

Sometimes extra variables/dimensionscan reduce size very much.

Regular n-gon in R2 has size n,

but is the projection of polytope Q

in higher dimension, of size O(logn)

Optimizing over P reduces to optimizing over Q.If Q has small size, this can be done efficiently!

How small can size(Q) be?

Can linear programs solve NP-hard problems? – p. 6/9

Extended formulations of polytopes

Sometimes extra variables/dimensionscan reduce size very much.

Regular n-gon in R2 has size n,

but is the projection of polytope Q

in higher dimension, of size O(logn)

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 ) = minsize(Q) | Q projects to P

Can linear programs solve NP-hard problems? – p. 6/9

Extended formulations of polytopes

Sometimes extra variables/dimensionscan reduce size very much.

Regular n-gon in R2 has size n,

but is the projection of polytope Q

in higher dimension, of size O(logn)

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 ) = minsize(Q) | Q projects to P

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

Can linear programs solve NP-hard problems? – p. 6/9

How to bound extension complexity?

Can linear programs solve NP-hard problems? – p. 7/9

How to bound extension complexity?

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

Sij = bi − Aivj

Can linear programs solve NP-hard problems? – p. 7/9

How to bound extension complexity?

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

Can linear programs solve NP-hard problems? – p. 7/9

How to bound extension complexity?

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

Yannakakis’88: xc(P ) = positive rank of S

Can linear programs solve NP-hard problems? – p. 7/9

How to bound extension complexity?

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

Yannakakis’88: xc(P ) = positive rank of S

Instead of considering all Q, can focus on slack matrix!

Can linear programs solve NP-hard problems? – p. 7/9

How to bound extension complexity?

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

Yannakakis’88: xc(P ) = positive rank of S

Instead of considering all Q, can focus on slack matrix!

We can use nondeterministic communicationcomplexity to lower bound rank+(S)

Can linear programs solve NP-hard problems? – p. 7/9

How to bound extension complexity?

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

Yannakakis’88: xc(P ) = positive rank of S

Instead of considering all Q, can focus on slack matrix!

We can use nondeterministic communicationcomplexity to lower bound rank+(S)

Big problem until now: which polytope to analyze,and how to analyze its slack matrix?

Can linear programs solve NP-hard problems? – p. 7/9

Lower bound for correlation polytope

Can linear programs solve NP-hard problems? – p. 8/9

Lower bound for correlation polytope

Correlation polytope: COR(n) = convbbT | b ∈ 0, 1n

Can linear programs solve NP-hard problems? – p. 8/9

Lower bound for correlation polytope

Correlation polytope: COR(n) = convbbT | b ∈ 0, 1n

Occurs naturally in NP-hard problems, e.g. MaxClique

Can linear programs solve NP-hard problems? – p. 8/9

Lower bound for correlation polytope

Correlation polytope: COR(n) = convbbT | b ∈ 0, 1n

Occurs naturally in NP-hard problems, e.g. MaxClique

We can find 2n valid constraints (indexed by a ∈ 0, 1n)

whose slacks w.r.t. vertex bbT are Mab = (1− aT b)2

Can linear programs solve NP-hard problems? – p. 8/9

Lower bound for correlation polytope

Correlation polytope: COR(n) = convbbT | b ∈ 0, 1n

Occurs naturally in NP-hard problems, e.g. MaxClique

We can find 2n valid constraints (indexed by a ∈ 0, 1n)

whose slacks w.r.t. vertex bbT are Mab = (1− aT b)2

Nondeterministic communication complexity of M wasalready analyzed in the quantum computing! (dW’00)

Can linear programs solve NP-hard problems? – p. 8/9

Lower bound for correlation polytope

Correlation polytope: COR(n) = convbbT | b ∈ 0, 1n

Occurs naturally in NP-hard problems, e.g. MaxClique

We can find 2n valid constraints (indexed by a ∈ 0, 1n)

whose slacks w.r.t. vertex bbT are Mab = (1− aT b)2

Nondeterministic communication complexity of M wasalready analyzed in the quantum computing! (dW’00)

Take slack matrix S for COR(n),

with 2n vertices bbT for columns,2n a-constraints for first 2n rows,remaining facets for other rows

S =

...

· · · Mab · · ·

...

...

Can linear programs solve NP-hard problems? – p. 8/9

Lower bound for correlation polytope

Correlation polytope: COR(n) = convbbT | b ∈ 0, 1n

Occurs naturally in NP-hard problems, e.g. MaxClique

We can find 2n valid constraints (indexed by a ∈ 0, 1n)

whose slacks w.r.t. vertex bbT are Mab = (1− aT b)2

Nondeterministic communication complexity of M wasalready analyzed in the quantum computing! (dW’00)

Take slack matrix S for COR(n),

with 2n vertices bbT for columns,2n a-constraints for first 2n rows,remaining facets for other rows

S =

...

· · · Mab · · ·

...

...

xc(COR(n))

Can linear programs solve NP-hard problems? – p. 8/9

Lower bound for correlation polytope

Correlation polytope: COR(n) = convbbT | b ∈ 0, 1n

Occurs naturally in NP-hard problems, e.g. MaxClique

We can find 2n valid constraints (indexed by a ∈ 0, 1n)

whose slacks w.r.t. vertex bbT are Mab = (1− aT b)2

Nondeterministic communication complexity of M wasalready analyzed in the quantum computing! (dW’00)

Take slack matrix S for COR(n),

with 2n vertices bbT for columns,2n a-constraints for first 2n rows,remaining facets for other rows

S =

...

· · · Mab · · ·

...

...

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

Can linear programs solve NP-hard problems? – p. 8/9

Lower bound for correlation polytope

Correlation polytope: COR(n) = convbbT | b ∈ 0, 1n

Occurs naturally in NP-hard problems, e.g. MaxClique

We can find 2n valid constraints (indexed by a ∈ 0, 1n)

whose slacks w.r.t. vertex bbT are Mab = (1− aT b)2

Nondeterministic communication complexity of M wasalready analyzed in the quantum computing! (dW’00)

Take slack matrix S for COR(n),

with 2n vertices bbT for columns,2n a-constraints for first 2n rows,remaining facets for other rows

S =

...

· · · Mab · · ·

...

...

xc(COR(n)) = rank+(S) ≥ rank+(M)

Can linear programs solve NP-hard problems? – p. 8/9

Lower bound for correlation polytope

Correlation polytope: COR(n) = convbbT | b ∈ 0, 1n

Occurs naturally in NP-hard problems, e.g. MaxClique

We can find 2n valid constraints (indexed by a ∈ 0, 1n)

whose slacks w.r.t. vertex bbT are Mab = (1− aT b)2

Nondeterministic communication complexity of M wasalready analyzed in the quantum computing! (dW’00)

Take slack matrix S for COR(n),

with 2n vertices bbT for columns,2n a-constraints for first 2n rows,remaining facets for other rows

S =

...

· · · Mab · · ·

...

...

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

Can linear programs solve NP-hard problems? – p. 8/9

Consequences for other polytopes

Can linear programs solve NP-hard problems? – p. 9/9

Consequences for other polytopes

Via reduction from the correlation polytope:

Can linear programs solve NP-hard problems? – p. 9/9

Consequences for other polytopes

Via reduction from the correlation polytope:

TSP-polytope has extension complexity ≥ 2√n

Can linear programs solve NP-hard problems? – p. 9/9

Consequences for other polytopes

Via reduction from the correlation polytope:

TSP-polytope has extension complexity ≥ 2√n

Recently improved to ≥ 2n by Rothvoss

Can linear programs solve NP-hard problems? – p. 9/9

Consequences for other polytopes

Via reduction from the correlation polytope:

TSP-polytope has extension complexity ≥ 2√n

Recently improved to ≥ 2n by Rothvoss

CUT-polytope has extension complexity ≥ 2n

Can linear programs solve NP-hard problems? – p. 9/9

Consequences for other polytopes

Via reduction from the correlation polytope:

TSP-polytope has extension complexity ≥ 2√n

Recently improved to ≥ 2n by Rothvoss

CUT-polytope has extension complexity ≥ 2n

For specific graphs, the stable-set polytope hasextension complexity ≥ 2n

Can linear programs solve NP-hard problems? – p. 9/9

Consequences for other polytopes

Via reduction from the correlation polytope:

TSP-polytope has extension complexity ≥ 2√n

Recently improved to ≥ 2n by Rothvoss

CUT-polytope has extension complexity ≥ 2n

For specific graphs, the stable-set polytope hasextension complexity ≥ 2n

So every linear program based on extendedformulations needs exponentially many constraints

Can linear programs solve NP-hard problems? – p. 9/9

Consequences for other polytopes

Via reduction from the correlation polytope:

TSP-polytope has extension complexity ≥ 2√n

Recently improved to ≥ 2n by Rothvoss

CUT-polytope has extension complexity ≥ 2n

For specific graphs, the stable-set polytope hasextension complexity ≥ 2n

So every linear program based on extendedformulations needs exponentially many constraints

This rules out many efficient algorithms for NP-hardproblems, and refutes all P=NP “proofs” à la Swart

Can linear programs solve NP-hard problems? – p. 9/9