Graph-coloring ideals Nullstellensatz certificates, Grbner bases - - PowerPoint PPT Presentation

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Graph-coloring ideals Nullstellensatz certificates, Grbner bases - - PowerPoint PPT Presentation

Feb 04, 2024 34 likes •808 views

Graph-coloring ideals Nullstellensatz certificates, Grbner bases for chordal graphs, and hardness of Grbner bases David Rolnick Massachusetts Institute of Technology drolnick@mit.edu Joint work with Jess De Loera, Susan Margulies,

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Graph-coloring ideals

Nullstellensatz certificates, Gröbner bases for chordal graphs, and hardness of Gröbner bases

David Rolnick Massachusetts Institute of Technology drolnick@mit.edu

Joint work with Jesús De Loera, Susan Margulies, Michael Pernpeintner, Eric Riedl, Gwen Spencer, Despina Stasi, Jon Swenson

International Symposium on Symbolic and Algebraic Computation Bath, July 2015

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 1 / 22

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Overview

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 2 / 22

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Graph-coloring

Graph-coloring problem: Proper coloring: no two neighboring vertices the same color Is there a proper coloring with k colors?

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 3 / 22

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Graph-coloring

Graph-coloring problem: Proper coloring: no two neighboring vertices the same color Is there a proper coloring with k colors? Approach: k-coloring ⇔ system of polynomial equations Solve the system or prove unsolvable

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 3 / 22

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The coloring ideal

Graph G = (V, E) Variable xi for each vertex i ∈ V Coloring ideal Ik(G) generated by: Vertex polynomials νi(x) := xk

i − 1,

∀i ∈ V Edge polynomials ηij(x) := xk

i − xk j

xi − xj , ∀ij ∈ E

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 4 / 22

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The coloring ideal

Graph G = (V, E) Variable xi for each vertex i ∈ V Coloring ideal Ik(G) generated by: Vertex polynomials νi(x) := xk

i − 1,

∀i ∈ V Edge polynomials ηij(x) := xk

i − xk j

xi − xj , ∀ij ∈ E Solutions x ⇔ proper k-colorings [Bayer 1982]

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 4 / 22

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The coloring ideal

Graph G = (V, E) Variable xi for each vertex i ∈ V Coloring ideal Ik(G) generated by: Vertex polynomials νi(x) := xk

i − 1,

∀i ∈ V Edge polynomials ηij(x) := xk

i − xk j

xi − xj , ∀ij ∈ E Solutions x ⇔ proper k-colorings [Bayer 1982] Need tool for finding solutions to a polynomial system

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 4 / 22

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Gröbner bases

Polynomial ideal I Leading terms of f ∈ I form leading term ideal L(I) Gröbner basis: g1, g2, . . . , gm ∈ I, leading terms generate L(I) Implies that {gi} form basis for I

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 5 / 22

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Gröbner bases

Polynomial ideal I Leading terms of f ∈ I form leading term ideal L(I) Gröbner basis: g1, g2, . . . , gm ∈ I, leading terms generate L(I) Implies that {gi} form basis for I Example:

I = x + z, x + y with lexicographic monomial order x > y > z {x + z, y − z} is a Groebner basis for I Check: x, y = L(I)

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 5 / 22

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Gröbner bases

Polynomial ideal I Leading terms of f ∈ I form leading term ideal L(I) Gröbner basis: g1, g2, . . . , gm ∈ I, leading terms generate L(I) Implies that {gi} form basis for I Example:

I = x + z, x + y with lexicographic monomial order x > y > z {x + z, y − z} is a Groebner basis for I Check: x, y = L(I)

Gröbner basis ⇒ solutions to ideal

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 5 / 22

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Gröbner basis of coloring ideal

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 6 / 22

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Gröbner basis of coloring ideal

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

Coloring ideal for k = 3: x3

1 − 1, x3 2 − 1, x3 3 − 1, x3 4 − 1, x2 1 + x1x2 + x2 2, x2 1 + x1x3 + x2 3,

x2

1 + x1x4 + x2 4, x2 2 + x2x3 + x2 3, x2 3 + x3x4 + x2 4

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 6 / 22

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Gröbner basis of coloring ideal

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

Coloring ideal for k = 3: x3

1 − 1, x3 2 − 1, x3 3 − 1, x3 4 − 1, x2 1 + x1x2 + x2 2, x2 1 + x1x3 + x2 3,

x2

1 + x1x4 + x2 4, x2 2 + x2x3 + x2 3, x2 3 + x3x4 + x2 4

Gröbner basis:

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 6 / 22

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Gröbner basis of coloring ideal

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

Coloring ideal for k = 3: x3

1 − 1, x3 2 − 1, x3 3 − 1, x3 4 − 1, x2 1 + x1x2 + x2 2, x2 1 + x1x3 + x2 3,

x2

1 + x1x4 + x2 4, x2 2 + x2x3 + x2 3, x2 3 + x3x4 + x2 4

Gröbner basis: {x1 + x3 + x4, x2 − x4, x3

2 + x3x4 + x4 2, x4 3 − 1}

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 6 / 22

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Gröbner basis of coloring ideal

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

Coloring ideal for k = 3: x3

1 − 1, x3 2 − 1, x3 3 − 1, x3 4 − 1, x2 1 + x1x2 + x2 2, x2 1 + x1x3 + x2 3,

x2

1 + x1x4 + x2 4, x2 2 + x2x3 + x2 3, x2 3 + x3x4 + x2 4

Gröbner basis: {x1 + x3 + x4, x2 − x4, x3

2 + x3x4 + x4 2, x4 3 − 1}

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 6 / 22

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Gröbner basis of coloring ideal

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

Coloring ideal for k = 3: x3

1 − 1, x3 2 − 1, x3 3 − 1, x3 4 − 1, x2 1 + x1x2 + x2 2, x2 1 + x1x3 + x2 3,

x2

1 + x1x4 + x2 4, x2 2 + x2x3 + x2 3, x2 3 + x3x4 + x2 4

Gröbner basis: {x1 + x3 + x4, x2 − x4, x3

2 + x3x4 + x4 2, x4 3 − 1}

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 6 / 22

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SLIDE 17

Gröbner basis of coloring ideal

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

Coloring ideal for k = 3: x3

1 − 1, x3 2 − 1, x3 3 − 1, x3 4 − 1, x2 1 + x1x2 + x2 2, x2 1 + x1x3 + x2 3,

x2

1 + x1x4 + x2 4, x2 2 + x2x3 + x2 3, x2 3 + x3x4 + x2 4

Gröbner basis: {x1 + x3 + x4, x2 − x4, x3

2 + x3x4 + x4 2, x4 3 − 1}

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 6 / 22

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Gröbner basis of coloring ideal

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

Coloring ideal for k = 3: x3

1 − 1, x3 2 − 1, x3 3 − 1, x3 4 − 1, x2 1 + x1x2 + x2 2, x2 1 + x1x3 + x2 3,

x2

1 + x1x4 + x2 4, x2 2 + x2x3 + x2 3, x2 3 + x3x4 + x2 4

Gröbner basis: {x1 + x3 + x4, x2 − x4, x3

2 + x3x4 + x4 2, x4 3 − 1}

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 6 / 22

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Computing Gröbner bases

Buchberger’s algorithm works but is slow Computation of Gröbner bases is EXPSPACE-complete [Kühnle and Mayr 1996] Even hard to write down: maximum degree can be large Mayr, Ritscher (2010): upper bound on maximum degree for r-dimensional ideal, n generators of degree d: 2 1 2dn−r + d 2r Ritscher (2009): example attaining maximum degree dn In practice, special cases often tractable

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 7 / 22

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Chordal graph algorithm

Definition A graph is chordal if it has no induced cycle of length ≥ 4.

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22

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Chordal graph algorithm

Definition A graph is chordal if it has no induced cycle of length ≥ 4.

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22

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Chordal graph algorithm

Definition A graph is chordal if it has no induced cycle of length ≥ 4.

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22

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Chordal graph algorithm

Definition A graph is chordal if it has no induced cycle of length ≥ 4.

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22

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Chordal graph algorithm

Definition A graph is chordal if it has no induced cycle of length ≥ 4. Chordal graphs admit a perfect elimination ordering: When a vertex is added, its neighborhood forms a clique

1 2 3 4 5 6 7 8

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22

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Chordal graph algorithm

Definition A graph is chordal if it has no induced cycle of length ≥ 4. Chordal graphs admit a perfect elimination ordering: When a vertex is added, its neighborhood forms a clique

1

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22

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Chordal graph algorithm

Definition A graph is chordal if it has no induced cycle of length ≥ 4. Chordal graphs admit a perfect elimination ordering: When a vertex is added, its neighborhood forms a clique

1 2

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22

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Chordal graph algorithm

Definition A graph is chordal if it has no induced cycle of length ≥ 4. Chordal graphs admit a perfect elimination ordering: When a vertex is added, its neighborhood forms a clique

1 2 3

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22

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Chordal graph algorithm

Definition A graph is chordal if it has no induced cycle of length ≥ 4. Chordal graphs admit a perfect elimination ordering: When a vertex is added, its neighborhood forms a clique

1 2 3 4

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22

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Chordal graph algorithm

Definition A graph is chordal if it has no induced cycle of length ≥ 4. Chordal graphs admit a perfect elimination ordering: When a vertex is added, its neighborhood forms a clique

1 2 3 4 5

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22

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Chordal graph algorithm

Definition A graph is chordal if it has no induced cycle of length ≥ 4. Chordal graphs admit a perfect elimination ordering: When a vertex is added, its neighborhood forms a clique

1 2 3 4 5 6

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22

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Chordal graph algorithm

Definition A graph is chordal if it has no induced cycle of length ≥ 4. Chordal graphs admit a perfect elimination ordering: When a vertex is added, its neighborhood forms a clique

1 2 3 4 5 6 7

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22

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Chordal graph algorithm

Definition A graph is chordal if it has no induced cycle of length ≥ 4. Chordal graphs admit a perfect elimination ordering: When a vertex is added, its neighborhood forms a clique

1 2 3 4 5 6 7 8

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22

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Chordal graph algorithm

Theorem (DMPRRSSS) Let G be a chordal graph on n vertices. Then there exists a Gröbner basis of size n for Ik(G), and it can be found efficiently.

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 9 / 22

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Chordal graph algorithm

Theorem (DMPRRSSS) Let G be a chordal graph on n vertices. Then there exists a Gröbner basis of size n for Ik(G), and it can be found efficiently.

1 2 3 4 5 6 7 8

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 9 / 22

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Chordal graph algorithm

Theorem (DMPRRSSS) Let G be a chordal graph on n vertices. Then there exists a Gröbner basis of size n for Ik(G), and it can be found efficiently.

1

Gröbner basis (k = 4) : {ν1(x1),

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 9 / 22

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Chordal graph algorithm

Theorem (DMPRRSSS) Let G be a chordal graph on n vertices. Then there exists a Gröbner basis of size n for Ik(G), and it can be found efficiently.

1 2

Gröbner basis (k = 4) : {ν1(x1), S3(x1, x2),

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 9 / 22

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Chordal graph algorithm

Theorem (DMPRRSSS) Let G be a chordal graph on n vertices. Then there exists a Gröbner basis of size n for Ik(G), and it can be found efficiently.

1 2 3

Gröbner basis (k = 4) : {ν1(x1), S3(x1, x2), S2(x1, x2, x3),

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 9 / 22

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Chordal graph algorithm

Theorem (DMPRRSSS) Let G be a chordal graph on n vertices. Then there exists a Gröbner basis of size n for Ik(G), and it can be found efficiently.

1 2 3 4

Gröbner basis (k = 4) : {ν1(x1), S3(x1, x2), S2(x1, x2, x3), S1(x1, x2, x3, x4),

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 9 / 22

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Chordal graph algorithm

Theorem (DMPRRSSS) Let G be a chordal graph on n vertices. Then there exists a Gröbner basis of size n for Ik(G), and it can be found efficiently.

1 2 3 4 5

Gröbner basis (k = 4) : {ν1(x1), S3(x1, x2), S2(x1, x2, x3), S1(x1, x2, x3, x4), S2(x3, x4, x5),

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 9 / 22

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Chordal graph algorithm

Theorem (DMPRRSSS) Let G be a chordal graph on n vertices. Then there exists a Gröbner basis of size n for Ik(G), and it can be found efficiently.

1 2 3 4 5 6

Gröbner basis (k = 4) : {ν1(x1), S3(x1, x2), S2(x1, x2, x3), S1(x1, x2, x3, x4), S2(x3, x4, x5), S3(x5, x6),

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 9 / 22

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Chordal graph algorithm

Theorem (DMPRRSSS) Let G be a chordal graph on n vertices. Then there exists a Gröbner basis of size n for Ik(G), and it can be found efficiently.

1 2 3 4 5 6 7

Gröbner basis (k = 4) : {ν1(x1), S3(x1, x2), S2(x1, x2, x3), S1(x1, x2, x3, x4), S2(x3, x4, x5), S3(x5, x6), S3(x6, x7),

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 9 / 22

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Chordal graph algorithm

Theorem (DMPRRSSS) Let G be a chordal graph on n vertices. Then there exists a Gröbner basis of size n for Ik(G), and it can be found efficiently.

1 2 3 4 5 6 7 8

Gröbner basis (k = 4) : {ν1(x1), S3(x1, x2), S2(x1, x2, x3), S1(x1, x2, x3, x4), S2(x3, x4, x5), S3(x5, x6), S3(x6, x7), S2(x6, x7, x8)}

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 9 / 22

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Chordal graph algorithm

Theorem (DMPRRSSS) Let G be a chordal graph on n vertices. Then there exists a Gröbner basis of size n for Ik(G), and it can be found efficiently. Gröbner basis (k = 4) : {ν1(x1), S3(x1, x2), S2(x1, x2, x3), S1(x1, x2, x3, x4), S2(x3, x4, x5), S3(x5, x6), S3(x6, x7), S2(x6, x7, x8)} Sm(y1, . . . , yt) :=

  • 1≤j1≤···≤jm≤t

yj1 · · · yjm .

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 9 / 22

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Chordal graph algorithm

Theorem (DMPRRSSS) Let G be a chordal graph on n vertices. Then there exists a Gröbner basis of size n for Ik(G), and it can be found efficiently. Complete homogeneous symmetric polynomials: Sm(y1, . . . , yt) :=

  • 1≤j1≤···≤jm≤t

yj1 · · · yjm .

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 10 / 22

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Chordal graph algorithm

Theorem (DMPRRSSS) Let G be a chordal graph on n vertices. Then there exists a Gröbner basis of size n for Ik(G), and it can be found efficiently. Complete homogeneous symmetric polynomials: Sm(y1, . . . , yt) :=

  • 1≤j1≤···≤jm≤t

yj1 · · · yjm . Lemma For a positive integer k, let ζ1, ζ2, . . . , ζk be the kth roots of unity in some order. Then, for every k > r, Sm(ζ1, ζ2, . . . , ζk−m, x) = (x − ζk−m+1)(x − ζk−m+2) · · · (x − ζk).

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 10 / 22

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Chordal graph algorithm

Sm(y1, . . . , yt) :=

  • 1≤j1≤···≤jm≤t

yj1 · · · yjm . Lemma For a positive integer k, let ζ1, ζ2, . . . , ζk be the kth roots of unity in some order. Then, for every k > r, Sm(ζ1, ζ2, . . . , ζk−m, x) = (x − ζk−m+1)(x − ζk−m+2) · · · (x − ζk). Proof of algorithm Perfect elimination order ⇒ polynomial Sm(x) for each vertex

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 10 / 22

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Chordal graph algorithm

Sm(y1, . . . , yt) :=

  • 1≤j1≤···≤jm≤t

yj1 · · · yjm . Lemma For a positive integer k, let ζ1, ζ2, . . . , ζk be the kth roots of unity in some order. Then, for every k > r, Sm(ζ1, ζ2, . . . , ζk−m, x) = (x − ζk−m+1)(x − ζk−m+2) · · · (x − ζk). Proof of algorithm Perfect elimination order ⇒ polynomial Sm(x) for each vertex These polynomials generate graph coloring ideal by induction

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 10 / 22

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Chordal graph algorithm

Sm(y1, . . . , yt) :=

  • 1≤j1≤···≤jm≤t

yj1 · · · yjm . Lemma For a positive integer k, let ζ1, ζ2, . . . , ζk be the kth roots of unity in some order. Then, for every k > r, Sm(ζ1, ζ2, . . . , ζk−m, x) = (x − ζk−m+1)(x − ζk−m+2) · · · (x − ζk). Proof of algorithm Perfect elimination order ⇒ polynomial Sm(x) for each vertex These polynomials generate graph coloring ideal by induction Form Gröbner basis, by considering S-pairs

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 10 / 22

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Hilbert’s Nullstellensatz

Theorem (Hilbert) Given a field K and f1, . . . , fs ∈ K[x1, . . . , xn], the system f1 = f2 = · · · = fs = 0 has no solutions in the algebraic closure of K iff there exist polynomials β1, . . . , βs ∈ K[x1, . . . , xn] such that 1 =

s

  • i=1

βifi. The set {βi} is a Nullstellensatz certificate.

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 11 / 22

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Hilbert’s Nullstellensatz

1 2 3 4

Certificate of infeasibility for 3-coloring ideal: 1 = ν4(x) + (−x1) · η12(x) + (−x2 − x4) · η13(x) + (−x1) · η14(x) + (−x1 − x4) · η23(x) + (−x2) · η24(x) + (−x1 − x2) · η34(x)

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 12 / 22

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Hilbert’s Nullstellensatz

1 2 3 4

Gröbner basis for coloring ideal: {1} Certificate of infeasibility for 3-coloring ideal: 1 = ν4(x) + (−x1) · η12(x) + (−x2 − x4) · η13(x) + (−x1) · η14(x) + (−x1 − x4) · η23(x) + (−x2) · η24(x) + (−x1 − x2) · η34(x)

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 12 / 22

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SLIDE 52

Hilbert’s Nullstellensatz

Theorem (Hilbert) Given a field K and f1, . . . , fs ∈ K[x1, . . . , xn], the system f1 = f2 = · · · = fs = 0 has no solutions in the algebraic closure of K iff there exist polynomials β1, . . . , βs ∈ K[x1, . . . , xn] such that 1 =

s

  • i=1

βifi. The set {βi} is a Nullstellensatz certificate. Degree of the certificate is minimum degree of the βi

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 13 / 22

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SLIDE 53

Hilbert’s Nullstellensatz

Theorem (Hilbert) Given a field K and f1, . . . , fs ∈ K[x1, . . . , xn], the system f1 = f2 = · · · = fs = 0 has no solutions in the algebraic closure of K iff there exist polynomials β1, . . . , βs ∈ K[x1, . . . , xn] such that 1 =

s

  • i=1

βifi. The set {βi} is a Nullstellensatz certificate. Degree of the certificate is minimum degree of the βi If degree small, find certificate by brute force over finite field K

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 13 / 22

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Certificates for Ik(G)

Theorem (DMPRRSSS) Given a non-k-colorable graph G, let d be the minimum degree of a Nullstellensatz certificate. d ≡ 1 mod k d ≥ k + 1 if k > 3.

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 14 / 22

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Certificates for Ik(G)

Theorem (DMPRRSSS) Given a non-k-colorable graph G, let d be the minimum degree of a Nullstellensatz certificate. d ≡ 1 mod k d ≥ k + 1 if k > 3. Brute force is inefficient for every G

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 14 / 22

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Certificates for Ik(G)

Example: degree-4 certificate over F2:

1 = (1 + x0x2x4 + x0x2x6 + x0x3x4 + x0x3x5 + x0x4x5 + x0x4x6 + x2

1 x4 + x2 1 x6

+ x1x3x4 + x1x3x5 + x1x5x6 + x2x3x4 + x2x3x6 + x3x5x6 + x4x5x6)(x3

0 + 1)

+ (x1 + x3 + x4 + x2

0x1x4 + x2 0x1x6 + x2 0 x2x4 + x2 0 x2x6 + x2 0 x3x4 + x2 0 x3x5

+ x2

0x5x6 + x0x1x3x4 + x0x1x3x6 + x0x2x3x4 + x0x2x3x6 + x0x2x4x5 + x0x2x4x6

+ x0x2x5x6 + x0x3x4x5 + x0x3x4x6 + x0x3x5x6 + x0x4x5x6+ + x1x3x4x5 + x1x3x4x6 + x1x4x5x6 + x2x3x4x5 + x2x3x4x6)(x2

0 + x0x1 + x2 1)

+ (x1 + x3 + x4 + x6 + x2

0 x1x4 + x2 0 x1x6 + x2 0 x4x5 + x2 0x4x6 + x2 0x5x6

+ x0x1x3x4 + x0x1x3x6 + x0x3x4x5 + x0x3x4x6 + x1x3x4x5 + x1x3x4x6 + x1x4x5x6 + x3x4x5x6)(x2

0 + x0x2 + x2 2) + · · · David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 15 / 22

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SLIDE 57

Certificates for Ik(G)

Example: degree-4 certificate over F2:

1 = (1 + x0x2x4 + x0x2x6 + x0x3x4 + x0x3x5 + x0x4x5 + x0x4x6 + x2

1 x4 + x2 1 x6

+ x1x3x4 + x1x3x5 + x1x5x6 + x2x3x4 + x2x3x6 + x3x5x6 + x4x5x6)(x3

0 + 1)

+ (x1 + x3 + x4 + x2

0x1x4 + x2 0x1x6 + x2 0 x2x4 + x2 0 x2x6 + x2 0 x3x4 + x2 0 x3x5

+ x2

0x5x6 + x0x1x3x4 + x0x1x3x6 + x0x2x3x4 + x0x2x3x6 + x0x2x4x5 + x0x2x4x6

+ x0x2x5x6 + x0x3x4x5 + x0x3x4x6 + x0x3x5x6 + x0x4x5x6+ + x1x3x4x5 + x1x3x4x6 + x1x4x5x6 + x2x3x4x5 + x2x3x4x6)(x2

0 + x0x1 + x2 1)

+ (x1 + x3 + x4 + x6 + x2

0 x1x4 + x2 0 x1x6 + x2 0 x4x5 + x2 0x4x6 + x2 0x5x6

+ x0x1x3x4 + x0x1x3x6 + x0x3x4x5 + x0x3x4x6 + x1x3x4x5 + x1x3x4x6 + x1x4x5x6 + x3x4x5x6)(x2

0 + x0x2 + x2 2) + · · · David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 15 / 22

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SLIDE 58

Certificates for Ik(G)

Example: degree-4 certificate over F2:

1 = (1 + x0x2x4 + x0x2x6 + x0x3x4 + x0x3x5 + x0x4x5 + x0x4x6 + x2

1 x4 + x2 1 x6

+ x1x3x4 + x1x3x5 + x1x5x6 + x2x3x4 + x2x3x6 + x3x5x6 + x4x5x6)(x3

0 + 1)

+ (x1 + x3 + x4 + x2

0x1x4 + x2 0x1x6 + x2 0 x2x4 + x2 0 x2x6 + x2 0 x3x4 + x2 0 x3x5

+ x2

0x5x6 + x0x1x3x4 + x0x1x3x6 + x0x2x3x4 + x0x2x3x6 + x0x2x4x5 + x0x2x4x6

+ x0x2x5x6 + x0x3x4x5 + x0x3x4x6 + x0x3x5x6 + x0x4x5x6+ + x1x3x4x5 + x1x3x4x6 + x1x4x5x6 + x2x3x4x5 + x2x3x4x6)(x2

0 + x0x1 + x2 1)

+ (x1 + x3 + x4 + x6 + x2

0 x1x4 + x2 0 x1x6 + x2 0 x4x5 + x2 0x4x6 + x2 0x5x6

+ x0x1x3x4 + x0x1x3x6 + x0x3x4x5 + x0x3x4x6 + x1x3x4x5 + x1x3x4x6 + x1x4x5x6 + x3x4x5x6)(x2

0 + x0x2 + x2 2) + · · · David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 15 / 22

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SLIDE 59

Certificates for Ik(G)

Example: degree-4 certificate over F2:

1 = (1 + x0x2x4 + x0x2x6 + x0x3x4 + x0x3x5 + x0x4x5 + x0x4x6 + x2

1 x4 + x2 1 x6

+ x1x3x4 + x1x3x5 + x1x5x6 + x2x3x4 + x2x3x6 + x3x5x6 + x4x5x6)(x3

0 + 1)

+ (x1 + x3 + x4 + x2

0x1x4 + x2 0x1x6 + x2 0 x2x4 + x2 0 x2x6 + x2 0 x3x4 + x2 0 x3x5

+ x2

0x5x6 + x0x1x3x4 + x0x1x3x6 + x0x2x3x4 + x0x2x3x6 + x0x2x4x5 + x0x2x4x6

+ x0x2x5x6 + x0x3x4x5 + x0x3x4x6 + x0x3x5x6 + x0x4x5x6+ + x1x3x4x5 + x1x3x4x6 + x1x4x5x6 + x2x3x4x5 + x2x3x4x6)(x2

0 + x0x1 + x2 1)

+ (x1 + x3 + x4 + x6 + x2

0 x1x4 + x2 0 x1x6 + x2 0 x4x5 + x2 0x4x6 + x2 0x5x6

+ x0x1x3x4 + x0x1x3x6 + x0x3x4x5 + x0x3x4x6 + x1x3x4x5 + x1x3x4x6 + x1x4x5x6 + x3x4x5x6)(x2

0 + x0x2 + x2 2) + · · · David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 15 / 22

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SLIDE 60

Certificates for Ik(G)

Example: degree-4 certificate over F2:

1 = (1 + x0x2x4 + x0x2x6 + x0x3x4 + x0x3x5 + x0x4x5 + x0x4x6 + x2

1 x4 + x2 1 x6

+ x1x3x4 + x1x3x5 + x1x5x6 + x2x3x4 + x2x3x6 + x3x5x6 + x4x5x6)(x3

0 + 1)

+ (x1 + x3 + x4 + x2

0x1x4 + x2 0x1x6 + x2 0 x2x4 + x2 0 x2x6 + x2 0 x3x4 + x2 0 x3x5

+ x2

0x5x6 + x0x1x3x4 + x0x1x3x6 + x0x2x3x4 + x0x2x3x6 + x0x2x4x5 + x0x2x4x6

+ x0x2x5x6 + x0x3x4x5 + x0x3x4x6 + x0x3x5x6 + x0x4x5x6+ + x1x3x4x5 + x1x3x4x6 + x1x4x5x6 + x2x3x4x5 + x2x3x4x6)(x2

0 + x0x1 + x2 1)

+ (x1 + x3 + x4 + x6 + x2

0 x1x4 + x2 0 x1x6 + x2 0 x4x5 + x2 0x4x6 + x2 0x5x6

+ x0x1x3x4 + x0x1x3x6 + x0x3x4x5 + x0x3x4x6 + x1x3x4x5 + x1x3x4x6 + x1x4x5x6 + x3x4x5x6)(x2

0 + x0x2 + x2 2) + · · · David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 15 / 22

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SLIDE 61

Certificates for Ik(G)

Theorem (DMPRRSSS) Given a non-k-colorable graph G, let d be the minimum degree of a certificate. d ≡ 1 mod k d ≥ k + 1 if k > 3.

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 16 / 22

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SLIDE 62

Certificates for Ik(G)

Theorem (DMPRRSSS) Given a non-k-colorable graph G, let d be the minimum degree of a certificate. d ≡ 1 mod k d ≥ k + 1 if k > 3. Graph k Possible degrees F2 F3 F5 F7 K4 3 1, 4, 7, 10, . . . 1 − 4 4 K5 4 5, 9, 13, . . . − 5 5 5 K6 5 6, 11, 16, . . . 6 6 − 11 K7 6 7, 13, 19, . . . − − 13 13 K8 7 8, 15, 22, . . . 8 ≥ 15 ≥ 15 − K9 8 9, 17, 25, . . . − ≥ 17 ≥ 17 ≥ 17 K10 9 10, 19, 28, . . . ≥ 19 − ≥ 19 ≥ 19 K11 10 11, 21, 31, . . . − ≥ 21 − ≥ 21

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 16 / 22

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SLIDE 63

Certificates for Ik(G)

Theorem (DMPRRSSS) Given a non-k-colorable graph G, let d be the minimum degree of a certificate. d ≡ 1 mod k d ≥ k + 1 if k > 3. Graph k Possible degrees F2 F3 F5 F7 K4 3 1, 4, 7, 10, . . . 1 − 4 4 K5 4 5, 9, 13, . . . − 5 5 5 K6 5 6, 11, 16, . . . 6 6 − 11 K7 6 7, 13, 19, . . . − − 13 13 K8 7 8, 15, 22, . . . 8 ≥ 15 ≥ 15 − K9 8 9, 17, 25, . . . − ≥ 17 ≥ 17 ≥ 17 K10 9 10, 19, 28, . . . ≥ 19 − ≥ 19 ≥ 19 K11 10 11, 21, 31, . . . − ≥ 21 − ≥ 21 Computation over Fp possible only if p, k relatively prime

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 16 / 22

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SLIDE 64

Certificates for Ik(G)

Conjecture For every field K, the minimum degree of a k-coloring certificate grows superlinearly in k.

Graph k Possible degrees F2 F3 F5 F7 K4 3 1, 4, 7, 10, . . . 1 − 4 4 K5 4 5, 9, 13, . . . − 5 5 5 K6 5 6, 11, 16, . . . 6 6 − 11 K7 6 7, 13, 19, . . . − − 13 13 K8 7 8, 15, 22, . . . 8 ≥ 15 ≥ 15 − K9 8 9, 17, 25, . . . − ≥ 17 ≥ 17 ≥ 17 K10 9 10, 19, 28, . . . ≥ 19 − ≥ 19 ≥ 19 K11 10 11, 21, 31, . . . − ≥ 21 − ≥ 21

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 17 / 22

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SLIDE 65

Inapproximability results

Definition Given a set of polynomials fi and an integer c. An independent set of variables do not pairwise co-occur in any fi. Gröbner problem: Find a Gröbner basis.

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 18 / 22

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SLIDE 66

Inapproximability results

Definition Given a set of polynomials fi and an integer c. An independent set of variables do not pairwise co-occur in any fi. Gröbner problem: Find a Gröbner basis. Weak c-partial Gröbner problem: Throw away c variables and the polynomials containing them, then find a Gröbner basis.

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 18 / 22

slide-67
SLIDE 67

Inapproximability results

Definition Given a set of polynomials fi and an integer c. An independent set of variables do not pairwise co-occur in any fi. Gröbner problem: Find a Gröbner basis. Weak c-partial Gröbner problem: Throw away c variables and the polynomials containing them, then find a Gröbner basis. Strong c-partial Gröbner problem: Throw away c independent sets

  • f variables and the polynomials containing them, then find a

Gröbner basis.

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 18 / 22

slide-68
SLIDE 68

Inapproximability results

Definition Given a set of polynomials fi and an integer c. An independent set of variables do not pairwise co-occur in any fi. Gröbner problem: Find a Gröbner basis. Weak c-partial Gröbner problem: Throw away c variables and the polynomials containing them, then find a Gröbner basis. Strong c-partial Gröbner problem: Throw away c independent sets

  • f variables and the polynomials containing them, then find a

Gröbner basis. Theorem (DMPRRSSS) The strong c-partial Gröbner problem is NP-hard for every c.

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 18 / 22

slide-69
SLIDE 69

Inapproximability results

Definition Given a set of polynomials fi and an integer c. An independent set of variables do not pairwise co-occur in any fi. Gröbner problem: Find a Gröbner basis. Weak c-partial Gröbner problem: Throw away c variables and the polynomials containing them, then find a Gröbner basis. Strong c-partial Gröbner problem: Throw away c independent sets

  • f variables and the polynomials containing them, then find a

Gröbner basis. Theorem (DMPRRSSS) The strong c-partial Gröbner problem is NP-hard for every c. Simpler proof holds for the weak c-partial Gröbner problem

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 18 / 22

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SLIDE 70

Inapproximability results

Theorem (DMPRRSSS) The strong c-partial Gröbner problem is NP-hard for every c.

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 19 / 22

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SLIDE 71

Inapproximability results

Theorem (DMPRRSSS) The strong c-partial Gröbner problem is NP-hard for every c. Proof idea: Remove c independent sets of vertices Corresponds to independent sets of variables in coloring ideal Gröbner basis ⇒ k-coloring of remaining vertices Gives (k + c)-coloring of graph

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 19 / 22

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SLIDE 72

Inapproximability results

Theorem (DMPRRSSS) The strong c-partial Gröbner problem is NP-hard for every c. Proof idea: Remove c independent sets of vertices Corresponds to independent sets of variables in coloring ideal Gröbner basis ⇒ k-coloring of remaining vertices Gives (k + c)-coloring of graph Theorem (Khanna, Linial, Safra 1993) It is NP-hard to color a k-chromatic graph with at most k + 2 k

3

  • − 1

colors.

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 19 / 22

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SLIDE 73

Technical details

Certain monomial orders are elimination orders Every lexicographic order x1 > · · · > xn is an elimination order For an elimination order, the Gröbner basis allows back-substitution, e.g. x3

1 + x2x3 − x2 3 − 1,

x3

2 − x2 + x2 3 + 1,

x2x2

3 − 2x3 3 + x3,

x3

3 + 1.

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 20 / 22

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SLIDE 74

Technical details

Certain monomial orders are elimination orders Every lexicographic order x1 > · · · > xn is an elimination order For an elimination order, the Gröbner basis allows back-substitution, e.g. x3

1 + x2x3 − x2 3 − 1,

x3

2 − x2 + x2 3 + 1,

x2x2

3 − 2x3 3 + x3,

x3

3 + 1.

Problem: What if one cannot solve for roots of unity, e.g. over a field other than R?

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 20 / 22

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SLIDE 75

Technical details

Certain monomial orders are elimination orders Every lexicographic order x1 > · · · > xn is an elimination order For an elimination order, the Gröbner basis allows back-substitution, e.g. x3

1 + x2x3 − x2 3 − 1,

x3

2 − x2 + x2 3 + 1,

x2x2

3 − 2x3 3 + x3,

x3

3 + 1.

Problem: What if one cannot solve for roots of unity, e.g. over a field other than R? Solution: Do not solve numerically, merely symbolically

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 20 / 22

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SLIDE 76

Summary of results

Polytime algorithm finding Gröbner basis of graph-coloring ideal in chordal graphs Complexity of Nullstellensatz certificate for general graphs Hardness of approximate Gröbner basis computation, ⇐ from hardness of approximate k-coloring

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 21 / 22

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SLIDE 77

Questions?

Acknowledgments This research was conducted through the AMS Mathematical Research Communities program, and was supported by the National Science Foundation under Grant Nos. DMS-1321794 and 1122374. Special thanks to Hannah Alpert, Agnes Szanto, Pablo Parrilo, Ellen Maycock, and the Simons Institute. 1 2 3 4 5 6 7 8

David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 22 / 22