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

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

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

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

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

The coloring ideal Graph G = ( V , E ) Variable x i for each vertex i ∈ V Coloring ideal I k ( G ) generated by: ν i ( x ) := x k Vertex polynomials i − 1 , ∀ i ∈ V x k i − x k j Edge polynomials η ij ( x ) := ∀ ij ∈ E , x i − x j David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 4 / 22

The coloring ideal Graph G = ( V , E ) Variable x i for each vertex i ∈ V Coloring ideal I k ( G ) generated by: ν i ( x ) := x k Vertex polynomials i − 1 , ∀ i ∈ V x k i − x k j Edge polynomials η ij ( x ) := ∀ ij ∈ E , x i − x j Solutions x ⇔ proper k -colorings [Bayer 1982] David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 4 / 22

The coloring ideal Graph G = ( V , E ) Variable x i for each vertex i ∈ V Coloring ideal I k ( G ) generated by: ν i ( x ) := x k Vertex polynomials i − 1 , ∀ i ∈ V x k i − x k j Edge polynomials η ij ( x ) := ∀ ij ∈ E , x i − x j 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

Gröbner bases Polynomial ideal I Leading terms of f ∈ I form leading term ideal L ( I ) Gröbner basis: g 1 , g 2 , . . . , g m ∈ I , leading terms generate L ( I ) Implies that { g i } form basis for I David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 5 / 22

Gröbner bases Polynomial ideal I Leading terms of f ∈ I form leading term ideal L ( I ) Gröbner basis: g 1 , g 2 , . . . , g m ∈ I , leading terms generate L ( I ) Implies that { g i } 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

Gröbner bases Polynomial ideal I Leading terms of f ∈ I form leading term ideal L ( I ) Gröbner basis: g 1 , g 2 , . . . , g m ∈ I , leading terms generate L ( I ) Implies that { g i } 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

Gröbner basis of coloring ideal 3 3 3 3 3 1 1 1 1 1 4 4 4 4 4 2 2 2 2 2 David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 6 / 22

Gröbner basis of coloring ideal 3 3 3 3 3 1 1 1 1 1 4 4 4 4 4 2 2 2 2 2 Coloring ideal for k = 3: � x 3 1 − 1 , x 3 2 − 1 , x 3 3 − 1 , x 3 4 − 1 , x 2 1 + x 1 x 2 + x 2 2 , x 2 1 + x 1 x 3 + x 2 3 , x 2 1 + x 1 x 4 + x 2 4 , x 2 2 + x 2 x 3 + x 2 3 , x 2 3 + x 3 x 4 + x 2 4 � David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 6 / 22

Gröbner basis of coloring ideal 3 3 3 3 3 1 1 1 1 1 4 4 4 4 4 2 2 2 2 2 Coloring ideal for k = 3: � x 3 1 − 1 , x 3 2 − 1 , x 3 3 − 1 , x 3 4 − 1 , x 2 1 + x 1 x 2 + x 2 2 , x 2 1 + x 1 x 3 + x 2 3 , x 2 1 + x 1 x 4 + x 2 4 , x 2 2 + x 2 x 3 + x 2 3 , x 2 3 + x 3 x 4 + x 2 4 � Gröbner basis: David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 6 / 22

Gröbner basis of coloring ideal 3 3 3 3 3 1 1 1 1 1 4 4 4 4 4 2 2 2 2 2 Coloring ideal for k = 3: � x 3 1 − 1 , x 3 2 − 1 , x 3 3 − 1 , x 3 4 − 1 , x 2 1 + x 1 x 2 + x 2 2 , x 2 1 + x 1 x 3 + x 2 3 , x 2 1 + x 1 x 4 + x 2 4 , x 2 2 + x 2 x 3 + x 2 3 , x 2 3 + x 3 x 4 + x 2 4 � Gröbner basis: 2 + x 3 x 4 + x 4 3 − 1 } 2 , x 4 { x 1 + x 3 + x 4 , x 2 − x 4 , x 3 David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 6 / 22

Gröbner basis of coloring ideal 3 3 3 3 3 1 1 1 1 1 4 4 4 4 4 2 2 2 2 2 Coloring ideal for k = 3: � x 3 1 − 1 , x 3 2 − 1 , x 3 3 − 1 , x 3 4 − 1 , x 2 1 + x 1 x 2 + x 2 2 , x 2 1 + x 1 x 3 + x 2 3 , x 2 1 + x 1 x 4 + x 2 4 , x 2 2 + x 2 x 3 + x 2 3 , x 2 3 + x 3 x 4 + x 2 4 � Gröbner basis: 2 + x 3 x 4 + x 4 3 − 1 } 2 , x 4 { x 1 + x 3 + x 4 , x 2 − x 4 , x 3 David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 6 / 22

Gröbner basis of coloring ideal 3 3 3 3 3 1 1 1 1 1 4 4 4 4 4 2 2 2 2 2 Coloring ideal for k = 3: � x 3 1 − 1 , x 3 2 − 1 , x 3 3 − 1 , x 3 4 − 1 , x 2 1 + x 1 x 2 + x 2 2 , x 2 1 + x 1 x 3 + x 2 3 , x 2 1 + x 1 x 4 + x 2 4 , x 2 2 + x 2 x 3 + x 2 3 , x 2 3 + x 3 x 4 + x 2 4 � Gröbner basis: 2 + x 3 x 4 + x 4 3 − 1 } 2 , x 4 { x 1 + x 3 + x 4 , x 2 − x 4 , x 3 David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 6 / 22

Gröbner basis of coloring ideal 3 3 3 3 3 1 1 1 1 1 4 4 4 4 4 2 2 2 2 2 Coloring ideal for k = 3: � x 3 1 − 1 , x 3 2 − 1 , x 3 3 − 1 , x 3 4 − 1 , x 2 1 + x 1 x 2 + x 2 2 , x 2 1 + x 1 x 3 + x 2 3 , x 2 1 + x 1 x 4 + x 2 4 , x 2 2 + x 2 x 3 + x 2 3 , x 2 3 + x 3 x 4 + x 2 4 � Gröbner basis: 2 + x 3 x 4 + x 4 3 − 1 } 2 , x 4 { x 1 + x 3 + x 4 , x 2 − x 4 , x 3 David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 6 / 22

Gröbner basis of coloring ideal 3 3 3 3 3 1 1 1 1 1 4 4 4 4 4 2 2 2 2 2 Coloring ideal for k = 3: � x 3 1 − 1 , x 3 2 − 1 , x 3 3 − 1 , x 3 4 − 1 , x 2 1 + x 1 x 2 + x 2 2 , x 2 1 + x 1 x 3 + x 2 3 , x 2 1 + x 1 x 4 + x 2 4 , x 2 2 + x 2 x 3 + x 2 3 , x 2 3 + x 3 x 4 + x 2 4 � Gröbner basis: 2 + x 3 x 4 + x 4 3 − 1 } 2 , x 4 { x 1 + x 3 + x 4 , x 2 − x 4 , x 3 David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 6 / 22

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 r � 1 2 d n − r + d 2 Ritscher (2009): example attaining maximum degree d n In practice, special cases often tractable David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 7 / 22

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

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

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

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

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 3 6 5 1 7 8 4 2 David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22

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

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

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 3 1 2 David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22

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 3 1 4 2 David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22

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 3 5 1 4 2 David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22