Chordal structure and polynomial systems Diego Cifuentes Laboratory for Information and Decision Systems Electrical Engineering and Computer Science Massachusetts Institute of Technology Joint work with Pablo A. Parrilo (MIT) UC Davis - November 2014 Cifuentes, Parrilo (MIT) Chordal structure and polynomial systems UC Davis - 2014 1 / 23 What it is solving? Decide if it is consistent. Find a solution. Describe all solutions. Find a Gr¨obner basis. Polynomial ideals Consider a system of m polynomial equations in n variables: fi (x0,..., xn−1) = 0, i = 1,..., m The objective is to “solve” these equations. Cifuentes, Parrilo (MIT) Chordal structure and polynomial systems UC Davis - 2014 2 / 23 Polynomial ideals Consider a system of m polynomial equations in n variables: fi (x0,..., xn−1) = 0, i = 1,..., m The objective is to “solve” these equations. What it is solving? Decide if it is consistent. Find a solution. Describe all solutions. Find a Gr¨obner basis. Cifuentes, Parrilo (MIT) Chordal structure and polynomial systems UC Davis - 2014 2 / 23 2 2 I = {(x0 x1x2+2x1+1) g1+(x1 +x2) g2+(x1+x2)g3+(x2x3)g4 : g1, g2, g3, g3} 2 I = {(x0 − 3) g1 + (x1 − 1) g2 + (x2 + 1)g3 + (x3)g4 : g1, g2, g3, g3} Gr¨obner basis: 2 I = hx0 − 3, x1 − 1, x2 + 1, x3i There are two solutions: √ √ V(I) = ( 3, 1, −1, 0), (− 3, 1, −1, 0) Polynomial ideals Example: 2 2 I = hx0 x1x2 + 2x1 + 1, x1 + x2, x1 + x2, x2x3i Cifuentes, Parrilo (MIT) Chordal structure and polynomial systems UC Davis - 2014 3 / 23 2 I = {(x0 − 3) g1 + (x1 − 1) g2 + (x2 + 1)g3 + (x3)g4 : g1, g2, g3, g3} Gr¨obner basis: 2 I = hx0 − 3, x1 − 1, x2 + 1, x3i There are two solutions: √ √ V(I) = ( 3, 1, −1, 0), (− 3, 1, −1, 0) Polynomial ideals Example: 2 2 I = hx0 x1x2 + 2x1 + 1, x1 + x2, x1 + x2, x2x3i 2 2 I = {(x0 x1x2+2x1+1) g1+(x1 +x2) g2+(x1+x2)g3+(x2x3)g4 : g1, g2, g3, g3} Cifuentes, Parrilo (MIT) Chordal structure and polynomial systems UC Davis - 2014 3 / 23 Gr¨obner basis: 2 I = hx0 − 3, x1 − 1, x2 + 1, x3i There are two solutions: √ √ V(I) = ( 3, 1, −1, 0), (− 3, 1, −1, 0) Polynomial ideals Example: 2 2 I = hx0 x1x2 + 2x1 + 1, x1 + x2, x1 + x2, x2x3i 2 2 I = {(x0 x1x2+2x1+1) g1+(x1 +x2) g2+(x1+x2)g3+(x2x3)g4 : g1, g2, g3, g3} 2 I = {(x0 − 3) g1 + (x1 − 1) g2 + (x2 + 1)g3 + (x3)g4 : g1, g2, g3, g3} Cifuentes, Parrilo (MIT) Chordal structure and polynomial systems UC Davis - 2014 3 / 23 Polynomial ideals Example: 2 2 I = hx0 x1x2 + 2x1 + 1, x1 + x2, x1 + x2, x2x3i 2 2 I = {(x0 x1x2+2x1+1) g1+(x1 +x2) g2+(x1+x2)g3+(x2x3)g4 : g1, g2, g3, g3} 2 I = {(x0 − 3) g1 + (x1 − 1) g2 + (x2 + 1)g3 + (x3)g4 : g1, g2, g3, g3} Gr¨obner basis: 2 I = hx0 − 3, x1 − 1, x2 + 1, x3i There are two solutions: √ √ V(I) = ( 3, 1, −1, 0), (− 3, 1, −1, 0) Cifuentes, Parrilo (MIT) Chordal structure and polynomial systems UC Davis - 2014 3 / 23 Gr¨obner basis: I = hx0 − x8, x1 − x8, x2 − x8, x3 + x8 + x9, x4 + x8 + x9, 2 2 3 x5 − x9, x6 + x8 + x9, x7 − x9, x8 + x8x9 + x9 , x9 − 1i There are six solutions: three choices for x9, two choices for x8. Polynomial ideals Example 2: 0 Let I be given by the equations: 6 7 3 xi − 1 = 0, 0 ≤ i ≤ 9 2 2 9 8 3 xi + xi xj + xj = 0, (i, j) edge 1 4 5 2 Cifuentes, Parrilo (MIT) Chordal structure and polynomial systems UC Davis - 2014 4 / 23 Polynomial ideals Example 2: 0 Let I be given by the equations: 6 7 3 xi − 1 = 0, 0 ≤ i ≤ 9 2 2 9 8 3 xi + xi xj + xj = 0, (i, j) edge 1 4 5 2 Gr¨obner basis: I = hx0 − x8, x1 − x8, x2 − x8, x3 + x8 + x9, x4 + x8 + x9, 2 2 3 x5 − x9, x6 + x8 + x9, x7 − x9, x8 + x8x9 + x9 , x9 − 1i There are six solutions: three choices for x9, two choices for x8. Cifuentes, Parrilo (MIT) Chordal structure and polynomial systems UC Davis - 2014 4 / 23 Finding a solution to a system of quadratic equations is NP-hard. Computing Gr¨obner bases may require (doubly) exponential time. Grobner¨ bases Given an ordering of the variables x0 > x1 > . > xn−1 there is a unique reduced lex Gr¨obner basis. The system is inconsistent iff the reduced Gr¨obner basis is h1i. If the system has finite solutions, we can find them recursively: solve a univariate polynomial in xn−1, for each solution x^n−1, solve a univariate polynomial in xn−2, etc. For an arbitrary ideal I, we can get the elimination ideals eliml (I) = I ∩ K[xl , xl+1,..., xn−1] Cifuentes, Parrilo (MIT) Chordal structure and polynomial systems UC Davis - 2014 5 / 23 Grobner¨ bases Given an ordering of the variables x0 > x1 > . > xn−1 there is a unique reduced lex Gr¨obner basis. The system is inconsistent iff the reduced Gr¨obner basis is h1i. If the system has finite solutions, we can find them recursively: solve a univariate polynomial in xn−1, for each solution x^n−1, solve a univariate polynomial in xn−2, etc. For an arbitrary ideal I, we can get the elimination ideals eliml (I) = I ∩ K[xl , xl+1,..., xn−1] Finding a solution to a system of quadratic equations is NP-hard. Computing Gr¨obner bases may require (doubly) exponential time. Cifuentes, Parrilo (MIT) Chordal structure and polynomial systems UC Davis - 2014 5 / 23 Example: 2 2 I = hx0 x1x2 + 2x1 + 1, x1 + x2, x1 + x2, x2x3i Polynomial systems and graphs A polynomial system defined by m equations in n variables: fi (x0,..., xn−1) = 0, i = 1,..., m Construct a graph G (“primal graph”) with n nodes, as: Nodes are variables {x0,..., xn−1}. For each equation, add a clique connecting the variables appearing in that equation Cifuentes, Parrilo (MIT) Chordal structure and polynomial systems UC Davis - 2014 6 / 23 Polynomial systems and graphs A polynomial system defined by m equations in n variables: fi (x0,..., xn−1) = 0, i = 1,..., m Construct a graph G (“primal graph”) with n nodes, as: Nodes are variables {x0,..., xn−1}. For each equation, add a clique connecting the variables appearing in that equation Example: 2 2 I = hx0 x1x2 + 2x1 + 1, x1 + x2, x1 + x2, x2x3i Cifuentes, Parrilo (MIT) Chordal structure and polynomial systems UC Davis - 2014 6 / 23 Can the graph structure help solve this system? For instance, to compute Groebner bases? Or, perhaps we can do something better? Preserve graph (sparsity) structure? Complexity aspects? Questions “Abstracted” the polynomial system to a graph. Cifuentes, Parrilo (MIT) Chordal structure and polynomial systems UC Davis - 2014 7 / 23 Questions “Abstracted” the polynomial system to a graph. Can the graph structure help solve this system? For instance, to compute Groebner bases? Or, perhaps we can do something better? Preserve graph (sparsity) structure? Complexity aspects? Cifuentes, Parrilo (MIT) Chordal structure and polynomial systems UC Davis - 2014 7 / 23 Remarkably (AFAIK) almost no work in computational algebraic geometry exploits this structure. We hope to change this... ;) Graphical modelling Pervasive idea in many areas, in particular: numerical linear algebra, graphical models, constraint satisfaction, database theory, . Key notions: chordality and treewidth. Many names: Arnborg, Beeri/Fagin/Maier/Yannakakis, Blair/Peyton, Bodlaender, Courcelle, Dechter, Lauritzen/Spiegelhalter, Pearl, Robertson/Seymour, . Cifuentes, Parrilo (MIT) Chordal structure and polynomial systems UC Davis - 2014 8 / 23 We hope to change this... ;) Graphical modelling Pervasive idea in many areas, in particular: numerical linear algebra, graphical models, constraint satisfaction, database theory, . Key notions: chordality and treewidth. Many names: Arnborg, Beeri/Fagin/Maier/Yannakakis, Blair/Peyton, Bodlaender, Courcelle, Dechter, Lauritzen/Spiegelhalter, Pearl, Robertson/Seymour, . Remarkably (AFAIK) almost no work in computational algebraic geometry exploits this structure. Cifuentes, Parrilo (MIT) Chordal structure and polynomial systems UC Davis - 2014 8 / 23 Graphical modelling Pervasive idea in many areas, in particular: numerical linear algebra, graphical models, constraint satisfaction, database theory, . Key notions: chordality and treewidth. Many names: Arnborg, Beeri/Fagin/Maier/Yannakakis, Blair/Peyton, Bodlaender, Courcelle, Dechter, Lauritzen/Spiegelhalter, Pearl, Robertson/Seymour, . Remarkably (AFAIK) almost no work in computational algebraic geometry exploits this structure. We hope to change this... ;) Cifuentes, Parrilo (MIT) Chordal structure and polynomial systems UC Davis - 2014 8 / 23 A chordal completion of G is a chordal graph with the same vertex set as G, and which contains all edges of G. The treewidth of a graph is the clique number (minus one) of its smallest chordal completion. Meta-theorem: NP-complete problems are “easy” on graphs of small treewidth. Chordality and treewidth Let G be a graph with vertices x0,..., xn−1. A vertex ordering x0 > x1 > ··· > xn−1 is a perfect elimination ordering if for each xl , the set Xl := {xl } ∪ {xm : xm is adjacent to xl , xl > xm} is such that the restriction G|Xl is a clique. A graph is chordal if it has a perfect elimination ordering. Cifuentes, Parrilo (MIT) Chordal structure and polynomial systems UC Davis - 2014 9 / 23 Meta-theorem: NP-complete problems are “easy” on graphs of small treewidth. Chordality and treewidth Let G be a graph with vertices x0,..., xn−1. A vertex ordering x0 > x1 > ··· > xn−1 is a perfect elimination ordering if for each xl , the set Xl := {xl } ∪ {xm : xm is adjacent to xl , xl > xm} is such that the restriction G|Xl is a clique.