Chordal structure and polynomial systemsdiegcif/files/2014-ChordalStructure.pdf · 2020. 10. 8. ·...

Chordal structure and polynomial systems

Diego Cifuentes

Laboratory for Information and Decision SystemsElectrical 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

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 Grobner 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.

What it is solving?Decide if it is consistent.Find a solution.Describe all solutions.Find a Grobner basis.


Polynomial idealsExample:

I = 〈x20 x1x2 + 2x1 + 1, x2

1 + x2, x1 + x2, x2x3〉

I = {(x20 x1x2+2x1+1) g1+(x2

1 +x2) g2+(x1+x2)g3+(x2x3)g4 : g1, g2, g3, g3}

I = {(x20 − 3) g1 + (x1 − 1) g2 + (x2 + 1)g3 + (x3)g4 : g1, g2, g3, g3}

Grobner basis:I = 〈x2

0 − 3, x1 − 1, x2 + 1, x3〉

There are two solutions:

V(I) = (√

3, 1,−1, 0), (−√

3, 1,−1, 0)



I = 〈x20 x1x2 + 2x1 + 1, x2

1 + x2, x1 + x2, x2x3〉

I = {(x20 x1x2+2x1+1) g1+(x2

1 +x2) g2+(x1+x2)g3+(x2x3)g4 : g1, g2, g3, g3}

I = {(x20 − 3) g1 + (x1 − 1) g2 + (x2 + 1)g3 + (x3)g4 : g1, g2, g3, g3}


0 − 3, x1 − 1, x2 + 1, x3〉


V(I) = (√

3, 1,−1, 0), (−√

3, 1,−1, 0)



I = 〈x20 x1x2 + 2x1 + 1, x2

1 + x2, x1 + x2, x2x3〉

I = {(x20 x1x2+2x1+1) g1+(x2

1 +x2) g2+(x1+x2)g3+(x2x3)g4 : g1, g2, g3, g3}

I = {(x20 − 3) g1 + (x1 − 1) g2 + (x2 + 1)g3 + (x3)g4 : g1, g2, g3, g3}


0 − 3, x1 − 1, x2 + 1, x3〉


V(I) = (√

3, 1,−1, 0), (−√

3, 1,−1, 0)



I = 〈x20 x1x2 + 2x1 + 1, x2

1 + x2, x1 + x2, x2x3〉

I = {(x20 x1x2+2x1+1) g1+(x2

1 +x2) g2+(x1+x2)g3+(x2x3)g4 : g1, g2, g3, g3}

I = {(x20 − 3) g1 + (x1 − 1) g2 + (x2 + 1)g3 + (x3)g4 : g1, g2, g3, g3}


0 − 3, x1 − 1, x2 + 1, x3〉


V(I) = (√

3, 1,−1, 0), (−√

3, 1,−1, 0)


Polynomial ideals

Example 2:

Let I be given by the equations:

x3i − 1 = 0, 0 ≤ i ≤ 9

x2i + xixj + x2

j = 0, (i , j) edge

0

1 2

3

4 5

6 7

89

Grobner basis:

I = 〈x0 − x8, x1 − x8, x2 − x8, x3 + x8 + x9, x4 + x8 + x9,

x5 − x9, x6 + x8 + x9, x7 − x9, x28 + x8x9 + x2

9 , x39 − 1〉

There are six solutions: three choices for x9, two choices for x8.


Polynomial ideals

Example 2:


x3i − 1 = 0, 0 ≤ i ≤ 9

x2i + xixj + x2

j = 0, (i , j) edge

0

1 2

3

4 5

6 7

89

Grobner basis:

I = 〈x0 − x8, x1 − x8, x2 − x8, x3 + x8 + x9, x4 + x8 + x9,

x5 − x9, x6 + x8 + x9, x7 − x9, x28 + x8x9 + x2

9 , x39 − 1〉

There are six solutions: three choices for x9, two choices for x8.


Grobner bases

Given an ordering of the variables x0 > x1 > . . . > xn−1 there is aunique reduced lex Grobner basis.The system is inconsistent iff the reduced Grobner basis is 〈1〉.If the system has finite solutions, we can find them recursively: solvea univariate polynomial in xn−1, for each solution xn−1, solve aunivariate 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 Grobner bases may require (doubly) exponential time.


Grobner bases

Given an ordering of the variables x0 > x1 > . . . > xn−1 there is aunique reduced lex Grobner basis.The system is inconsistent iff the reduced Grobner basis is 〈1〉.If the system has finite solutions, we can find them recursively: solvea univariate polynomial in xn−1, for each solution xn−1, solve aunivariate 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 Grobner bases may require (doubly) exponential time.


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 inthat equation

Example:

I = 〈x20 x1x2 + 2x1 + 1, x2

1 + x2, x1 + x2, x2x3〉


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 inthat equation

Example:

I = 〈x20 x1x2 + 2x1 + 1, x2

1 + x2, x1 + x2, x2x3〉


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?


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?


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 geometryexploits this structure.

We hope to change this... ;)


Graphical modelling







Graphical modelling







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 iffor 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.

A chordal completion of G is a chordal graph with the same vertex set asG , and which contains all edges of G .The treewidth of a graph is the clique number (minus one) of its smallestchordal completion.

Meta-theorem: NP-complete problems are “easy” on graphs of smalltreewidth.
















Bad news? (I)

Subset sum problem, with data A = {a1, . . . , an} ⊂ Z.Is there a subset of A that adds up to S?

Letting si be the partial sums, we can write a polynomial system:

0 = s0

0 = (si − si−1)(si − si−1 − ai )

S = sn

The graph associated with these equations is a path

s0 — s1 — s2 —· · ·— sn

But, subset sum is NP-complete...


Bad news? (II)

For linear equations, “good” elimination preserves graph structure(perfect!)

For polynomials, however, Groebner bases can destroy chordality.

Ex: ConsiderI = 〈x0x2 − 1, x1x2 − 1〉,

whose associated graph is the path x0 — x2 — x1 .

Every Groebner basis must contain the polynomial x0 − x1, breaking thesparsity structure.


Bad news? (II)







Bad news? (II)







Our results

A chordal elimination algorithm, to exploit graphical structure.Conditions under which chordal elimination succeeds.Recursive method for computing elimination ideals of maximal cliquesFor a certain class, complexity is linear in number of variables!(exponential in treewidth)Implementation and experimental results


Chordal elimination (sketch)

Given equations, construct graph G , a chordal completion, and a perfectelimination ordering.

Will produce a decreasing sequence of ideals I = I0 ⊇ I1 ⊇ · · · ⊇ In−1.

Given current ideal Il , split the generators

Il = Jl︸︷︷︸∈K[Xl ]

+ Kl+1︸︷︷︸6∈K[Xl ]

and eliminate variable xl

Il+1 = eliml+1(Jl ) + Kl+1

“Ideally” (!), Il should be the l-th elimination ideal eliml (I)...

Notice that by chordality, graph structure is always preserved!


When does chordal elimination succeed?

We need conditions for this to work, i.e., for V(Il ) = V(eliml (I)).

Thm 1: Let I be an ideal and assume that for each l such that Xl is amaximal clique of G , the ideal Jl ⊆ K[Xl ] is zero dimensional. Then,chordal elimination succeeds.

In particular, finite fields Fq, and 0/1 problems.

Def: A polynomial f is simplicial if for each variable xl , the monomial mlof largest degree in xl is unique and has the form ml = xdl .

Thm 2: Let I = 〈f1, . . . , fs〉 be an ideal such that for each 1 ≤ i ≤ s, fi isgeneric simplicial. Then, chordal elimination succeeds.

[Intuition: interaction of (iterated) “closure/extension thm” + chordality]


















Naive chordal elimination can fail

I = 〈x0x2 + 1, x21 + x2, x1 + x2, x2x3〉

Groebner basis:

{x0 − 1, x1 − 1, x2 + 1, x3}

0

2

3

1

Elimination:

x0x2 + 1 }

x21 + x2

x1 + x2x2x3

x21 + x2

x1 + x2

}→ x2

2 + x2

x2x3 x2x3

→ 0

We got I3 = 〈0〉, but really elim3(I) = 〈x3〉.


Elimination ideals of maximal cliques

In general, Groebner bases can be very large, and destroy chordality.

Can we do something nearly as good, preserving graph structure?

Idea: Compute elimination ideals Hl := I ∩K[Xl ] for the maximal cliques.(A chordal graph has at most n maximal cliques)

For some purposes, ∪lgb(Hl ) has same information as gb(I), and ismuch smaller/sparser.Compute the maximal clique ideals K[Xl ] from the output of thechordal elimination algorithm, in a structure-preserving way.

[Intuition: variety has “small” coordinate projections, can compute those,and glue them]


Elimination ideals of maximal cliques

In general, Groebner bases can be very large, and destroy chordality.

Can we do something nearly as good, preserving graph structure?

Idea: Compute elimination ideals Hl := I ∩K[Xl ] for the maximal cliques.(A chordal graph has at most n maximal cliques)

For some purposes, ∪lgb(Hl ) has same information as gb(I), and ismuch smaller/sparser.Compute the maximal clique ideals K[Xl ] from the output of thechordal elimination algorithm, in a structure-preserving way.

[Intuition: variety has “small” coordinate projections, can compute those,and glue them]


Example: graph colorings


x3i − 1 = 0, 0 ≤ i ≤ 8

x9 − 1 = 0x2

i + xixj + x2j = 0, (i , j) blue edge

0

1 2

3

4 5

6 7

89

Graph G (blue) and its chordal completion G (green).There are 7 maximal cliques:

X0 = {x0, x6, x7}, X1 = {x1, x4, x9}, X2 = {x2, x3, x5}, X3 = {x3, x5, x7, x8},X4 = {x4, x5, x8, x9}, X5 = {x5, x7, x8, x9}, X6 = {x6, x7, x8, x9}


Elimination tree of the graph G .The root/sink is 9.

0

1 2

3

4 5

6 7

89

Some of the clique elimination ideals:

H0 = 〈x0 + x6 + 1, x26 + x6 + 1, x7 − 1〉

H5 = 〈x5 − 1, x7 − 1, x28 + x8 + 1, x9 − 1〉

H6 = 〈x6 + x8 + 1, x7 − 1, x28 + x8 + 1, x9 − 1〉

The corresponding varieties are:

H0 : {x0, x6, x7} →{ζ, ζ2, 1

},

{ζ2, ζ, 1

}H5 : {x5, x7, x8, x9} → {1, 1, ζ, 1} ,

{1, 1, ζ2, 1

}H6 : {x6, x7, x8, x9} →

{ζ2, 1, ζ, 1

},{ζ, 1, ζ2, 1

}Cifuentes, Parrilo (MIT) Chordal structure and polynomial systems UC Davis - 2014 18 / 23

Complexity

For “nice” cases, complexity is linear in number of variables n, number ofequations s, and exponential in treewidth κ.

Thm: Let I be such that each (maximal) H j is q-dominated. Thecomplexity of computing Il is O(s + lqακ). We can find all eliminationideals in O(nqακ).

E.g., we recover known results on linear-time colorability for boundedtreewidth:Cor: Let G be a graph and G a chordal completion with largest clique ofsize κ. We can describe all q-colorings of G in O(nqακ).


Implementation and examples

Implemented in Sage, using Singular and PolyBoRi (for F2).

Graph colorings (counting q-colorings)Cryptography (“baby” AES, Cid et al.)Sensor Network localizationDiscretization of polynomial equations


Results: Crypto - AES variant (Cid et al.) - F2[x ]Performance on SR(n, 1, 2, 4) for chordal elimination, and computing(lex/degrevlex) Grobner bases (PolyBoRi).

n Variables Equations Seed ChordElim LexGB DegrevlexGB

6 176 3200 575.516 402.255 256.253

1 609.529 284.216 144.316

2 649.408 258.965 133.367

10 288 5280 941.068 > 1100, aborted 1279.879

1 784.709 > 1400, aborted 1150.332

2 1124.942 > 3600, aborted > 2500, aborted

For small problems standard Grobner bases outperform chordal elimination,particularly using degrevlex order.Nevertheless, chordal elimination scales better, being faster than bothmethods for n = 10.In addition, standard Grobner bases have higher memory requirements,which is reflected in the many experiments that aborted for this reason.


Results: Sensor network localization - Q[x ]

Find positions, given a few known fixed anchors and pairwise distances.Comparison with Singular: DegrevlexGB, LexFGLM

Natural graph structure

‖xi − xj‖2 = d2ij ij ∈ A

‖xi − ak‖2 = e2ij ik ∈ B

Simplicial, therefore exactelimination

Underconstrained regime:chordal is much better

Overconstrained regime:competitive (plot)


Summary

Chordal structure can notably simplify polynomial system solvingUnder assumptions (treewidth + algebraic structure), tractable!Yields practical, competitive, implementable algorithms

If you want to know more:D. Cifuentes, P.A. Parrilo, Exploiting chordal structure in polynomial ideals: a Groebnerbasis approach. arXiv:1411.1745.D. Cifuentes, Exploiting chordal structure in systems of polynomial equations, S.M. thesis,MIT, 2014.

Thanks for your attention!


Summary

Chordal structure can notably simplify polynomial system solvingUnder assumptions (treewidth + algebraic structure), tractable!Yields practical, competitive, implementable algorithms

If you want to know more:D. Cifuentes, P.A. Parrilo, Exploiting chordal structure in polynomial ideals: a Groebnerbasis approach. arXiv:1411.1745.D. Cifuentes, Exploiting chordal structure in systems of polynomial equations, S.M. thesis,MIT, 2014.

Thanks for your attention!


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