An algorithm for computing resultant polytopes

An output-sensitive algorithm for computing(projections of) resultant polytopes

Vissarion FisikopoulosJoint work with I.Z. Emiris, C. Konaxis and L. Penaranda

Department of Informatics, University of Athens

SoCG, Chapel Hill, NC, USA, 18.Jun.2012

An interesting class of polytopes: resultant polytopes

I Geometry: Minkowski summands of secondary polytopes,equivalence classes of secondary vertices, generalization of Birkhoffpolytopes

I Motivation: useful to express the solvability of polynomial systemsI Applications: discriminant and resultant computation, implicitization

of parametric hypersurfaces

Enneper’s Minimal Surface

Existing work

I Theory of resultants, secondary polytopes, Cayley trick [GKZ ’94]

I TOPCOM [Rambau ’02] computes all vertices of secondary polytope.

I [Michiels & Verschelde DCG’99] define and enumerate coarseequivalence classes of secondary polytope vertices.

I [Michiels & Cools DCG’00] describe a decomposition of Σ(A) inMinkoski summands, including N(R).

I Tropical geometry [Sturmfels-Yu ’08] leads to algorithms for theresultant polytope (GFan library) [Jensen-Yu ’11] and thediscriminant polytope (TropLi software) [Rincón ’12].

What is a resultant polytope?

I Given n + 1 point sets A0,A1, . . . ,An ⊂ Zn

I A =∪n

i=0(Ai × {ei}) ⊂ Z2n where ei = (0, . . . , 1, . . . , 0) ⊂ Zn

I A =∪n

I Given T a triangulation of conv(A), a cell is a-mixed if it is theMinkowski sum of n 1-dimensional segments from Aj, j 6= i, andsome vertex a ∈ Ai.

a3, 1 a4, 1

a1, 0 a2, 0

I A =∪n

I ρT(a) =∑

a−mixedσ∈T:a∈σ

vol(σ) ∈ N, a ∈ A

ρT = (0, 2, 1, 0)

a3, 1 a4, 1

a1, 0 a2, 0

I A =∪n

I ρT(a) =∑

a−mixedσ∈T:a∈σ

vol(σ) ∈ N, a ∈ A

I Resultant polytope N(R) = conv(ρT : T triang. of conv(A))

A N(R)

Connection with Algebra

I The Newton polytope of f, N(f), is the convex hull of the set ofexponents of its monomials with non-zero coefficient.

I The resultant R is the polynomial in the coefficients of a system ofpolynomials which is zero iff the system has a common solution.

N(R) R(a, b, c, d, e) = ad2b+ c2b2 − 2caeb+ a2e2

f0(x) = ax2 + b

f1(x) = cx2 + dx+ e

f0(x, y) = ax+ by + c

f1(x, y) = dx+ ey + f

f2(x, y) = gx+ hy + iA2

a b cd e fg h i

4-dimensional Birkhoff polytope

R(a, b, c, d, e, f, g, h, i) =

f0(x, y) = axy2 + x4y + c

f1(x, y) = dx+ ey

f2(x, y) = gx2 + hy + iA2

NP-hard to compute the resultantin the general case

The idea of the algorithmInput: A ∈ Z2n defined by A0,A1, . . . ,An ⊂ Zn

Simplistic method:I compute the secondary polytope Σ(A)

I many-to-one relation between vertices of Σ(A) and N(R) verticesCannot enumerate 1 representative per class by walking on secondaryedges

The idea of the algorithmInput: A ∈ Z2n defined by A0,A1, . . . ,An ⊂ Zn

New Algorithm:I Vertex oracle: given a direction vector compute a vertex of N(R)I Output sensitive: computes only one triangulation of A per N(R)

vertex + one per N(R) facetI Computes projections of N(R) or Σ(A)

The Oracle

Input: A ⊂ Z2n, direction w ∈ (R|A|)×

Output: vertex ∈ N(R), extremal wrt w

1. use w as a lifting to construct regular subdivision S of A

face of Σ(A)

The Oracle

1. use w as a lifting to construct regular subdivision S of A2. refine S into triangulation T of A

face of Σ(A)

The Oracle

1. use w as a lifting to construct regular subdivision S of A2. refine S into triangulation T of A3. return ρT ∈ N|A|

face of Σ(A)

The Oracle

1. use w as a lifting to construct regular subdivision S of A2. refine S into triangulation T of A3. return ρT ∈ N|A|

Oracle property: its output is a vertex of the target polytope (Lem. 5).

Incremental AlgorithmInput: A

Output: H-rep. QH, V-rep. QV of Q = N(R)

1. initialization step

initialization:I Q ⊂ N(R)I dim(Q)=dim(N(R))

1. initialization step2. all hyperplanes of QH are illegal

2 kinds of hyperplanes of QH :I legal if it supports facet ⊂ N(R)I illegal otherwise

1. initialization step2. all hyperplanes of QH are illegal3. while ∃ illegal hyperplane H ⊂ QH with outer normal w do

I call oracle for w and compute v, QV ← QV ∪ {v}

Extending an illegal facet

I call oracle for w and compute v, QV ← QV ∪ {v}I if v /∈ QV ∩ H then QH ← CH(QV ∪ {v}) else H is legal

Extending an illegal facet

Validating a legal facet

At any step, Q is an innerapproximation . . .

At any step, Q is an innerapproximation . . . from which wecan compute an outer approximationQo.

Complexity

TheoremWe compute the Vertex- and Halfspace-representations of N(R), as wellas a triangulation T of N(R), in

O∗(m5 |vtx(N(R))| · |T|2),

where m = dim N(R), and |T| the number of full-dim faces of T.

Elements of proofI Computation is done in dimension m = |A| − 2n + 1.I At most ≤ vtx(N(R)) + fct(N(R)) oracle calls (Lem. 9).I Beneath-and-Beyond algorithm for converting V-rep. to H-rep

[Joswig ’02].

ResPol package

I CGAL, triangulation [Boissonnat,Devillers,Hornus]extreme points d [Gartner] (preprocessing step)

I Hashing of determinantal predicates: optimizing sequences of similardeterminants

I http://sourceforge.net/projects/respol

I Applications of ResPol on I.Emiris talk this afternoon (CGAL, anOpen Gate to Computational Geometry!)

Output-sensitivity

I oracle calls ≤ vtx(N(R)) + fct(N(R))I output vertices bound polynomially the output triangulation sizeI subexponential runtime wrt to input points (L), output vertices (R)

Hashing and Gfan

I hashing determinants speeds ≤ 10-100x when dim(N(R)) = 3, 4

I faster than Gfan [Yu-Jensen’11] for dimN(R) ≤ 6, else competitive

dim(N(R)) = 4:

Ongoing and future work

I approximate resultant polytopes (dim(N(R)) ≥ 7) using approximatevolume computation

I combinatorial characterization of 4-dimensional resultant polytopesI computation of discriminant polytopes

More on I.Emiris talk this afternoon (CGAL, an Open Gate toComputational Geometry!)

(figure courtesy of M.Joswig)

Facet and vertex graph of the largest 4-dimensional resultant polytope

Ongoing and future work

(figure courtesy of M.Joswig)

Facet and vertex graph of the largest 4-dimensional resultant polytope

Thank You !

Convex hull implementations

I From V- to H-rep. of N(R).I triangulation (on/off-line), polymake beneath-beyond,

cdd, lrs

0 500 1000 1500 2000 2500 3000

Number of points

bbcddlrs

triang_off

triang_on

dim(N(R)) = 4

An algorithm for computing resultant polytopes

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