Completely log-concave polynomials and matroids Cynthia Vinzant - - PowerPoint PPT Presentation

Completely log-concave polynomials and matroids

⇒ Cynthia Vinzant (North Carolina State University) joint work with Nima Anari, KuiKui Liu, Shayan Oveis Gharan (Stanford) (U. Washington) (U. Washington)

Cynthia Vinzant Completely log-concave polynomials and matroids

Matroids

A matroid on ground set [n] = {1, . . . , n} is a nonempty collection I of independent subsets of [n] satisfying:

◮ If S ⊆ T and T ∈ I, then S ∈ I. ◮ If S, T ∈ I and |T| > |S|, then ∃i ∈ T\S with S ∪ {i} ∈ I.

Examples:

◮ linear independence of vectors v1, . . . , vn ∈ Rd ◮ cyclic independence of n edges in a graph

Independence poly. gM(y, z1, . . . , zn) =

I∈I yn−|I| i∈I zi

Cynthia Vinzant Completely log-concave polynomials and matroids

Results and other work

Mason’s conjecture: Let Ik = # indep. sets of matroid M of size k. (i) I2

k ≥ Ik−1 · Ik+1

(log-concavity) (ii) I2

k ≥

k+1

k

• · Ik−1 · Ik+1

(iii)

• Ik

n

k

• 2

≥ Ik−1 n

k−1

· Ik+1 n

k+1

• (ultra log-concavity)

Cynthia Vinzant Completely log-concave polynomials and matroids

Results and other work

Mason’s conjecture: Let Ik = # indep. sets of matroid M of size k. (i) I2

k ≥ Ik−1 · Ik+1

(log-concavity) (ii) I2

k ≥

k+1

k

• · Ik−1 · Ik+1

(iii)

• Ik

n

k

• 2

≥ Ik−1 n

k−1

· Ik+1 n

k+1

• (ultra log-concavity)

Adiprasito, Huh, Katz use combinatorial Hodge theory to prove (i) Huh, Schr¨

• ter, Wang use ↑ to prove (ii)

Anari, Liu, Oveis Gharan, V. use complete log-concavity to prove (iii) Br¨ and´ en, Huh independently use Lorentz polynomials to prove (iii)

Cynthia Vinzant Completely log-concave polynomials and matroids

Complete log-concavity

f ∈ R[z1, . . . , zn] is log-concave on Rn

>0 if f ≡ 0 or

f (x) ≥ 0 for all x ∈ Rn

≥0

and log(f ) is concave on Rn

>0.

Cynthia Vinzant Completely log-concave polynomials and matroids

Complete log-concavity

f ∈ R[z1, . . . , zn] is log-concave on Rn

>0 if f ≡ 0 or

f (x) ≥ 0 for all x ∈ Rn

≥0

and log(f ) is concave on Rn

>0.

For v = (v1, . . . , vn) ∈ Rn, let Dv = n

i=1 vi ∂f ∂zi .

f ∈ R[z1, . . . , zn] is completely log-concave (CLC) on Rn

>0 if

for all k ∈ N, v1, . . . , vk ∈ Rn

≥0,

Dv1 · · · Dvkf is log-concave on Rn

≥0.

Cynthia Vinzant Completely log-concave polynomials and matroids

Complete log-concavity

f ∈ R[z1, . . . , zn] is log-concave on Rn

>0 if f ≡ 0 or

f (x) ≥ 0 for all x ∈ Rn

≥0

and log(f ) is concave on Rn

>0.

For v = (v1, . . . , vn) ∈ Rn, let Dv = n

i=1 vi ∂f ∂zi .

f ∈ R[z1, . . . , zn] is completely log-concave (CLC) on Rn

>0 if

for all k ∈ N, v1, . . . , vk ∈ Rn

≥0,

Dv1 · · · Dvkf is log-concave on Rn

≥0.

Example: f = d

i=1(z + ri)

⇒ log(f )′′ = d

i=1 −1 (z+ri)2 ≤ 0

Cynthia Vinzant Completely log-concave polynomials and matroids

Example: stable polynomials

f ∈ R[z1, . . . , zn]d is stable if f (tv + w) ∈ R[t] is real rooted for all v ∈ Rn

≥0, w ∈ Rn.

⇒ f is completely log-concave

Cynthia Vinzant Completely log-concave polynomials and matroids

Example: stable polynomials

f ∈ R[z1, . . . , zn]d is stable if f (tv + w) ∈ R[t] is real rooted for all v ∈ Rn

≥0, w ∈ Rn.

⇒ f is completely log-concave Example: det(n

i=1 zivivT i ) = I∈([n]

d ) det(vi : i ∈ I)2

i∈I zi

Cynthia Vinzant Completely log-concave polynomials and matroids

Example: stable polynomials

f ∈ R[z1, . . . , zn]d is stable if f (tv + w) ∈ R[t] is real rooted for all v ∈ Rn

≥0, w ∈ Rn.

⇒ f is completely log-concave Example: det(n

i=1 zivivT i ) = I∈([n]

d ) det(vi : i ∈ I)2

i∈I zi

Choe, Oxley, Sokal, Wagner: If f =

I∈([n]

d ) cI

• i∈I zi is stable,

then supp(f ) = {I : cI = 0} are the bases of a matroid on [n]. Br¨ and´ en: Fano matroid = support of a stable polynomial f

Cynthia Vinzant Completely log-concave polynomials and matroids

Equivalent conditions and univariate characterization

Gurvits: f is strongly log-concave (SLC) if ∂αf = ( ∂

∂z1 )α1 · · · ( ∂ ∂zn )αnf

is log-concave on Rn

≥0.

Cynthia Vinzant Completely log-concave polynomials and matroids

Equivalent conditions and univariate characterization

Gurvits: f is strongly log-concave (SLC) if ∂αf = ( ∂

∂z1 )α1 · · · ( ∂ ∂zn )αnf

is log-concave on Rn

≥0.

Theorem (ALOV): For f ∈ R[z1, . . . , zn]d, f CLC ⇔ f SLC ⇔

• ∂αf is indecomposable for all |α| ≤ d − 2

and ∂αf is CLC for all |α| = d − 2

Cynthia Vinzant Completely log-concave polynomials and matroids

Equivalent conditions and univariate characterization

Gurvits: f is strongly log-concave (SLC) if ∂αf = ( ∂

∂z1 )α1 · · · ( ∂ ∂zn )αnf

is log-concave on Rn

≥0.

Theorem (ALOV): For f ∈ R[z1, . . . , zn]d, f CLC ⇔ f SLC ⇔

• ∂αf is indecomposable for all |α| ≤ d − 2

and ∂αf is CLC for all |α| = d − 2 (d=2) f = zTQz is CLC ⇔ Qij ≥ 0 and Q has 1 pos. eig. value.

Cynthia Vinzant Completely log-concave polynomials and matroids

Equivalent conditions and univariate characterization

Gurvits: f is strongly log-concave (SLC) if ∂αf = ( ∂

∂z1 )α1 · · · ( ∂ ∂zn )αnf

is log-concave on Rn

≥0.

Theorem (ALOV): For f ∈ R[z1, . . . , zn]d, f CLC ⇔ f SLC ⇔

• ∂αf is indecomposable for all |α| ≤ d − 2

and ∂αf is CLC for all |α| = d − 2 (d=2) f = zTQz is CLC ⇔ Qij ≥ 0 and Q has 1 pos. eig. value.

• Cor. (Gurvits/ALOV)

n

• k=0

akyn−kzk is CLC ⇔

• ak

n

k

• 2

≥ ak−1 n

k−1

· ak+1 n

k+1

• Cynthia Vinzant

Completely log-concave polynomials and matroids

Complete log-concavity for matroids

• Theorem. gM(y, z1, . . . , zn) =

I∈I yn−|I| i∈I zi is CLC.

(just check rank-two matroids M)

Cor: gM(y, z, . . . , z) = n

k=0 Ikyn−kzk is CLC.

Cor: {Ik}k is ultra log-concave (Mason’s conjecture)

Cynthia Vinzant Completely log-concave polynomials and matroids

Other results

Theorem: For any matroid M, the solution to the concave program τ = max

p∈PM n

• i=1

pi log 1

pi + (1 − pi) log 1 1−pi

can be computed in polynomial time and β = eτ satisfies 2O(−r)β ≤ # bases of M ≤ β.

Cynthia Vinzant Completely log-concave polynomials and matroids

Other results

Theorem: For any matroid M, the solution to the concave program τ = max

p∈PM n

• i=1

pi log 1

pi + (1 − pi) log 1 1−pi

can be computed in polynomial time and β = eτ satisfies 2O(−r)β ≤ # bases of M ≤ β. Theorem: The natural Markov Chain P(B, B′) on the bases of any rank-r matroid on [n] mixes quickly: min{t ∈ N : ||Pt(B, ·) − π||1 ≤ ǫ} ≤ r2 log(n/ǫ).

Cynthia Vinzant Completely log-concave polynomials and matroids

Sum up: completely log-concave polynomials

◮ log-concavity of polynomial as functions

⇒ log-concavity of coefficients

◮ many matroid polynomials are completely log-concave ◮ the theory of stable polynomials extends to CLC polynomials

⇒

Cynthia Vinzant Completely log-concave polynomials and matroids

Sum up: completely log-concave polynomials

◮ log-concavity of polynomial as functions

⇒ log-concavity of coefficients

◮ many matroid polynomials are completely log-concave ◮ the theory of stable polynomials extends to CLC polynomials

⇒ Thanks!

Cynthia Vinzant Completely log-concave polynomials and matroids

References

◮

Karim Adiprasito, June Huh, Eric Katz, Hodge theory for combinatorial geometries, Annals of Mathematics 188(2), 2018.

◮

Nima Anari, Shayan Oveis Gharan, Cynthia Vinzant, Log-Concave Polynomials I: Entropy and a Deterministic Approximation Algorithm for Counting Bases of Matroids, arXiv:1807.00929

◮

Nima Anari, KuiKui Liu, Shayan Oveis Gharan, Cynthia Vinzant, Log-Concave Polynomials II: High-Dimensional Walks and an FPRAS for Counting Bases of a Matroid, arXiv:1811.01816

◮

Nima Anari, KuiKui Liu, Shayan Oveis Gharan, Cynthia Vinzant, Log-Concave Polynomials III: Mason’s Ultra-Log-Concavity Conjecture for Independent Sets of Matroids, arXiv:1811.01600

◮

Petter Br¨ and´ en, Polynomials with the half-plane property and matroid theory, Advances in Mathematics 216(1), 2007.

◮

Petter Br¨ and´ en, June Huh, Hodge-Riemann relations for Potts model partition functions, arXiv:1811.01696

◮

Young-Bin Choe, James Oxley, Alan Sokal, David Wagner, Homogeneous multivariate polynomials with the half-plane property, Advances in Applied Mathematics, 32(1-2), 2004.

◮

Leonid Gurvits, On multivariate Newton-like inequalities, Advances in combinatorial mathematics, 61–78, 2009.

◮

June Huh, Benjamin Schr¨

• ter, Botong Wang, Correlation bounds for fields and matroids, arXiv:1806.02675

Cynthia Vinzant Completely log-concave polynomials and matroids