Extended formulations, non-negative factorizations and ... · Extended formulations, non-negative...

Extended formulations, non-negativefactorizations and randomized communication

protocols

Michele Conforti1 Yuri Faenza1 Samuel Fiorini2

Roland Grappe1 Hans Raj Tiwary2

1Universita di Padova

2Universite Libre de Bruxelles

What is this talk about?

In 91, Mihalis Yannakakis published an important paper.

Theorem (Y91)Every symmetric extended formulation for the perfect matching polytope

has exponential size.

What we do:

I Prove a result of the type: every 〈other condition〉 extendedformulation for the perfect matching polytope has 〈certain size〉

I Develop a new way of interpreting extended formulations

What we do:

Extension complexityGetting started

P ⊆ Rd polytope

Q ⊆ Re polyhedron &π : Re → Rd linear map

define extension of P if π(Q) = P

Definitionxc(P) := extension complexity of P

:= min #facets in an extension of P

Example:

xc(regular 8-gon) 6 6

P ⊆ Rd polytope

Example:

P ⊆ Rd polytope

Example:

Extension complexityMore examples, and things for your to do list

Known results:

I xc(regular n-gon) = Θ(log n) [BN01]

I xc(n-permutohedron) = Θ(n log n) [G10]

I xc(spanning tree polytope of Kn) = O(n3) [M87]

I xc(stable set polytope of perfect graph G ) = nO(log n) [Y91]

Open questions:

I xc(nonregular n-gon) = ?

I xc(0/1 polytope P in Rd) at most polynomial in d? [K]

I xc(perfect matching polytope of Kn) = ?

I . . .

Extension complexityMore examples, and things for your to do list

Known results:

I xc(regular n-gon) = Θ(log n) [BN01]

I xc(n-permutohedron) = Θ(n log n) [G10]

I xc(spanning tree polytope of Kn) = O(n3) [M87]

I xc(stable set polytope of perfect graph G ) = nO(log n) [Y91]

Open questions:

I xc(nonregular n-gon) = ?

I xc(0/1 polytope P in Rd) at most polynomial in d? [K]

I xc(perfect matching polytope of Kn) = ?

I . . .

Slack matrices

P = convv1, . . . , vn= x ∈ Rd : aT

1 x − b1 > 0, . . . , aTmx − bm > 0

non-redundant inner/outer descriptions of P (assuming P full-dim.)

DefinitionS(P) := slack matrix of P

:= m × n matrix (sij) with sij = aTi vj − bi > 0

= slack of jth vertex w.r.t. ith facet

Non-negative rank

S ∈ Rm×n+

Definitionrank+(S) := non-negative rank of S

:= min r s.t. S = AB with A ∈ Rm×r+ and B ∈ Rr×n

= min r s.t. S is sum of r non-neg. rank 1 matrices

Theorem (Y91, FKPT10+)xc(P) = rank+(S(P))

Why like this result?

I elegant mathematical fact

I connects several research fields togetherI polyhedral combinatoricsI applied linear algebraI communication complexity

Non-negative rank

S ∈ Rm×n+

Non-negative rank. . . of a slack matrix

S ∈ Rm×n+

Non-negative rank. . . of a slack matrix

S ∈ Rm×n+

Deterministic communication protocolsThe tale of Alice and Bob

f : X × Y → 0, 1 boolean function (matrix)

Two players:

I Alice knows x ∈ X

I Bob knows y ∈ Y

want to compute f (x , y) by exchanging bits

Goal: Minimize complexity := #bits exchanged

Deterministic communication protocolsAn example of a protocol

y1 y2 y3 y4

x1 0 0 0 1x2 0 0 0 1x3 0 0 0 0x4 0 1 1 1

Bob Bob

x ∈ x1, x2 x ∈ x3, x4

y ∈ y1, y2, y3 y ∈ y4 y ∈ y2, y3, y4 y ∈ y1

x ∈ x4 x ∈ x3

Deterministic communication protocolsAn example of a protocol

y1 y2 y3 y4

x1 0 0 0 1x2 0 0 0 1x3 0 0 0 0x4 0 1 1 1

Bob Bob

x ∈ x1, x2 x ∈ x3, x4

y ∈ y1, y2, y3 y ∈ y4 y ∈ y2, y3, y4 y ∈ y1

x ∈ x4 x ∈ x3

Deterministic communication protocolsCases where Alice and Bob provide an extended formulation

Assume: P = STAB(G ) for G perfect graph

=⇒ S(P) is binary

Slack matrix: rows correspond to cliques Kcolumns correspond to stable sets Sslack of S w.r.t K is 1− χS(K ) = 1− |S ∩ K |

∃ complexity c protocol for computing S(P) =⇒ xc(P) 6 2c

Theorem (Y91)For all n-vertex perfect graphs G,

xc(STAB(G )) 6 2O(log2 n) = nO(log n)

Proof.∃O(log2 n) complexity protocol for computing S(STAB(G ))

Assume: P = STAB(G ) for G perfect graph =⇒ S(P) is binary

Deterministic communication protocolsPerfect without claw

Example: P = STAB(G ) for G claw-free perfect graph

∃ 3 log n + O(1) complexity protocol for S(P) =⇒ xc(P) = O(n3)

Alice Bob

vtx u ∈ K

Alice Bob

vtx u ∈ K

Alice Bob

vtx u ∈ K

Alice Bob

u N(u)

vtx u ∈ K

Alice Bob

u N(u)

S ∩ N(u)

Alice Bob

u N(u)

S ∩ N(u)

Alice Bob

u N(u)

Alice Bob

u N(u)

Deterministic communication protocolsHow good are they?

Limitations:

1. binary slack matrices S(P)

2. deterministic protocols imply disjoint rectangles

Possible solutions:

1. Alice and Bob could output non-binary values

2. randomized protocols would allow overlapping rectangles

Deterministic communication protocolsHow good are they?

Limitations:

1. binary slack matrices S(P)

2. deterministic protocols imply disjoint rectangles

Possible solutions:

1. Alice and Bob could output non-binary values

2. randomized protocols would allow overlapping rectangles

Our main resultA tight relationship between extended formulations and communication complexity

TheoremIf c is the minimum complexity of a randomized communication protocolcomputing S(P) on average, then

xc(P) = rank+(S(P)) = Θ(2c)

Randomized communication protocolsIntroducing the model

Similar to deterministic protocols except that

I Alice and Bob can use private random bits to make a choice

1− pi (x)pi (x)

I The value output on given input (x , y) is a random variable

We say that the protocol computes f on average if: ∀(x , y) ∈ X × Y ,

E [value output by the protocol on input (x , y)] = f (x , y)

Randomized communication protocolsIntroducing the model

Similar to deterministic protocols except that

I Alice and Bob can use private random bits to make a choice

1− pi (x)pi (x)

I The value output on given input (x , y) is a random variable

We say that the protocol computes f on average if: ∀(x , y) ∈ X × Y ,

E [value output by the protocol on input (x , y)] = f (x , y)

Randomized communication protocolsGo randomized

The previous deterministic protocol can be viewed as a randomizedprotocol where all the probabilities are either zero or one

Bob Bob

( 1 , 1, 0, 0)T (0, 0, 1, 1)T

(1, 1, 1, 0) (0, 0, 0, 1) (0, 1, 1, 1) (1, 0, 0, 0)

(0, 0, 0, 1)T (1, 1, 1, 0)T

In general can use arbitrary probabilities, and nonnegative values!

Proof of the theoremFactorization =⇒ protocol

Write S(P) = TU, whereT ∈ Rm×r

+ row stochastic (w.l.o.g.)U ∈ Rr×n

r = rank+(S(P))+1

Protocol:

I Alice gets row index i , Bob gets column index j

I Alice picks random column index k w.p. tik , sends it to Bob

I Bob outputs value ukj

Expected value on input (i , j):r∑

tikukj = sij

Complexity: log rank+(S(P)) + O(1)

r = rank+(S(P))+1

Protocol:

tikukj = sij

r = rank+(S(P))+1

Protocol:

tikukj = sij

New lower bound for the perfect matching polytope of Kn

Now: P = perfect matching polytope of Kn

Slack matrix: rows correspond to odd sets Scolumns correspond to matchings Mslack of M w.r.t. S is |M ∩ δ(S)|−1

Theorem (“small randomness implies large size”)

Consider an extension for P and a corresponding randomized protocol. Ifthe probability that the protocol outputs a non-zero value, given a pair(S ,M) with non-zero slack, is at least p(n) 1/n, then the protocol hascomplexity Ω(np(n)) and the extended formulation has size 2Ω(np(n))

Particular case: If a non-zero slack is detected with constant probability,then extension has exponential size

Even more particular case: If the protocol is deterministic,then extension has exponential size

Proof sketch (reduction from SET DISJOINTNESS)

A BA ∩ B = ∅

|δ(S) ∩M| = 1 MS

A BA ∩ B = ∅

A BA ∩ B 6= ∅

|δ(S) ∩M| > 1 MS

A BA ∩ B 6= ∅

Proof sketch (wrapping up)

I repeat protocol Θ(1/p(n)) times on pair (S ,M), independently

I declare A and B non-disjoint as soon as know that slack(S ,M) 6= 0

I otherwise, declare A and B disjoint

Finally, use

Fact. computing DISJ with high probability needs Ω(n) bits

Proof sketch (wrapping up)

I repeat protocol Θ(1/p(n)) times on pair (S ,M), independently

I declare A and B non-disjoint as soon as know that slack(S ,M) 6= 0

I otherwise, declare A and B disjoint

Finally, use

Fact. computing DISJ with high probability needs Ω(n) bits

Concluding remarks / questions

I “Small randomness implies large size” also holds for spanning trees!!

I ∃ nice characterization of polytopes with binary slack matrix?

I What is the communication complexity of the relation

(S ,M, e) : S odd set,M perfect matching, e ∈ δ(S) ∩M ?

I (“log rank conjecture”-inspired) For which polytopes do we have

randomized-cc(support of S(P)) 6 poly(log rank+ S(P)) ?

Thank You!

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