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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 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 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 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
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
PP
Qππ
Q
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
PP
Qππ
Q
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
PP
Qππ
Q
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.)
vj
Hi
sij
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+
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. . . of a slack matrix
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. . . of a slack matrix
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
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
Alice
Alice
Bob Bob
0 1 0
1 0
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
Alice
Alice
Bob Bob
0 1 0
1 0
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 ))
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 ))
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 ))
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 ))
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 ))
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)
K S
Alice Bob
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)
K S
Alice Bob
u
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)
vtx u ∈ K
K S
Alice Bob
u
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)
vtx u ∈ K
K S
Alice Bob
u u
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)
vtx u ∈ K
K S
Alice Bob
u N(u)
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)
vtx u ∈ K
K S
Alice Bob
u N(u)
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)
S ∩ N(u)
K S
Alice Bob
u N(u)
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)
S ∩ N(u)
SK
Alice Bob
u N(u)
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)
0/1
SK
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
i
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
i
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
Alice
Alice
Bob Bob
0 1 0
1 0
( 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∑
k=1
tikukj = sij
Complexity: log rank+(S(P)) + O(1)
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∑
k=1
tikukj = sij
Complexity: log rank+(S(P)) + O(1)
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∑
k=1
tikukj = sij
Complexity: log rank+(S(P)) + O(1)
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
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
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
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 B
Proof sketch (reduction from SET DISJOINTNESS)
A BA ∩ B = ∅
Proof sketch (reduction from SET DISJOINTNESS)
S
A BA ∩ B = ∅
Proof sketch (reduction from SET DISJOINTNESS)
MS
A BA ∩ B = ∅
Proof sketch (reduction from SET DISJOINTNESS)
|δ(S) ∩M| = 1 MS
A BA ∩ B = ∅
Proof sketch (reduction from SET DISJOINTNESS)
A B
Proof sketch (reduction from SET DISJOINTNESS)
A BA ∩ B 6= ∅
Proof sketch (reduction from SET DISJOINTNESS)
S
A BA ∩ B 6= ∅
Proof sketch (reduction from SET DISJOINTNESS)
MS
A BA ∩ B 6= ∅
Proof sketch (reduction from SET DISJOINTNESS)
|δ(S) ∩M| > 1 MS
A BA ∩ B 6= ∅
Proof sketch (wrapping up)
Next,
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)
Next,
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)) ?
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)) ?
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)) ?
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!