Maximum Matching and Linear Programming in Fixed-Point ...
Transcript of Maximum Matching and Linear Programming in Fixed-Point ...
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Maximum Matching and Linear Programmingin Fixed-Point Logic with Counting
University of Cambridge Computer Laboratory
26 June 2013
Matthew Anderson Anuj Dawar Bjarki Holm
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Motivation
Question:Is there is logical characterisation of P on unordered structures?
FP ⊆ FPC ⊆ CPT(Card) ⊆ P
• P ⊆ FP? No, parity 6∈ FP.
• P ⊆ FPC? No, [Cai-Furer-Immerman ’92].
• P ⊆ CPT(Card)?
Choiceless computation is powerful:
• CircuitValueProblem ∈ FP.
• BipartitePM ∈ FPC [Blass-Gurevich-Shelah ’02] andconjectured PM 6∈ CPT(Card).
We show:
• MaximumMatching ∈ FPC.
• LinearProgramming ∈ FPC.
![Page 3: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/3.jpg)
Motivation
Question:Is there is logical characterisation of P on unordered structures?
FP ⊆ FPC ⊆ CPT(Card) ⊆ P
• P ⊆ FP? No, parity 6∈ FP.
• P ⊆ FPC? No, [Cai-Furer-Immerman ’92].
• P ⊆ CPT(Card)?
Choiceless computation is powerful:
• CircuitValueProblem ∈ FP.
• BipartitePM ∈ FPC [Blass-Gurevich-Shelah ’02] andconjectured PM 6∈ CPT(Card).
We show:
• MaximumMatching ∈ FPC.
• LinearProgramming ∈ FPC.
![Page 4: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/4.jpg)
Motivation
Question:Is there is logical characterisation of P on unordered structures?
FP ⊆ FPC ⊆ CPT(Card) ⊆ P
• P ⊆ FP? No, parity 6∈ FP.
• P ⊆ FPC? No, [Cai-Furer-Immerman ’92].
• P ⊆ CPT(Card)?
Choiceless computation is powerful:
• CircuitValueProblem ∈ FP.
• BipartitePM ∈ FPC [Blass-Gurevich-Shelah ’02] andconjectured PM 6∈ CPT(Card).
We show:
• MaximumMatching ∈ FPC.
• LinearProgramming ∈ FPC.
![Page 5: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/5.jpg)
Motivation
Question:Is there is logical characterisation of P on unordered structures?
FP ⊆ FPC ⊆ CPT(Card) ⊆ P
• P ⊆ FP? No, parity 6∈ FP.
• P ⊆ FPC? No, [Cai-Furer-Immerman ’92].
• P ⊆ CPT(Card)?
Choiceless computation is powerful:
• CircuitValueProblem ∈ FP.
• BipartitePM ∈ FPC [Blass-Gurevich-Shelah ’02] andconjectured PM 6∈ CPT(Card).
We show:
• MaximumMatching ∈ FPC.
• LinearProgramming ∈ FPC.
![Page 6: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/6.jpg)
Motivation
Question:Is there is logical characterisation of P on unordered structures?
FP ⊆ FPC ⊆ CPT(Card) ⊆ P
• P ⊆ FP? No, parity 6∈ FP.
• P ⊆ FPC? No, [Cai-Furer-Immerman ’92].
• P ⊆ CPT(Card)?
Choiceless computation is powerful:
• CircuitValueProblem ∈ FP.
• BipartitePM ∈ FPC [Blass-Gurevich-Shelah ’02] andconjectured PM 6∈ CPT(Card).
We show:
• MaximumMatching ∈ FPC.
• LinearProgramming ∈ FPC.
![Page 7: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/7.jpg)
Motivation
Question:Is there is logical characterisation of P on unordered structures?
FP ⊆ FPC ⊆ CPT(Card) ⊆ P
• P ⊆ FP? No, parity 6∈ FP.
• P ⊆ FPC? No, [Cai-Furer-Immerman ’92].
• P ⊆ CPT(Card)?
Choiceless computation is powerful:
• CircuitValueProblem ∈ FP.
• BipartitePM ∈ FPC [Blass-Gurevich-Shelah ’02] andconjectured PM 6∈ CPT(Card).
We show:
• MaximumMatching ∈ FPC.
• LinearProgramming ∈ FPC.
![Page 8: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/8.jpg)
Motivation
Question:Is there is logical characterisation of P on unordered structures?
FP ⊆ FPC ⊆ CPT(Card) ⊆ P
• P ⊆ FP? No, parity 6∈ FP.
• P ⊆ FPC? No, [Cai-Furer-Immerman ’92].
• P ⊆ CPT(Card)?
Choiceless computation is powerful:
• CircuitValueProblem ∈ FP.
• BipartitePM ∈ FPC [Blass-Gurevich-Shelah ’02] andconjectured PM 6∈ CPT(Card).
We show:
• MaximumMatching ∈ FPC.
• LinearProgramming ∈ FPC.
![Page 9: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/9.jpg)
Motivation
Question:Is there is logical characterisation of P on unordered structures?
FP ⊆ FPC ⊆ CPT(Card) ⊆ P
• P ⊆ FP? No, parity 6∈ FP.
• P ⊆ FPC? No, [Cai-Furer-Immerman ’92].
• P ⊆ CPT(Card)?
Choiceless computation is powerful:
• CircuitValueProblem ∈ FP.
• BipartitePM ∈ FPC [Blass-Gurevich-Shelah ’02] andconjectured PM 6∈ CPT(Card).
We show:
• MaximumMatching ∈ FPC.
• LinearProgramming ∈ FPC.
![Page 10: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/10.jpg)
Structures, Numbers, and Matrices
Vocabulary τ
Finite τ -structures fin[τ ]
Vocabularies for encoding numerical data in structures:
τQ Encodes the binary expansion of a rational number q ∈ Q in adomain of ordered bits B .
τvec Encodes a vector v ∈ QI as a set of rational numbers indexedby a separate domain I .
τmat Encodes a matrix M ∈ QI×J as a set of rational numbersindexed by a pair of separate domains I and J .
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Structures, Numbers, and Matrices
Vocabulary τ
Finite τ -structures fin[τ ]
Vocabularies for encoding numerical data in structures:
τQ Encodes the binary expansion of a rational number q ∈ Q in adomain of ordered bits B .
τvec Encodes a vector v ∈ QI as a set of rational numbers indexedby a separate domain I .
τmat Encodes a matrix M ∈ QI×J as a set of rational numbersindexed by a pair of separate domains I and J .
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Structures, Numbers, and Matrices
Vocabulary τ
Finite τ -structures fin[τ ]
Vocabularies for encoding numerical data in structures:
τQ Encodes the binary expansion of a rational number q ∈ Q in adomain of ordered bits B .
τvec Encodes a vector v ∈ QI as a set of rational numbers indexedby a separate domain I .
τmat Encodes a matrix M ∈ QI×J as a set of rational numbersindexed by a pair of separate domains I and J .
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FPC and Interpretations
FPC Inflationary fixed-point logic extended with the ability toexpress the size of definable sets.
• Assume standard syntax and semantics.
• FPC[τ ] defines relations over dom(A) ] [|dom(A)|+ 1]invariant to automorphisms of A ∈ fin[τ ].
Immerman-Vardi Theorem
Every polynomial-time decidable property of ordered structures isdefinable in FPC (indeed, in FP).
An FPC interpretation of τ in σ is a function fin[σ]→ fin[τ ]defined by a sequence of FPC[σ] formulas.
FPC interpretations can express many standard linear algebraicoperations, e.g., multiplication, inverse, and rank [Holm ’10].
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FPC and Interpretations
FPC Inflationary fixed-point logic extended with the ability toexpress the size of definable sets.
• Assume standard syntax and semantics.
• FPC[τ ] defines relations over dom(A) ] [|dom(A)|+ 1]invariant to automorphisms of A ∈ fin[τ ].
Immerman-Vardi Theorem
Every polynomial-time decidable property of ordered structures isdefinable in FPC (indeed, in FP).
An FPC interpretation of τ in σ is a function fin[σ]→ fin[τ ]defined by a sequence of FPC[σ] formulas.
FPC interpretations can express many standard linear algebraicoperations, e.g., multiplication, inverse, and rank [Holm ’10].
![Page 15: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/15.jpg)
FPC and Interpretations
FPC Inflationary fixed-point logic extended with the ability toexpress the size of definable sets.
• Assume standard syntax and semantics.
• FPC[τ ] defines relations over dom(A) ] [|dom(A)|+ 1]invariant to automorphisms of A ∈ fin[τ ].
Immerman-Vardi Theorem
Every polynomial-time decidable property of ordered structures isdefinable in FPC (indeed, in FP).
An FPC interpretation of τ in σ is a function fin[σ]→ fin[τ ]defined by a sequence of FPC[σ] formulas.
FPC interpretations can express many standard linear algebraicoperations, e.g., multiplication, inverse, and rank [Holm ’10].
![Page 16: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/16.jpg)
FPC and Interpretations
FPC Inflationary fixed-point logic extended with the ability toexpress the size of definable sets.
• Assume standard syntax and semantics.
• FPC[τ ] defines relations over dom(A) ] [|dom(A)|+ 1]invariant to automorphisms of A ∈ fin[τ ].
Immerman-Vardi Theorem
Every polynomial-time decidable property of ordered structures isdefinable in FPC (indeed, in FP).
An FPC interpretation of τ in σ is a function fin[σ]→ fin[τ ]defined by a sequence of FPC[σ] formulas.
FPC interpretations can express many standard linear algebraicoperations, e.g., multiplication, inverse, and rank [Holm ’10].
![Page 17: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/17.jpg)
Convex Optimisation – Geometry
Consider the Euclidean space QV indexed by a set V .
Constraint
• For a ∈ QV , b ∈ Q, {x ∈ QV | a>x ≤ b}.
• Size 〈a, b〉 := 〈b〉+∑
v∈V 〈av 〉.
Polytope
• For A ∈ QC×V , b ∈ QC , PA,b := {x ∈ QV | Ax ≤ b}.
• Size 〈PA,b〉 := maxr∈C 〈Ar , br 〉.
E.g: A =
−1 3
3 84 −32 −6
, b =
6
2069
.
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Convex Optimisation – Geometry
Consider the Euclidean space QV indexed by a set V .
Constraint
• For a ∈ QV , b ∈ Q, {x ∈ QV | a>x ≤ b}.
• Size 〈a, b〉 := 〈b〉+∑
v∈V 〈av 〉.Polytope
• For A ∈ QC×V , b ∈ QC , PA,b := {x ∈ QV | Ax ≤ b}.
• Size 〈PA,b〉 := maxr∈C 〈Ar , br 〉.
E.g: A =
−1 3
3 84 −32 −6
, b =
6
2069
.
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Convex Optimisation – Geometry
Consider the Euclidean space QV indexed by a set V .
Constraint
• For a ∈ QV , b ∈ Q, {x ∈ QV | a>x ≤ b}.
• Size 〈a, b〉 := 〈b〉+∑
v∈V 〈av 〉.Polytope
• For A ∈ QC×V , b ∈ QC , PA,b := {x ∈ QV | Ax ≤ b}.
• Size 〈PA,b〉 := maxr∈C 〈Ar , br 〉.
E.g: A =
−1 3
3 84 −32 −6
, b =
6
2069
.
1
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Convex Optimisation – Geometry
Consider the Euclidean space QV indexed by a set V .
Constraint
• For a ∈ QV , b ∈ Q, {x ∈ QV | a>x ≤ b}.
• Size 〈a, b〉 := 〈b〉+∑
v∈V 〈av 〉.
Polytope
• For A ∈ QC×V , b ∈ QC , PA,b := {x ∈ QV | Ax ≤ b}.
• Size 〈PA,b〉 := maxr∈C 〈Ar , br 〉.
E.g: A =
−1 3
3 84 −32 −6
, b =
6
2069
.
1
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Convex Optimisation – Geometry
Consider the Euclidean space QV indexed by a set V .
Constraint
• For a ∈ QV , b ∈ Q, {x ∈ QV | a>x ≤ b}.
• Size 〈a, b〉 := 〈b〉+∑
v∈V 〈av 〉.
Polytope
• For A ∈ QC×V , b ∈ QC , PA,b := {x ∈ QV | Ax ≤ b}.
• Size 〈PA,b〉 := maxr∈C 〈Ar , br 〉.
E.g: A =
−1 3
3 84 −32 −6
, b =
6
2069
.
1
![Page 22: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/22.jpg)
Convex Optimisation – Geometry
Consider the Euclidean space QV indexed by a set V .
Constraint
• For a ∈ QV , b ∈ Q, {x ∈ QV | a>x ≤ b}.
• Size 〈a, b〉 := 〈b〉+∑
v∈V 〈av 〉.
Polytope
• For A ∈ QC×V , b ∈ QC , PA,b := {x ∈ QV | Ax ≤ b}.
• Size 〈PA,b〉 := maxr∈C 〈Ar , br 〉.
E.g: A =
−1 33 8
4 −32 −6
, b =
620
69
.
1
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Convex Optimisation – Geometry
Consider the Euclidean space QV indexed by a set V .
Constraint
• For a ∈ QV , b ∈ Q, {x ∈ QV | a>x ≤ b}.
• Size 〈a, b〉 := 〈b〉+∑
v∈V 〈av 〉.
Polytope
• For A ∈ QC×V , b ∈ QC , PA,b := {x ∈ QV | Ax ≤ b}.
• Size 〈PA,b〉 := maxr∈C 〈Ar , br 〉.
E.g: A =
−1 33 84 −32 −6
, b =
62069
.
1
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Convex Optimisation – Geometry
Consider the Euclidean space QV indexed by a set V .
Constraint
• For a ∈ QV , b ∈ Q, {x ∈ QV | a>x ≤ b}.• Size 〈a, b〉 := 〈b〉+
∑v∈V 〈av 〉.
Polytope
• For A ∈ QC×V , b ∈ QC , PA,b := {x ∈ QV | Ax ≤ b}.
• Size 〈PA,b〉 := maxr∈C 〈Ar , br 〉.
E.g: A =
−1 33 84 −32 −6
, b =
62069
.
1
![Page 25: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/25.jpg)
Convex Optimisation – Geometry
Consider the Euclidean space QV indexed by a set V .
Constraint
• For a ∈ QV , b ∈ Q, {x ∈ QV | a>x ≤ b}.• Size 〈a, b〉 := 〈b〉+
∑v∈V 〈av 〉.
Polytope
• For A ∈ QC×V , b ∈ QC , PA,b := {x ∈ QV | Ax ≤ b}.• Size 〈PA,b〉 := maxr∈C 〈Ar , br 〉.
E.g: A =
−1 33 84 −32 −6
, b =
62069
.
1
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Convex Optimisation – Linear Optimisation
Linear Optimisation Problem
Given: A polytope P ⊆ QV and objective vector k ∈ QV .Determine:
1 x ∈ P with k>x = max{k>y | y ∈ P},2 P = ∅, or
3 P is unbounded in the direction of k .
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Convex Optimisation – Linear Optimisation
Linear Optimisation Problem
Given: A polytope P ⊆ QV and objective vector k ∈ QV .Determine:
1 x ∈ P with k>x = max{k>y | y ∈ P},2 P = ∅, or
3 P is unbounded in the direction of k .
1
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Convex Optimisation – Linear Optimisation
Linear Optimisation Problem
Given: A polytope P ⊆ QV and objective vector k ∈ QV .Determine:
1 x ∈ P with k>x = max{k>y | y ∈ P},2 P = ∅, or
3 P is unbounded in the direction of k .
k
1
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Convex Optimisation – Linear Optimisation
Linear Optimisation Problem
Given: A polytope P ⊆ QV and objective vector k ∈ QV .Determine:
1 x ∈ P with k>x = max{k>y | y ∈ P},2 P = ∅, or
3 P is unbounded in the direction of k .
k
1x
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Convex Optimisation – Linear Optimisation
Linear Optimisation Problem
Given: A polytope P ⊆ QV and objective vector k ∈ QV .Determine:
1 x ∈ P with k>x = max{k>y | y ∈ P},2 P = ∅, or
3 P is unbounded in the direction of k .
1
k
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Convex Optimisation – Linear Optimisation
Linear Optimisation Problem
Given: A polytope P ⊆ QV and objective vector k ∈ QV .Determine:
1 x ∈ P with k>x = max{k>y | y ∈ P},2 P = ∅, or
3 P is unbounded in the direction of k .
1
kUnbounded!
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Convex Optimisation – Separation
Separation Problem
Given: A polytope P ⊆ QV and point x ∈ QV .Determine:
1 x ∈ P , or
2 c ∈ QV with c>x > max{c>y | y ∈ P}.
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Convex Optimisation – Separation
Separation Problem
Given: A polytope P ⊆ QV and point x ∈ QV .Determine:
1 x ∈ P , or
2 c ∈ QV with c>x > max{c>y | y ∈ P}.
1
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Convex Optimisation – Separation
Separation Problem
Given: A polytope P ⊆ QV and point x ∈ QV .Determine:
1 x ∈ P , or
2 c ∈ QV with c>x > max{c>y | y ∈ P}.
1
x
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Convex Optimisation – Separation
Separation Problem
Given: A polytope P ⊆ QV and point x ∈ QV .Determine:
1 x ∈ P , or
2 c ∈ QV with c>x > max{c>y | y ∈ P}.
1
x
x ∈ P
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Convex Optimisation – Separation
Separation Problem
Given: A polytope P ⊆ QV and point x ∈ QV .Determine:
1 x ∈ P , or
2 c ∈ QV with c>x > max{c>y | y ∈ P}.
1
x
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Convex Optimisation – Separation
Separation Problem
Given: A polytope P ⊆ QV and point x ∈ QV .Determine:
1 x ∈ P , or
2 c ∈ QV with c>x > max{c>y | y ∈ P}.
1
x
c
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Convex Optimisation – Separation
Separation Problem
Given: A polytope P ⊆ QV and point x ∈ QV .Determine:
1 x ∈ P , or
2 c ∈ QV with c>x > max{c>y | y ∈ P}.
1
x
c
c′′
c′
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Convex Optimisation – Separation
Separation Problem
Given: A polytope P ⊆ QV and point x ∈ QV .Determine:
1 x ∈ P , or
2 c ∈ QV with c>x > max{c>y | y ∈ P}.
For polytopes in ordered spaces, the separation and optimisationproblem are polynomial-time equivalent (via the ellipsoid method[Khachiyan ’79]).
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Convex Optimisation – Separation
Separation Problem
Given: A polytope P ⊆ QV and point x ∈ QV .Determine:
1 x ∈ P , or
2 c ∈ QV with c>x > max{c>y | y ∈ P}.
Typical algorithm for solving separation on explicit polytope PA,b .
SEP(A ∈ QC×V , b ∈ QC , x ∈ QV ):
1 If Ax ≤ b, return “x ∈ P”.
2 Pick r ∈ C with Arx > br .
3 Return Ar .
1 If Ax ≤ b, return “x ∈ P”.
2 c ←∑{r∈C | Arx>br}Ar .
3 If c = 0V , return 1V .
4 Return c.
⇒{
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Convex Optimisation – Separation
Separation Problem
Given: A polytope P ⊆ QV and point x ∈ QV .Determine:
1 x ∈ P , or
2 c ∈ QV with c>x > max{c>y | y ∈ P}.
Typical algorithm for solving separation on explicit polytope PA,b .
SEP(A ∈ QC×V , b ∈ QC , x ∈ QV ):
1 If Ax ≤ b, return “x ∈ P”.
2 Pick r ∈ C with Arx > br .
3 Return Ar .
1 If Ax ≤ b, return “x ∈ P”.
2 c ←∑{r∈C | Arx>br}Ar .
3 If c = 0V , return 1V .
4 Return c.
⇒{
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Convex Optimisation – Separation
Separation Problem
Given: A polytope P ⊆ QV and point x ∈ QV .Determine:
1 x ∈ P , or
2 c ∈ QV with c>x > max{c>y | y ∈ P}.
Typical algorithm for solving separation on explicit polytope PA,b .
SEP(A ∈ QC×V , b ∈ QC , x ∈ QV ):
1 If Ax ≤ b, return “x ∈ P”.
2 Pick r ∈ C with Arx > br .
3 Return Ar .
1 If Ax ≤ b, return “x ∈ P”.
2 c ←∑{r∈C | Arx>br}Ar .
3 If c = 0V , return 1V .
4 Return c.
⇒{
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Convex Optimisation – Representation
Representation
A representation (τ, ν) of a class P of polytopes is
• a vocabulary τ , and
• an onto function ν : fin[τ ]→ P which is isomorphisminvariant, i.e., A ∼= B ⇒ ν(A) ∼= ν(B), ∀A,B ∈ fin[τ ].
Explicit representation takes fin[τmat ] τvec] to the class of allpolytopes via ν : (A, b) 7→ PA,b .
A representation (τ, ν) is well described if for all A ∈ fin[τ ],〈ν(A)〉 = poly(|A|).
• The explicit representation is trivially well described.
• There are well-described representations with an exponentialnumber of constraints.
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Convex Optimisation – Representation
Representation
A representation (τ, ν) of a class P of polytopes is
• a vocabulary τ , and
• an onto function ν : fin[τ ]→ P which is isomorphisminvariant, i.e., A ∼= B ⇒ ν(A) ∼= ν(B), ∀A,B ∈ fin[τ ].
Explicit representation takes fin[τmat ] τvec] to the class of allpolytopes via ν : (A, b) 7→ PA,b .
A representation (τ, ν) is well described if for all A ∈ fin[τ ],〈ν(A)〉 = poly(|A|).
• The explicit representation is trivially well described.
• There are well-described representations with an exponentialnumber of constraints.
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Convex Optimisation – Representation
Representation
A representation (τ, ν) of a class P of polytopes is
• a vocabulary τ , and
• an onto function ν : fin[τ ]→ P which is isomorphisminvariant, i.e., A ∼= B ⇒ ν(A) ∼= ν(B), ∀A,B ∈ fin[τ ].
Explicit representation takes fin[τmat ] τvec] to the class of allpolytopes via ν : (A, b) 7→ PA,b .
A representation (τ, ν) is well described if for all A ∈ fin[τ ],〈ν(A)〉 = poly(|A|).
• The explicit representation is trivially well described.
• There are well-described representations with an exponentialnumber of constraints.
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Convex Optimisation – Representation
Representation
A representation (τ, ν) of a class P of polytopes is
• a vocabulary τ , and
• an onto function ν : fin[τ ]→ P which is isomorphisminvariant, i.e., A ∼= B ⇒ ν(A) ∼= ν(B), ∀A,B ∈ fin[τ ].
Explicit representation takes fin[τmat ] τvec] to the class of allpolytopes via ν : (A, b) 7→ PA,b .
A representation (τ, ν) is well described if for all A ∈ fin[τ ],〈ν(A)〉 = poly(|A|).
• The explicit representation is trivially well described.
• There are well-described representations with an exponentialnumber of constraints.
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Convex Optimisation – Representation
Representation
A representation (τ, ν) of a class P of polytopes is
• a vocabulary τ , and
• an onto function ν : fin[τ ]→ P which is isomorphisminvariant, i.e., A ∼= B ⇒ ν(A) ∼= ν(B), ∀A,B ∈ fin[τ ].
Explicit representation takes fin[τmat ] τvec] to the class of allpolytopes via ν : (A, b) 7→ PA,b .
A representation (τ, ν) is well described if for all A ∈ fin[τ ],〈ν(A)〉 = poly(|A|).
• The explicit representation is trivially well described.
• There are well-described representations with an exponentialnumber of constraints.
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Convex Optimisation – Representation, contd.
Expressing Linear Optimisation
Let Pτ,ν be a class of polytopes given by a representation (τ, ν).The linear optimisation problem for Pτ,ν is expressible in FPC ifthere is an FPC interpretation
fin[τ ] τvec]→ fin[τQ ] τvec]
which takes
(A ∈ fin[τ ], vector k) 7→ (rational flag f , point x )
such that
1 f = 0⇒ x ∈ ν(A) with k>x = max{k>y | y ∈ ν(A)},2 f = 1⇒ ν(A) = ∅, or
3 f = 2⇒ ν(A) is unbounded in the direction of k .
An analogous definition can be made for the separation problem.
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Convex Optimisation – Representation, contd.
Expressing Linear Optimisation
Let Pτ,ν be a class of polytopes given by a representation (τ, ν).The linear optimisation problem for Pτ,ν is expressible in FPC ifthere is an FPC interpretation
fin[τ ] τvec]→ fin[τQ ] τvec]which takes
(A ∈ fin[τ ], vector k) 7→ (rational flag f , point x )
such that
1 f = 0⇒ x ∈ ν(A) with k>x = max{k>y | y ∈ ν(A)},2 f = 1⇒ ν(A) = ∅, or
3 f = 2⇒ ν(A) is unbounded in the direction of k .
An analogous definition can be made for the separation problem.
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Convex Optimisation – Representation, contd.
Expressing Linear Optimisation
Let Pτ,ν be a class of polytopes given by a representation (τ, ν).The linear optimisation problem for Pτ,ν is expressible in FPC ifthere is an FPC interpretation
fin[τ ] τvec]→ fin[τQ ] τvec]which takes
(A ∈ fin[τ ], vector k) 7→ (rational flag f , point x )
such that
1 f = 0⇒ x ∈ ν(A) with k>x = max{k>y | y ∈ ν(A)},2 f = 1⇒ ν(A) = ∅, or
3 f = 2⇒ ν(A) is unbounded in the direction of k .
An analogous definition can be made for the separation problem.
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Convex Optimisation – Representation, contd.
Expressing Linear Optimisation
Let Pτ,ν be a class of polytopes given by a representation (τ, ν).The linear optimisation problem for Pτ,ν is expressible in FPC ifthere is an FPC interpretation
fin[τ ] τvec]→ fin[τQ ] τvec]which takes
(A ∈ fin[τ ], vector k) 7→ (rational flag f , point x )
such that
1 f = 0⇒ x ∈ ν(A) with k>x = max{k>y | y ∈ ν(A)},2 f = 1⇒ ν(A) = ∅, or
3 f = 2⇒ ν(A) is unbounded in the direction of k .
An analogous definition can be made for the separation problem.
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Main Result – Linear Programming ∈ FPC
Theorem (c.f., e.g., [GLS88, Theorem 6.4.9])
Let P be a class of well-described polytopes. Then,
linear optimisation on P ≤pT separation on P
We prove an FPC analog.
Theorem
Let Pτ,ν be a class of well-describe polytopes given by τ -structuresand the function ν. Then,
linear optimisation on Pτ,ν ≤FPC separation on Pτ,ν
Corollary (Linear Programming ∈ FPC)
There is an FPC interpretation expressing the linear optimisationproblem w.r.t. the explicit representation.
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Main Result – Linear Programming ∈ FPC
Theorem (c.f., e.g., [GLS88, Theorem 6.4.9])
Let P be a class of well-described polytopes. Then,
linear optimisation on P ≤pT separation on P
We prove an FPC analog.
Theorem
Let Pτ,ν be a class of well-describe polytopes given by τ -structuresand the function ν. Then,
linear optimisation on Pτ,ν ≤FPC separation on Pτ,ν
Corollary (Linear Programming ∈ FPC)
There is an FPC interpretation expressing the linear optimisationproblem w.r.t. the explicit representation.
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Main Result – Linear Programming ∈ FPC
Theorem (c.f., e.g., [GLS88, Theorem 6.4.9])
Let P be a class of well-described polytopes. Then,
linear optimisation on P ≤pT separation on P
We prove an FPC analog.
Theorem
Let Pτ,ν be a class of well-describe polytopes given by τ -structuresand the function ν. Then,
linear optimisation on Pτ,ν ≤FPC separation on Pτ,ν
Corollary (Linear Programming ∈ FPC)
There is an FPC interpretation expressing the linear optimisationproblem w.r.t. the explicit representation.
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Main Result – Maximum Matching ∈ FPC
Maximum Matching Problem
Given: A graph G = (V ,E ) by an incidence matrix {0, 1}V×E .Determine: M ⊆ E such that
1 for all e 6= e ′ ∈ M , |e ∩ e ′| = 0, and
2 |M | is maximum.
There is no canonical maximum matching!
This answers an open question of [Blass-Gurevich-Shelah ’99].
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Main Result – Maximum Matching ∈ FPC
Maximum Matching Problem
Given: A graph G = (V ,E ) by an incidence matrix {0, 1}V×E .Determine: M ⊆ E such that
1 for all e 6= e ′ ∈ M , |e ∩ e ′| = 0, and
2 |M | is maximum.
There is no canonical maximum matching!
This answers an open question of [Blass-Gurevich-Shelah ’99].
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Main Result – Maximum Matching ∈ FPC
Maximum Matching Problem
Given: A graph G = (V ,E ) by an incidence matrix {0, 1}V×E .Determine: M ⊆ E such that
1 for all e 6= e ′ ∈ M , |e ∩ e ′| = 0, and
2 |M | is maximum.
There is no canonical maximum matching!
This answers an open question of [Blass-Gurevich-Shelah ’99].
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Main Result – Maximum Matching ∈ FPC
Maximum Matching Problem
Given: A graph G = (V ,E ) by an incidence matrix {0, 1}V×E .Determine: M ⊆ E such that
1 for all e 6= e ′ ∈ M , |e ∩ e ′| = 0, and
2 |M | is maximum.
There is no canonical maximum matching!
This answers an open question of [Blass-Gurevich-Shelah ’99].
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Main Result – Maximum Matching ∈ FPC
Maximum Matching Problem
Given: A graph G = (V ,E ) by an incidence matrix {0, 1}V×E .Determine: M ⊆ E such that
1 for all e 6= e ′ ∈ M , |e ∩ e ′| = 0, and
2 |M | is maximum.
There is no canonical maximum matching!
This answers an open question of [Blass-Gurevich-Shelah ’99].
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Main Result – Maximum Matching ∈ FPC
Maximum Matching Problem
Given: A graph G = (V ,E ) by an incidence matrix {0, 1}V×E .Determine: M ⊆ E such that
1 for all e 6= e ′ ∈ M , |e ∩ e ′| = 0, and
2 |M | is maximum.
There is no canonical maximum matching!
This answers an open question of [Blass-Gurevich-Shelah ’99].
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Main Result – Maximum Matching ∈ FPC
Maximum Matching Problem
Given: A graph G = (V ,E ) by an incidence matrix {0, 1}V×E .Determine: M ⊆ E such that
1 for all e 6= e ′ ∈ M , |e ∩ e ′| = 0, and
2 |M | is maximum.
There is no canonical maximum matching!
This answers an open question of [Blass-Gurevich-Shelah ’99].
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Main Result – Maximum Matching ∈ FPC
Maximum Matching Problem
Given: A graph G = (V ,E ) by an incidence matrix {0, 1}V×E .Determine: M ⊆ E such that
1 for all e 6= e ′ ∈ M , |e ∩ e ′| = 0, and
2 |M | is maximum.
There is no canonical maximum matching!
This answers an open question of [Blass-Gurevich-Shelah ’99].
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Main Result – Maximum Matching ∈ FPC
Maximum Matching Problem
Given: A graph G = (V ,E ) by an incidence matrix {0, 1}V×E .Determine: M ⊆ E such that
1 for all e 6= e ′ ∈ M , |e ∩ e ′| = 0, and
2 |M | is maximum.
There is no canonical maximum matching!
Theorem
There is an FPC interpretation fin[τmat]→ fin[τQ] which takes aτmat-structure coding a graph G to an integer m indicating thesize of a maximum matching in G .
This answers an open question of [Blass-Gurevich-Shelah ’99].
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Main Result – Maximum Matching ∈ FPC
Maximum Matching Problem
Given: A graph G = (V ,E ) by an incidence matrix {0, 1}V×E .Determine: M ⊆ E such that
1 for all e 6= e ′ ∈ M , |e ∩ e ′| = 0, and
2 |M | is maximum.
There is no canonical maximum matching!
Theorem
There is an FPC interpretation fin[τmat]→ fin[τQ] which takes aτmat-structure coding a graph G to an integer m indicating thesize of a maximum matching in G .
This answers an open question of [Blass-Gurevich-Shelah ’99].
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Proof Overview
Optimization
Separation
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Proof Overview
Optimization
Separation
ExplicitSeparation
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Proof Overview
LinProgOptimization
Separation
ExplicitSeparation
Optimization
SeparationExplicit
Separation
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Proof Overview
LinProgOptimization
Separation
ExplicitSeparation
MaxFlowOptimization
SeparationExplicit
Separation
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Proof Overview
MinCut
Canonical (s, t)-MinCut
LinProgOptimization
Separation
ExplicitSeparation
MaxFlowOptimization
SeparationExplicit
Separation
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Proof Overview
MinCut
MinOddCut
Some canonical MinCut
Canonical (s, t)-MinCut
is MinOddCutLinProgOptimization
Separation
ExplicitSeparation
MaxFlowOptimization
SeparationExplicit
Separation
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Proof Overview
MinCut
MinOddCut
[Padberg-Rao ’82]
Some canonical MinCut
Canonical (s, t)-MinCut
is MinOddCutLinProgOptimization
Separation
ExplicitSeparation
MaxMatchSeparation
MaxFlowOptimization
SeparationExplicit
Separation
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Proof Overview
MinCut
MinOddCut
MaxMatch
[Padberg-Rao ’82]
Some canonical MinCut
Canonical (s, t)-MinCut
is MinOddCutLinProgOptimization
Separation
ExplicitSeparation
MaxMatchSeparation
MaxFlowOptimization
SeparationExplicit
Separation
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Proof Sketch – Optimisation to Separation
Suppose we have a bijection σ : V → [|V |].
• σ induces an isometry QV → Q|V |.• σ + SEP on P + Immerman-Vardi Thm ⇒ OPT on P .
Difficulty: We don’t (or can’t) know σ.
• Elements of V are initially indistinguishable.
Observation: Solving the separation problem may differentiate V .
• Let SEP(P , 0V ) = c.
• Suppose for some u, v ∈ V , cu 6= cv .
• Learn a relative ordering of u and v because cu , cv ∈ Q.
We can use such c to construct an approximate σ.
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Proof Sketch – Optimisation to Separation
Suppose we have a bijection σ : V → [|V |].• σ induces an isometry QV → Q|V |.
• σ + SEP on P + Immerman-Vardi Thm ⇒ OPT on P .
Difficulty: We don’t (or can’t) know σ.
• Elements of V are initially indistinguishable.
Observation: Solving the separation problem may differentiate V .
• Let SEP(P , 0V ) = c.
• Suppose for some u, v ∈ V , cu 6= cv .
• Learn a relative ordering of u and v because cu , cv ∈ Q.
We can use such c to construct an approximate σ.
![Page 75: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/75.jpg)
Proof Sketch – Optimisation to Separation
Suppose we have a bijection σ : V → [|V |].• σ induces an isometry QV → Q|V |.• σ + SEP on P + Immerman-Vardi Thm ⇒ OPT on P .
Difficulty: We don’t (or can’t) know σ.
• Elements of V are initially indistinguishable.
Observation: Solving the separation problem may differentiate V .
• Let SEP(P , 0V ) = c.
• Suppose for some u, v ∈ V , cu 6= cv .
• Learn a relative ordering of u and v because cu , cv ∈ Q.
We can use such c to construct an approximate σ.
![Page 76: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/76.jpg)
Proof Sketch – Optimisation to Separation
Suppose we have a bijection σ : V → [|V |].• σ induces an isometry QV → Q|V |.• σ + SEP on P + Immerman-Vardi Thm ⇒ OPT on P .
Difficulty: We don’t (or can’t) know σ.
• Elements of V are initially indistinguishable.
Observation: Solving the separation problem may differentiate V .
• Let SEP(P , 0V ) = c.
• Suppose for some u, v ∈ V , cu 6= cv .
• Learn a relative ordering of u and v because cu , cv ∈ Q.
We can use such c to construct an approximate σ.
![Page 77: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/77.jpg)
Proof Sketch – Optimisation to Separation
Suppose we have a bijection σ : V → [|V |].• σ induces an isometry QV → Q|V |.• σ + SEP on P + Immerman-Vardi Thm ⇒ OPT on P .
Difficulty: We don’t (or can’t) know σ.
• Elements of V are initially indistinguishable.
Observation: Solving the separation problem may differentiate V .
• Let SEP(P , 0V ) = c.
• Suppose for some u, v ∈ V , cu 6= cv .
• Learn a relative ordering of u and v because cu , cv ∈ Q.
We can use such c to construct an approximate σ.
![Page 78: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/78.jpg)
Proof Sketch – Optimisation to Separation
Suppose we have a bijection σ : V → [|V |].• σ induces an isometry QV → Q|V |.• σ + SEP on P + Immerman-Vardi Thm ⇒ OPT on P .
Difficulty: We don’t (or can’t) know σ.
• Elements of V are initially indistinguishable.
Observation: Solving the separation problem may differentiate V .
• Let SEP(P , 0V ) = c.
• Suppose for some u, v ∈ V , cu 6= cv .
• Learn a relative ordering of u and v because cu , cv ∈ Q.
We can use such c to construct an approximate σ.
![Page 79: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/79.jpg)
Proof Sketch – Optimisation to Separation
Suppose we have a bijection σ : V → [|V |].• σ induces an isometry QV → Q|V |.• σ + SEP on P + Immerman-Vardi Thm ⇒ OPT on P .
Difficulty: We don’t (or can’t) know σ.
• Elements of V are initially indistinguishable.
Observation: Solving the separation problem may differentiate V .
• Let SEP(P , 0V ) = c.
• Suppose for some u, v ∈ V , cu 6= cv .
• Learn a relative ordering of u and v because cu , cv ∈ Q.
We can use such c to construct an approximate σ.
![Page 80: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/80.jpg)
Proof Sketch – Optimisation to Separation
Suppose we have a bijection σ : V → [|V |].• σ induces an isometry QV → Q|V |.• σ + SEP on P + Immerman-Vardi Thm ⇒ OPT on P .
Difficulty: We don’t (or can’t) know σ.
• Elements of V are initially indistinguishable.
Observation: Solving the separation problem may differentiate V .
• Let SEP(P , 0V ) = c.
• Suppose for some u, v ∈ V , cu 6= cv .
• Learn a relative ordering of u and v because cu , cv ∈ Q.
We can use such c to construct an approximate σ.
![Page 81: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/81.jpg)
Proof Sketch – Optimisation to Separation, contd.
Suppose we have σ : V → [n], for n ≤ |V |.
We say c ∈ QV agrees with σ, if σ(u) = σ(v)⇒ cu = cv .
Fold P ⊆ QV into Pσ ⊆ Qn .
• Pσ is a polytope.
• 〈Pσ〉 = poly(〈P〉).
• An optimum of Pσ gives anoptimum of P .
• SEP(Pσ, x ) reduces toSEP(P , x−σ) = c, but...only if c agrees with σ.
σ = ({u, v}, {w})u
v
w
u = v
P
Pσ
x
xσ
![Page 82: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/82.jpg)
Proof Sketch – Optimisation to Separation, contd.
Suppose we have σ : V → [n], for n ≤ |V |.We say c ∈ QV agrees with σ, if σ(u) = σ(v)⇒ cu = cv .
Fold P ⊆ QV into Pσ ⊆ Qn .
• Pσ is a polytope.
• 〈Pσ〉 = poly(〈P〉).
• An optimum of Pσ gives anoptimum of P .
• SEP(Pσ, x ) reduces toSEP(P , x−σ) = c, but...only if c agrees with σ.
σ = ({u, v}, {w})u
v
w
u = v
P
Pσ
x
xσ
![Page 83: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/83.jpg)
Proof Sketch – Optimisation to Separation, contd.
Suppose we have σ : V → [n], for n ≤ |V |.We say c ∈ QV agrees with σ, if σ(u) = σ(v)⇒ cu = cv .
Fold P ⊆ QV into Pσ ⊆ Qn .
• Pσ is a polytope.
• 〈Pσ〉 = poly(〈P〉).
• An optimum of Pσ gives anoptimum of P .
• SEP(Pσ, x ) reduces toSEP(P , x−σ) = c, but...only if c agrees with σ.
σ = ({u, v}, {w})u
v
w
u = v
P
Pσ
x
xσ
![Page 84: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/84.jpg)
Proof Sketch – Optimisation to Separation, contd.
Suppose we have σ : V → [n], for n ≤ |V |.We say c ∈ QV agrees with σ, if σ(u) = σ(v)⇒ cu = cv .
Fold P ⊆ QV into Pσ ⊆ Qn .
• Pσ is a polytope.
• 〈Pσ〉 = poly(〈P〉).
• An optimum of Pσ gives anoptimum of P .
• SEP(Pσ, x ) reduces toSEP(P , x−σ) = c, but...only if c agrees with σ.
σ = ({u, v}, {w})u
v
w
u = v
P
Pσ
x
xσ
![Page 85: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/85.jpg)
Proof Sketch – Optimisation to Separation, contd.
Suppose we have σ : V → [n], for n ≤ |V |.We say c ∈ QV agrees with σ, if σ(u) = σ(v)⇒ cu = cv .
Fold P ⊆ QV into Pσ ⊆ Qn .
• Pσ is a polytope.
• 〈Pσ〉 = poly(〈P〉).
• An optimum of Pσ gives anoptimum of P .
• SEP(Pσ, x ) reduces toSEP(P , x−σ) = c, but...only if c agrees with σ.
σ = ({u, v}, {w})u
v
w
u = v
P
Pσ
x
xσ
![Page 86: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/86.jpg)
Proof Sketch – Optimisation to Separation, contd.
Suppose we have σ : V → [n], for n ≤ |V |.We say c ∈ QV agrees with σ, if σ(u) = σ(v)⇒ cu = cv .
Fold P ⊆ QV into Pσ ⊆ Qn .
• Pσ is a polytope.
• 〈Pσ〉 = poly(〈P〉).
• An optimum of Pσ gives anoptimum of P .
• SEP(Pσ, x ) reduces toSEP(P , x−σ) = c, but...only if c agrees with σ.
σ = ({u, v}, {w})u
v
w
u = v
P
Pσ
x
xσ
![Page 87: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/87.jpg)
Proof Sketch – Optimisation to Separation, contd.
Suppose we have σ : V → [n], for n ≤ |V |.We say c ∈ QV agrees with σ, if σ(u) = σ(v)⇒ cu = cv .
Fold P ⊆ QV into Pσ ⊆ Qn .
• Pσ is a polytope.
• 〈Pσ〉 = poly(〈P〉).
• An optimum of Pσ gives anoptimum of P .
• SEP(Pσ, x ) reduces toSEP(P , x−σ) = c, but...only if c agrees with σ.
σ = ({u, v}, {w})u
v
w
u = v
P
Pσ
x
xσ
![Page 88: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/88.jpg)
Proof Sketch – Optimisation to Separation, contd.
Suppose we have σ : V → [n], for n ≤ |V |.We say c ∈ QV agrees with σ, if σ(u) = σ(v)⇒ cu = cv .
Fold P ⊆ QV into Pσ ⊆ Qn .
• Pσ is a polytope.
• 〈Pσ〉 = poly(〈P〉).
• An optimum of Pσ gives anoptimum of P .
• SEP(Pσ, x ) reduces toSEP(P , x−σ) = c, but...only if c agrees with σ.
σ = ({u, v}, {w})u
v
w
u = v
P
Pσ
x
xσ
![Page 89: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/89.jpg)
Proof Sketch – Optimisation to Separation, contd.
Suppose we have σ : V → [n], for n ≤ |V |.We say c ∈ QV agrees with σ, if σ(u) = σ(v)⇒ cu = cv .
Fold P ⊆ QV into Pσ ⊆ Qn .
• Pσ is a polytope.
• 〈Pσ〉 = poly(〈P〉).
• An optimum of Pσ gives anoptimum of P .
• SEP(Pσ, x ) reduces toSEP(P , x−σ) = c, but...only if c agrees with σ.
σ = ({u, v}, {w})u
v
w
u = v
P
Pσ
x
xσ
![Page 90: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/90.jpg)
Proof Sketch – Optimisation to Separation, contd.
Suppose we have σ : V → [n], for n ≤ |V |.We say c ∈ QV agrees with σ, if σ(u) = σ(v)⇒ cu = cv .
Fold P ⊆ QV into Pσ ⊆ Qn .
• Pσ is a polytope.
• 〈Pσ〉 = poly(〈P〉).
• An optimum of Pσ gives anoptimum of P .
• SEP(Pσ, x ) reduces toSEP(P , x−σ) = c, but...only if c agrees with σ.
σ = ({u, v}, {w})u
v
w
u = v
P
Pσ
x
xσ
![Page 91: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/91.jpg)
Proof Sketch – Optimisation to Separation, contd.
Suppose we have σ : V → [n], for n ≤ |V |.We say c ∈ QV agrees with σ, if σ(u) = σ(v)⇒ cu = cv .
Fold P ⊆ QV into Pσ ⊆ Qn .
• Pσ is a polytope.
• 〈Pσ〉 = poly(〈P〉).
• An optimum of Pσ gives anoptimum of P .
• SEP(Pσ, x ) reduces toSEP(P , x−σ) = c, but...only if c agrees with σ.
σ = ({u, v}, {w})u
v
w
u = v
P
Pσ
x
xσ
![Page 92: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/92.jpg)
Proof Sketch – Optimisation to Separation, contd.
Key Idea Attempt to optimise on Pσ.
• If c = SEP(P , x−σ) always agrees, return eventual optimum.
• Else, refine disagreement of c and σ into σ′ and try again.
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Proof Sketch – Optimisation to Separation, contd.
Key Idea Attempt to optimise on Pσ.
• If c = SEP(P , x−σ) always agrees, return eventual optimum.
• Else, refine disagreement of c and σ into σ′ and try again.
![Page 94: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/94.jpg)
Proof Sketch – Optimisation to Separation, contd.
Key Idea Attempt to optimise on Pσ.
• If c = SEP(P , x−σ) always agrees, return eventual optimum.
• Else, refine disagreement of c and σ into σ′ and try again.
![Page 95: Maximum Matching and Linear Programming in Fixed-Point ...](https://reader034.fdocuments.net/reader034/viewer/2022052016/62866032cf3a341167535e79/html5/thumbnails/95.jpg)
Proof Sketch – Optimisation to Separation, contd.
Key Idea Attempt to optimise on Pσ.
• If c = SEP(P , x−σ) always agrees, return eventual optimum.
• Else, refine disagreement of c and σ into σ′ and try again.
P 0
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Proof Sketch – Optimisation to Separation, contd.
Key Idea Attempt to optimise on Pσ.
• If c = SEP(P , x−σ) always agrees, return eventual optimum.
• Else, refine disagreement of c and σ into σ′ and try again.
P 0 P σ
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Proof Sketch – Optimisation to Separation, contd.
Key Idea Attempt to optimise on Pσ.
• If c = SEP(P , x−σ) always agrees, return eventual optimum.
• Else, refine disagreement of c and σ into σ′ and try again.
P 0 P σ P σ′
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Proof Sketch – Optimisation to Separation, contd.
Key Idea Attempt to optimise on Pσ.
• If c = SEP(P , x−σ) always agrees, return eventual optimum.
• Else, refine disagreement of c and σ into σ′ and try again.
· · ·P 0 P σ P σ′ P
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Summary
Prove FPC analog of P-reduction from optimisation to separation.
Theorem (Main)
Let Pτ,ν be a class of well-describe polytopes given by τ -structuresand the function ν. Then,
linear optimisation on Pτ,ν ≤FPC separation on Pτ,ν
And use it to prove several optimisation problems are in FPC.
Theorem
The follow problems are expressible in FPC:
• LinProg,
• MaxFlow / MinCut,
• MinOddCut, and
• MaxMatch.
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Summary
Prove FPC analog of P-reduction from optimisation to separation.
Theorem (Main)
Let Pτ,ν be a class of well-describe polytopes given by τ -structuresand the function ν. Then,
linear optimisation on Pτ,ν ≤FPC separation on Pτ,ν
And use it to prove several optimisation problems are in FPC.
Theorem
The follow problems are expressible in FPC:
• LinProg,
• MaxFlow / MinCut,
• MinOddCut, and
• MaxMatch.
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Open Questions
• Extend our main reduction to:• quadratic programs,• semidefinite programs, or• convex programs.
• What other problems can be put in FPC?
• Is linear programming complete for FPC under FOinterpretations?
• Do our results provide a route to proving integrality gaps forhierarchies of linear programming relaxations usinginexpressibility in FPC?
Thanks!