Even delta-matroids and the complexity of planar Boolean CSPs - - PowerPoint PPT Presentation

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Even delta-matroids and the complexity of planar Boolean CSPs - - PowerPoint PPT Presentation

Dec 28, 2023 118 likes •670 views

Even delta-matroids and the complexity of planar Boolean CSPs Alexandr Kazda, Vladimir Kolmogorov, Michal Rol nek A K, V K, M R (IST Austria) Edge CSP for even -matroids 1 / 12 History Our world: CSP( { 0 , 1 } , ) where contains

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SLIDE 1

Even delta-matroids and the complexity of planar Boolean CSPs

Alexandr Kazda, Vladimir Kolmogorov, Michal Rol´ ınek

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 1 / 12

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SLIDE 2

History

Our world: CSP({0, 1}, Γ) where Γ contains constants {0} and {1}. We limit the instance shape – each variable appears at most k times. For which Γs do we get easier CSP?

  • T. Feder: Fanout limitations on constraint systems, 2001.
  • V. Dalmau, D. Ford: Generalized satisfiability with k occurences per

variable: A study through delta-matroid parity, 2003. Only interesting case: k = 2.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 2 / 12

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SLIDE 3

History

Our world: CSP({0, 1}, Γ) where Γ contains constants {0} and {1}. We limit the instance shape – each variable appears at most k times. For which Γs do we get easier CSP?

  • T. Feder: Fanout limitations on constraint systems, 2001.
  • V. Dalmau, D. Ford: Generalized satisfiability with k occurences per

variable: A study through delta-matroid parity, 2003. Only interesting case: k = 2.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 2 / 12

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SLIDE 4

History

Our world: CSP({0, 1}, Γ) where Γ contains constants {0} and {1}. We limit the instance shape – each variable appears at most k times. For which Γs do we get easier CSP?

  • T. Feder: Fanout limitations on constraint systems, 2001.
  • V. Dalmau, D. Ford: Generalized satisfiability with k occurences per

variable: A study through delta-matroid parity, 2003. Only interesting case: k = 2.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 2 / 12

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SLIDE 5

History

Our world: CSP({0, 1}, Γ) where Γ contains constants {0} and {1}. We limit the instance shape – each variable appears at most k times. For which Γs do we get easier CSP?

  • T. Feder: Fanout limitations on constraint systems, 2001.
  • V. Dalmau, D. Ford: Generalized satisfiability with k occurences per

variable: A study through delta-matroid parity, 2003. Only interesting case: k = 2.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 2 / 12

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SLIDE 6

History

Our world: CSP({0, 1}, Γ) where Γ contains constants {0} and {1}. We limit the instance shape – each variable appears at most k times. For which Γs do we get easier CSP?

  • T. Feder: Fanout limitations on constraint systems, 2001.
  • V. Dalmau, D. Ford: Generalized satisfiability with k occurences per

variable: A study through delta-matroid parity, 2003. Only interesting case: k = 2.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 2 / 12

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SLIDE 7

History

Our world: CSP({0, 1}, Γ) where Γ contains constants {0} and {1}. We limit the instance shape – each variable appears at most k times. For which Γs do we get easier CSP?

  • T. Feder: Fanout limitations on constraint systems, 2001.
  • V. Dalmau, D. Ford: Generalized satisfiability with k occurences per

variable: A study through delta-matroid parity, 2003. Only interesting case: k = 2.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 2 / 12

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SLIDE 8

Edge CSP

Wlog each variable appears in exactly two constrains. We can draw instances of this CSP as graphs with variables = edges. Some people call this binary CSP, we prefer edge CSP. Feder: The only new case is when all relations in Γ are ∆-matroids.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 3 / 12

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SLIDE 9

Edge CSP

Wlog each variable appears in exactly two constrains. We can draw instances of this CSP as graphs with variables = edges. Some people call this binary CSP, we prefer edge CSP. Feder: The only new case is when all relations in Γ are ∆-matroids.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 3 / 12

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SLIDE 10

Edge CSP

Wlog each variable appears in exactly two constrains. We can draw instances of this CSP as graphs with variables = edges. Some people call this binary CSP, we prefer edge CSP. Feder: The only new case is when all relations in Γ are ∆-matroids.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 3 / 12

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SLIDE 11

Edge CSP

Wlog each variable appears in exactly two constrains. We can draw instances of this CSP as graphs with variables = edges. Some people call this binary CSP, we prefer edge CSP. Feder: The only new case is when all relations in Γ are ∆-matroids.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 3 / 12

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SLIDE 12

Edge CSP

Wlog each variable appears in exactly two constrains. We can draw instances of this CSP as graphs with variables = edges. Some people call this binary CSP, we prefer edge CSP. Feder: The only new case is when all relations in Γ are ∆-matroids.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 3 / 12

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SLIDE 13

∆-matroids

AKA “generalized matroids” R = ∅ is an even ∆-matroid if all tuples in R have the same parity and for all α, β ∈ R and for all u variables such that α(u) = β(u) there exists v = u such that α(v) = β(v) and α ⊕ u ⊕ v ∈ R: ∆-matroids: No parity restriction, enought to have α ⊕ u ∈ R instead

  • f α ⊕ u ⊕ v ∈ R.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 4 / 12

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SLIDE 14

∆-matroids

AKA “generalized matroids” R = ∅ is an even ∆-matroid if all tuples in R have the same parity and for all α, β ∈ R and for all u variables such that α(u) = β(u) there exists v = u such that α(v) = β(v) and α ⊕ u ⊕ v ∈ R: ∆-matroids: No parity restriction, enought to have α ⊕ u ∈ R instead

  • f α ⊕ u ⊕ v ∈ R.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 4 / 12

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SLIDE 15

∆-matroids

AKA “generalized matroids” R = ∅ is an even ∆-matroid if all tuples in R have the same parity and for all α, β ∈ R and for all u variables such that α(u) = β(u) there exists v = u such that α(v) = β(v) and α ⊕ u ⊕ v ∈ R: α = 1 1 β = 1 1 1 1 ∆-matroids: No parity restriction, enought to have α ⊕ u ∈ R instead

  • f α ⊕ u ⊕ v ∈ R.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 4 / 12

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SLIDE 16

∆-matroids

AKA “generalized matroids” R = ∅ is an even ∆-matroid if all tuples in R have the same parity and for all α, β ∈ R and for all u variables such that α(u) = β(u) there exists v = u such that α(v) = β(v) and α ⊕ u ⊕ v ∈ R: α = 1 1 β = 1 1 1 1 ∆-matroids: No parity restriction, enought to have α ⊕ u ∈ R instead

  • f α ⊕ u ⊕ v ∈ R.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 4 / 12

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SLIDE 17

∆-matroids

AKA “generalized matroids” R = ∅ is an even ∆-matroid if all tuples in R have the same parity and for all α, β ∈ R and for all u variables such that α(u) = β(u) there exists v = u such that α(v) = β(v) and α ⊕ u ⊕ v ∈ R: α = 1 1 β = 1 1 1 1 ∆-matroids: No parity restriction, enought to have α ⊕ u ∈ R instead

  • f α ⊕ u ⊕ v ∈ R.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 4 / 12

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SLIDE 18

∆-matroids

AKA “generalized matroids” R = ∅ is an even ∆-matroid if all tuples in R have the same parity and for all α, β ∈ R and for all u variables such that α(u) = β(u) there exists v = u such that α(v) = β(v) and α ⊕ u ⊕ v ∈ R: α = 1 1 β = 1 1 1 1 α ⊕ u ⊕ v = 1 1 1 1 ∆-matroids: No parity restriction, enought to have α ⊕ u ∈ R instead

  • f α ⊕ u ⊕ v ∈ R.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 4 / 12

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SLIDE 19

∆-matroids

AKA “generalized matroids” R = ∅ is an even ∆-matroid if all tuples in R have the same parity and for all α, β ∈ R and for all u variables such that α(u) = β(u) there exists v = u such that α(v) = β(v) and α ⊕ u ⊕ v ∈ R: α = 1 1 β = 1 1 1 1 α ⊕ u ⊕ v = 1 1 1 1 ∆-matroids: No parity restriction, enought to have α ⊕ u ∈ R instead

  • f α ⊕ u ⊕ v ∈ R.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 4 / 12

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SLIDE 20

Good and bad news about ∆-matroids

Intersection of two even ∆-matroids need not be a ∆-matroid:        (0 0) (1 1 0) (0 1 1) (1 1 1 1)        ∩        (0 0) (1 1 0) (0 1 1) (1 1 1 1)        = (0 0) (1 1 1 1)

  • If there is any way to use polymorphisms here, we did not find it.

However, (even) ∆-matroids are closed under primitive positive definitions where each bound variable appears exactly twice and each free variable exactly once.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 5 / 12

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SLIDE 21

Good and bad news about ∆-matroids

Intersection of two even ∆-matroids need not be a ∆-matroid:        (0 0) (1 1 0) (0 1 1) (1 1 1 1)        ∩        (0 0) (1 1 0) (0 1 1) (1 1 1 1)        = (0 0) (1 1 1 1)

  • If there is any way to use polymorphisms here, we did not find it.

However, (even) ∆-matroids are closed under primitive positive definitions where each bound variable appears exactly twice and each free variable exactly once.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 5 / 12

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SLIDE 22

Good and bad news about ∆-matroids

Intersection of two even ∆-matroids need not be a ∆-matroid:        (0 0) (1 1 0) (0 1 1) (1 1 1 1)        ∩        (0 0) (1 1 0) (0 1 1) (1 1 1 1)        = (0 0) (1 1 1 1)

  • If there is any way to use polymorphisms here, we did not find it.

However, (even) ∆-matroids are closed under primitive positive definitions where each bound variable appears exactly twice and each free variable exactly once.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 5 / 12

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SLIDE 23

Good and bad news about ∆-matroids

Intersection of two even ∆-matroids need not be a ∆-matroid:        (0 0) (1 1 0) (0 1 1) (1 1 1 1)        ∩        (0 0) (1 1 0) (0 1 1) (1 1 1 1)        = (0 0) (1 1 1 1)

  • If there is any way to use polymorphisms here, we did not find it.

However, (even) ∆-matroids are closed under primitive positive definitions where each bound variable appears exactly twice and each free variable exactly once.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 5 / 12

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SLIDE 24

Edge CSPs generalize perfect matchings

Perfect matchings in graphs: Given G, assign 0 or 1 to each edge so that each vertex of G is incident to exactly one edge labelled by 1. Known to be polynomial (J. Edmonds, 1965). Perfect matchings correspond to edge CSP with constraints of the form {(1 . . . 0) (0 1 . . . 0) (0 1 . . . 0) ... (0 . . . 1)} These are even ∆-matroids!

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 6 / 12

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SLIDE 25

Edge CSPs generalize perfect matchings

Perfect matchings in graphs: Given G, assign 0 or 1 to each edge so that each vertex of G is incident to exactly one edge labelled by 1. Known to be polynomial (J. Edmonds, 1965). Perfect matchings correspond to edge CSP with constraints of the form {(1 . . . 0) (0 1 . . . 0) (0 1 . . . 0) ... (0 . . . 1)} These are even ∆-matroids!

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 6 / 12

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SLIDE 26

Edge CSPs generalize perfect matchings

Perfect matchings in graphs: Given G, assign 0 or 1 to each edge so that each vertex of G is incident to exactly one edge labelled by 1. Known to be polynomial (J. Edmonds, 1965). Perfect matchings correspond to edge CSP with constraints of the form {(1 . . . 0) (0 1 . . . 0) (0 1 . . . 0) ... (0 . . . 1)} These are even ∆-matroids!

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 6 / 12

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SLIDE 27

Edge CSPs generalize perfect matchings

Perfect matchings in graphs: Given G, assign 0 or 1 to each edge so that each vertex of G is incident to exactly one edge labelled by 1. Known to be polynomial (J. Edmonds, 1965). Perfect matchings correspond to edge CSP with constraints of the form {(1 . . . 0) (0 1 . . . 0) (0 1 . . . 0) ... (0 . . . 1)} These are even ∆-matroids!

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 6 / 12

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SLIDE 28

Edge CSPs generalize perfect matchings

Perfect matchings in graphs: Given G, assign 0 or 1 to each edge so that each vertex of G is incident to exactly one edge labelled by 1. Known to be polynomial (J. Edmonds, 1965). Perfect matchings correspond to edge CSP with constraints of the form {(1 . . . 0) (0 1 . . . 0) (0 1 . . . 0) ... (0 . . . 1)} These are even ∆-matroids!

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 6 / 12

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SLIDE 29

Planar CSPs

  • Z. Dvoˇ

r´ ak and M. Kupec: On Planar Boolean CSP, 2015. CSP({0, 1}, Γ) with incidence graphs of instances planar. Constraints – faces of a planar graph, variables – vertices. Dvoˇ r´ ak and Kupec show that all interesting cases of planar CSP can be reduced to edge CSP with ∆-matroid constraints. If there is a polynomial algorithm for edge CSP with even ∆-matroid constraints, we have a dichotomy for planar CSP.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 7 / 12

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SLIDE 30

Planar CSPs

  • Z. Dvoˇ

r´ ak and M. Kupec: On Planar Boolean CSP, 2015. CSP({0, 1}, Γ) with incidence graphs of instances planar. Constraints – faces of a planar graph, variables – vertices. Dvoˇ r´ ak and Kupec show that all interesting cases of planar CSP can be reduced to edge CSP with ∆-matroid constraints. If there is a polynomial algorithm for edge CSP with even ∆-matroid constraints, we have a dichotomy for planar CSP.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 7 / 12

slide-31
SLIDE 31

Planar CSPs

  • Z. Dvoˇ

r´ ak and M. Kupec: On Planar Boolean CSP, 2015. CSP({0, 1}, Γ) with incidence graphs of instances planar. Constraints – faces of a planar graph, variables – vertices. Dvoˇ r´ ak and Kupec show that all interesting cases of planar CSP can be reduced to edge CSP with ∆-matroid constraints. If there is a polynomial algorithm for edge CSP with even ∆-matroid constraints, we have a dichotomy for planar CSP.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 7 / 12

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SLIDE 32

Planar CSPs

  • Z. Dvoˇ

r´ ak and M. Kupec: On Planar Boolean CSP, 2015. CSP({0, 1}, Γ) with incidence graphs of instances planar. Constraints – faces of a planar graph, variables – vertices. Dvoˇ r´ ak and Kupec show that all interesting cases of planar CSP can be reduced to edge CSP with ∆-matroid constraints. If there is a polynomial algorithm for edge CSP with even ∆-matroid constraints, we have a dichotomy for planar CSP.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 7 / 12

slide-33
SLIDE 33

Planar CSPs

  • Z. Dvoˇ

r´ ak and M. Kupec: On Planar Boolean CSP, 2015. CSP({0, 1}, Γ) with incidence graphs of instances planar. Constraints – faces of a planar graph, variables – vertices. Dvoˇ r´ ak and Kupec show that all interesting cases of planar CSP can be reduced to edge CSP with ∆-matroid constraints. If there is a polynomial algorithm for edge CSP with even ∆-matroid constraints, we have a dichotomy for planar CSP.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 7 / 12

slide-34
SLIDE 34

Planar CSPs

  • Z. Dvoˇ

r´ ak and M. Kupec: On Planar Boolean CSP, 2015. CSP({0, 1}, Γ) with incidence graphs of instances planar. Constraints – faces of a planar graph, variables – vertices. Dvoˇ r´ ak and Kupec show that all interesting cases of planar CSP can be reduced to edge CSP with ∆-matroid constraints. If there is a polynomial algorithm for edge CSP with even ∆-matroid constraints, we have a dichotomy for planar CSP.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 7 / 12

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SLIDE 35

Our strategy

We generalize Edmond’s blossom algorithm for perfect matchings. Edge labeling f assigns 0 or 1 to each half-edge: Pair {v, C} where v lies in constraint C so that all constraints are satisfied. Variable is consistent in f if both half edges corresponding to v have the same labels. Edge labeling with all variables consistent = a solution of the instance. We want to augment a given labeling f : Find g labeling with fewer inconsistencies. If f is an edge labeling that can be improved, there is an augmenting f -walk p from one inconsistent variable to another.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 8 / 12

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SLIDE 36

Our strategy

We generalize Edmond’s blossom algorithm for perfect matchings. Edge labeling f assigns 0 or 1 to each half-edge: Pair {v, C} where v lies in constraint C so that all constraints are satisfied. Variable is consistent in f if both half edges corresponding to v have the same labels. Edge labeling with all variables consistent = a solution of the instance. We want to augment a given labeling f : Find g labeling with fewer inconsistencies. If f is an edge labeling that can be improved, there is an augmenting f -walk p from one inconsistent variable to another.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 8 / 12

slide-37
SLIDE 37

Our strategy

We generalize Edmond’s blossom algorithm for perfect matchings. Edge labeling f assigns 0 or 1 to each half-edge: Pair {v, C} where v lies in constraint C so that all constraints are satisfied. Variable is consistent in f if both half edges corresponding to v have the same labels. Edge labeling with all variables consistent = a solution of the instance. We want to augment a given labeling f : Find g labeling with fewer inconsistencies. If f is an edge labeling that can be improved, there is an augmenting f -walk p from one inconsistent variable to another.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 8 / 12

slide-38
SLIDE 38

Our strategy

We generalize Edmond’s blossom algorithm for perfect matchings. Edge labeling f assigns 0 or 1 to each half-edge: Pair {v, C} where v lies in constraint C so that all constraints are satisfied. Variable is consistent in f if both half edges corresponding to v have the same labels. Edge labeling with all variables consistent = a solution of the instance. We want to augment a given labeling f : Find g labeling with fewer inconsistencies. If f is an edge labeling that can be improved, there is an augmenting f -walk p from one inconsistent variable to another.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 8 / 12

slide-39
SLIDE 39

Our strategy

We generalize Edmond’s blossom algorithm for perfect matchings. Edge labeling f assigns 0 or 1 to each half-edge: Pair {v, C} where v lies in constraint C so that all constraints are satisfied. Variable is consistent in f if both half edges corresponding to v have the same labels. Edge labeling with all variables consistent = a solution of the instance. We want to augment a given labeling f : Find g labeling with fewer inconsistencies. If f is an edge labeling that can be improved, there is an augmenting f -walk p from one inconsistent variable to another.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 8 / 12

slide-40
SLIDE 40

Our strategy

We generalize Edmond’s blossom algorithm for perfect matchings. Edge labeling f assigns 0 or 1 to each half-edge: Pair {v, C} where v lies in constraint C so that all constraints are satisfied. Variable is consistent in f if both half edges corresponding to v have the same labels. Edge labeling with all variables consistent = a solution of the instance. We want to augment a given labeling f : Find g labeling with fewer inconsistencies. If f is an edge labeling that can be improved, there is an augmenting f -walk p from one inconsistent variable to another.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 8 / 12

slide-41
SLIDE 41

Our strategy

We generalize Edmond’s blossom algorithm for perfect matchings. Edge labeling f assigns 0 or 1 to each half-edge: Pair {v, C} where v lies in constraint C so that all constraints are satisfied. Variable is consistent in f if both half edges corresponding to v have the same labels. Edge labeling with all variables consistent = a solution of the instance. We want to augment a given labeling f : Find g labeling with fewer inconsistencies. If f is an edge labeling that can be improved, there is an augmenting f -walk p from one inconsistent variable to another.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 8 / 12

slide-42
SLIDE 42

Example

1 1 (1 1 1 1) (1 1 0) (0 1 1) (0 0) (1 0) (0 1) 1 1 1 1 u u′ C D E

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 9 / 12

slide-43
SLIDE 43

Sketch of the algorithm

Take f , search from all inconsistent variables, building a forest of visited variables and constraints. If we can find f -walks u . . . Cv and u′ . . . Dv for u, u′ inconsistent, we can augment and make u, u′ consistent. If we find f -walks u . . . Cv and u . . . Dv, we have found a blossom. This we contract and re-run the algorithm on a smaller instance. If we don’t find any of the above, then f can not be augmented.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 10 / 12

slide-44
SLIDE 44

Sketch of the algorithm

Take f , search from all inconsistent variables, building a forest of visited variables and constraints. If we can find f -walks u . . . Cv and u′ . . . Dv for u, u′ inconsistent, we can augment and make u, u′ consistent. If we find f -walks u . . . Cv and u . . . Dv, we have found a blossom. This we contract and re-run the algorithm on a smaller instance. If we don’t find any of the above, then f can not be augmented.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 10 / 12

slide-45
SLIDE 45

Sketch of the algorithm

Take f , search from all inconsistent variables, building a forest of visited variables and constraints. If we can find f -walks u . . . Cv and u′ . . . Dv for u, u′ inconsistent, we can augment and make u, u′ consistent. If we find f -walks u . . . Cv and u . . . Dv, we have found a blossom. This we contract and re-run the algorithm on a smaller instance. If we don’t find any of the above, then f can not be augmented.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 10 / 12

slide-46
SLIDE 46

Sketch of the algorithm

Take f , search from all inconsistent variables, building a forest of visited variables and constraints. If we can find f -walks u . . . Cv and u′ . . . Dv for u, u′ inconsistent, we can augment and make u, u′ consistent. If we find f -walks u . . . Cv and u . . . Dv, we have found a blossom. This we contract and re-run the algorithm on a smaller instance. If we don’t find any of the above, then f can not be augmented.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 10 / 12

slide-47
SLIDE 47

Sketch of the algorithm

Take f , search from all inconsistent variables, building a forest of visited variables and constraints. If we can find f -walks u . . . Cv and u′ . . . Dv for u, u′ inconsistent, we can augment and make u, u′ consistent. If we find f -walks u . . . Cv and u . . . Dv, we have found a blossom. This we contract and re-run the algorithm on a smaller instance. If we don’t find any of the above, then f can not be augmented.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 10 / 12

slide-48
SLIDE 48

Consequences and future work

We have finished the classification started by Dvoˇ r´ ak and Kupec. To get dichotomy for edge CSPs, all that is needed is to generalize

  • ur argument from even ∆-matroids to all ∆-matroids.

We can go beyond even ∆-matroids and cover many previously known polynomial classes, but there still remains a large gap. We are now begining to look at valued version of edge CSP for even ∆-matroids. Generalization to value sets larger than 2 is going to be hard.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 11 / 12

slide-49
SLIDE 49

Consequences and future work

We have finished the classification started by Dvoˇ r´ ak and Kupec. To get dichotomy for edge CSPs, all that is needed is to generalize

  • ur argument from even ∆-matroids to all ∆-matroids.

We can go beyond even ∆-matroids and cover many previously known polynomial classes, but there still remains a large gap. We are now begining to look at valued version of edge CSP for even ∆-matroids. Generalization to value sets larger than 2 is going to be hard.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 11 / 12

slide-50
SLIDE 50

Consequences and future work

We have finished the classification started by Dvoˇ r´ ak and Kupec. To get dichotomy for edge CSPs, all that is needed is to generalize

  • ur argument from even ∆-matroids to all ∆-matroids.

We can go beyond even ∆-matroids and cover many previously known polynomial classes, but there still remains a large gap. We are now begining to look at valued version of edge CSP for even ∆-matroids. Generalization to value sets larger than 2 is going to be hard.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 11 / 12

slide-51
SLIDE 51

Consequences and future work

We have finished the classification started by Dvoˇ r´ ak and Kupec. To get dichotomy for edge CSPs, all that is needed is to generalize

  • ur argument from even ∆-matroids to all ∆-matroids.

We can go beyond even ∆-matroids and cover many previously known polynomial classes, but there still remains a large gap. We are now begining to look at valued version of edge CSP for even ∆-matroids. Generalization to value sets larger than 2 is going to be hard.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 11 / 12

slide-52
SLIDE 52

Consequences and future work

We have finished the classification started by Dvoˇ r´ ak and Kupec. To get dichotomy for edge CSPs, all that is needed is to generalize

  • ur argument from even ∆-matroids to all ∆-matroids.

We can go beyond even ∆-matroids and cover many previously known polynomial classes, but there still remains a large gap. We are now begining to look at valued version of edge CSP for even ∆-matroids. Generalization to value sets larger than 2 is going to be hard.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 11 / 12

slide-53
SLIDE 53

Consequences and future work

We have finished the classification started by Dvoˇ r´ ak and Kupec. To get dichotomy for edge CSPs, all that is needed is to generalize

  • ur argument from even ∆-matroids to all ∆-matroids.

We can go beyond even ∆-matroids and cover many previously known polynomial classes, but there still remains a large gap. We are now begining to look at valued version of edge CSP for even ∆-matroids. Generalization to value sets larger than 2 is going to be hard.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 11 / 12

slide-54
SLIDE 54

Thank you for your attention.

A K, V K, M R (IST Austria) Edge CSP for even ∆-matroids 12 / 12