TREE-TRANSVERSE MATCHINGS AND LOCAL CONSISTENCY Ross Churchley - - PowerPoint PPT Presentation

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TREE-TRANSVERSE MATCHINGS AND LOCAL CONSISTENCY Ross Churchley - - PowerPoint PPT Presentation

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Introduction Small tree-transverse matchings Local Consistency Open Questions TREE-TRANSVERSE MATCHINGS AND LOCAL CONSISTENCY Ross Churchley Simon Fraser University 13 June 2013 Introduction Small tree-transverse matchings Local

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Introduction Small tree-transverse matchings Local Consistency Open Questions

TREE-TRANSVERSE MATCHINGS AND LOCAL CONSISTENCY

Ross Churchley

Simon Fraser University

13 June 2013

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Introduction Small tree-transverse matchings Local Consistency Open Questions

H-TRANSVERSE MATCHINGS

Df

An H-TRANSVERSE MATCHING of a graph is a matching which covers all copies of H A C4-transverse matching.

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Introduction Small tree-transverse matchings Local Consistency Open Questions

H-TRANSVERSE MATCHINGS

Df

An H-TRANSVERSE MATCHING of a graph is a matching which covers all copies of H Not a C4-transverse matching.

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Introduction Small tree-transverse matchings Local Consistency Open Questions

MOTIVATION

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Introduction Small tree-transverse matchings Local Consistency Open Questions

MOTIVATION — GRAPH RAMSEY THEORY

Df

An H-TRANSVERSE MATCHING of a graph is a matching which covers all copies of H H-transverse matching

  • edge-colouring with no blue P3

and no red H.

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Introduction Small tree-transverse matchings Local Consistency Open Questions

MOTIVATION — GRAPH RAMSEY THEORY

Df

An H-TRANSVERSE MATCHING of a graph is a matching which covers all copies of H H-transverse matching

  • edge-colouring with no blue P3

and no red H.

Tm

For every fixed H, only finitely many complete graphs have an H-transverse matching.

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Introduction Small tree-transverse matchings Local Consistency Open Questions

MOTIVATION — TATAMI TILINGS

Df

An H-TRANSVERSE MATCHING of a graph is a matching which covers all copies of H a TATAMI TILING where no four tiles meet

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Introduction Small tree-transverse matchings Local Consistency Open Questions

MOTIVATION — TATAMI TILINGS

Df

An H-TRANSVERSE MATCHING of a graph is a matching which covers all copies of H a TATAMI TILING where no four tiles meet

  • C4-transverse matching in the

dual graph

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Introduction Small tree-transverse matchings Local Consistency Open Questions

MOTIVATION — MONOPOLAR PARTITIONS

Df

An H-TRANSVERSE MATCHING of a graph is a matching which covers all copies of H Line graph L(G) has partition (ind. set, disjoint cliques)

  • G has a P4-transverse matching
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Introduction Small tree-transverse matchings Local Consistency Open Questions

THE H-TRANSVERSE MATCHING PROBLEM

Q

Does a given graph G admit an H-transverse matching?

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Introduction Small tree-transverse matchings Local Consistency Open Questions

THE H-TRANSVERSE MATCHING PROBLEM

Q

Does a given graph G admit an H-transverse matching? H = Cn+4 NP-c

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Introduction Small tree-transverse matchings Local Consistency Open Questions

THE H-TRANSVERSE MATCHING PROBLEM

Q

Does a given graph G admit an H-transverse matching? H = Cn+4 3-connected NP-c NP-c

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Introduction Small tree-transverse matchings Local Consistency Open Questions

THE H-TRANSVERSE MATCHING PROBLEM

Q

Does a given graph G admit an H-transverse matching? H = Cn+4 3-connected diam 4+ trees NP-c NP-c NP-c

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Introduction Small tree-transverse matchings Local Consistency Open Questions

THE H-TRANSVERSE MATCHING PROBLEM

Q

Does a given graph G admit an H-transverse matching? H = Cn+4 3-connected diam 4+ trees small trees NP-c NP-c NP-c P / ???

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Introduction Small tree-transverse matchings Local Consistency Open Questions

H = P4

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Introduction Small tree-transverse matchings Local Consistency Open Questions

P4-TRANSVERSE MATCHINGS — PROPAGATION

Huang and Xu: certain paths “propagate” the inclusion of an edge:

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Introduction Small tree-transverse matchings Local Consistency Open Questions

P4-TRANSVERSE MATCHINGS — PROPAGATION

Huang and Xu: certain paths “propagate” the inclusion of an edge:

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Introduction Small tree-transverse matchings Local Consistency Open Questions

P4-TRANSVERSE MATCHINGS — PROPAGATION

Huang and Xu: certain paths “propagate” the inclusion of an edge:

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Introduction Small tree-transverse matchings Local Consistency Open Questions

P4-TRANSVERSE MATCHINGS — PROPAGATION

Huang and Xu: certain paths “propagate” the inclusion of an edge:

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Introduction Small tree-transverse matchings Local Consistency Open Questions

P4-TRANSVERSE MATCHINGS — PROPAGATION

Huang and Xu: certain paths “propagate” the inclusion of an edge: Some structures “force” an edge:

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Introduction Small tree-transverse matchings Local Consistency Open Questions

P4-TRANSVERSE MATCHINGS — PROPAGATION

Huang and Xu: certain paths “propagate” the inclusion of an edge: Some structures “force” an edge:

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Introduction Small tree-transverse matchings Local Consistency Open Questions

P4-TRANSVERSE MATCHINGS — PROPAGATION

Huang and Xu: certain paths “propagate” the inclusion of an edge: Some structures “force” an edge:

Tm

Any maximal “compatible” set is a P4-transverse match- ing (provided one exists) —Churchley + Huang

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Introduction Small tree-transverse matchings Local Consistency Open Questions

ANOTHER ALGORITHM

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Introduction Small tree-transverse matchings Local Consistency Open Questions

P4-TRANSVERSE MATCHINGS — REDUCTION

Work on triangle-transverse matchings by Churchley, Huang, Xu inspired another solution.

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Introduction Small tree-transverse matchings Local Consistency Open Questions

P4-TRANSVERSE MATCHINGS — REDUCTION

Work on triangle-transverse matchings by Churchley, Huang, Xu inspired another solution.

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Introduction Small tree-transverse matchings Local Consistency Open Questions

P4-TRANSVERSE MATCHINGS — REDUCTION

Work on triangle-transverse matchings by Churchley, Huang, Xu inspired another solution.

Tm

Finding a P4-transverse matching amounts to finding

  • ne covering specific vertices.
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Introduction Small tree-transverse matchings Local Consistency Open Questions

CHAIRS

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Introduction Small tree-transverse matchings Local Consistency Open Questions

CHAIR-TRANSVERSE MATCHINGS

The previous reduction works, but has many more cases.

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Introduction Small tree-transverse matchings Local Consistency Open Questions

CHAIR-TRANSVERSE MATCHINGS

The previous reduction works, but has many more cases.

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Introduction Small tree-transverse matchings Local Consistency Open Questions

CHAIR-TRANSVERSE MATCHINGS

The previous reduction works, but has many more cases. Ad hoc case analysis is not very nice. Can we interpret these?

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Introduction Small tree-transverse matchings Local Consistency Open Questions

LOCAL CONSISTENCY

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Introduction Small tree-transverse matchings Local Consistency Open Questions

The concept of LOCAL CONSISTENCY CHECKS from the study of CSPs unifies these approaches.

  • 1. View the problem as a

satisfaction problem. edges ↔ variables matching ↔ true variables

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Introduction Small tree-transverse matchings Local Consistency Open Questions

The concept of LOCAL CONSISTENCY CHECKS from the study of CSPs unifies these approaches.

  • 1. View the problem as a

satisfaction problem. edges ↔ variables matching ↔ true variables e f ↓ (e ∨ f)

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Introduction Small tree-transverse matchings Local Consistency Open Questions

The concept of LOCAL CONSISTENCY CHECKS from the study of CSPs unifies these approaches.

  • 1. View the problem as a

satisfaction problem. edges ↔ variables matching ↔ true variables e f ↓ (e ∨ f) e f g ↓ (e ∨ f ∨ g)

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Introduction Small tree-transverse matchings Local Consistency Open Questions

The concept of LOCAL CONSISTENCY CHECKS from the study of CSPs unifies these approaches.

  • 1. View the problem as a

satisfaction problem.

  • 2. Consider the constraints
  • n each set of k variables.

If they imply any simpler constraints, add them. e1 f g e2 (e1 ∨ f ∨ g) ∧ (e2 ∨ f ∨ g) ∧ (e1 ∨ e2)

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Introduction Small tree-transverse matchings Local Consistency Open Questions

The concept of LOCAL CONSISTENCY CHECKS from the study of CSPs unifies these approaches.

  • 1. View the problem as a

satisfaction problem.

  • 2. Consider the constraints
  • n each set of k variables.

If they imply any simpler constraints, add them. e1 f g e2 e1 f g (e1 ∨ f ∨ g) ∧ (e2 ∨ f ∨ g) ∧ (e1 ∨ e2)

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Introduction Small tree-transverse matchings Local Consistency Open Questions

The concept of LOCAL CONSISTENCY CHECKS from the study of CSPs unifies these approaches.

  • 1. View the problem as a

satisfaction problem.

  • 2. Consider the constraints
  • n each set of k variables.

If they imply any simpler constraints, add them. e1 f g e2 f g e2 (e1 ∨ f ∨ g) ∧ (e2 ∨ f ∨ g) ∧ (e1 ∨ e2)

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Introduction Small tree-transverse matchings Local Consistency Open Questions

The concept of LOCAL CONSISTENCY CHECKS from the study of CSPs unifies these approaches.

  • 1. View the problem as a

satisfaction problem.

  • 2. Consider the constraints
  • n each set of k variables.

If they imply any simpler constraints, add them. e1 f g e2 e1 e2 (e1 ∨ f ∨ g) ∧ (e2 ∨ f ∨ g) ∧ (e1 ∨ e2)

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Introduction Small tree-transverse matchings Local Consistency Open Questions

The concept of LOCAL CONSISTENCY CHECKS from the study of CSPs unifies these approaches.

  • 1. View the problem as a

satisfaction problem.

  • 2. Consider the constraints
  • n each set of k variables.

If they imply any simpler constraints, add them. e1 f g e2 (e1 ∨ f ∨ g) ∧ (e2 ∨ f ∨ g) ∧ (e1 ∨ e2) ⇓ (f ∨ g)

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Introduction Small tree-transverse matchings Local Consistency Open Questions

The concept of LOCAL CONSISTENCY CHECKS from the study of CSPs unifies these approaches.

  • 1. View the problem as a

satisfaction problem.

  • 2. Consider the constraints
  • n each set of k variables.

If they imply any simpler constraints, add them.

  • 3. Continue until equivalent

to some well-known problem. Only three cases: two simplify to clauses of size 2.

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Introduction Small tree-transverse matchings Local Consistency Open Questions

The concept of LOCAL CONSISTENCY CHECKS from the study of CSPs unifies these approaches.

  • 1. View the problem as a

satisfaction problem.

  • 2. Consider the constraints
  • n each set of k variables.

If they imply any simpler constraints, add them.

  • 3. Continue until equivalent

to some well-known problem. Only three cases: two simplify to clauses of size 2. The remaining P4 clauses are satisfied by any maximal solution to the 2SAT instance.

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Introduction Small tree-transverse matchings Local Consistency Open Questions

The concept of LOCAL CONSISTENCY CHECKS from the study of CSPs unifies these approaches.

  • 1. View the problem as a

satisfaction problem.

  • 2. Consider the constraints
  • n each set of k variables.

If they imply any simpler constraints, add them.

  • 3. Continue until equivalent

to some well-known problem. e f h g ↓ (e ∨ f ∨ g ∨ h)

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Introduction Small tree-transverse matchings Local Consistency Open Questions

The concept of LOCAL CONSISTENCY CHECKS from the study of CSPs unifies these approaches.

  • 1. View the problem as a

satisfaction problem.

  • 2. Consider the constraints
  • n each set of k variables.

If they imply any simpler constraints, add them.

  • 3. Continue until equivalent

to some well-known problem. e′ f h g e ⇓ (f ∨ h)

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Introduction Small tree-transverse matchings Local Consistency Open Questions

The concept of LOCAL CONSISTENCY CHECKS from the study of CSPs unifies these approaches.

  • 1. View the problem as a

satisfaction problem.

  • 2. Consider the constraints
  • n each set of k variables.

If they imply any simpler constraints, add them.

  • 3. Continue until equivalent

to some well-known problem. Unlike the P4-transverse matching problem, the clauses may be larger than two variables.

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Introduction Small tree-transverse matchings Local Consistency Open Questions

The concept of LOCAL CONSISTENCY CHECKS from the study of CSPs unifies these approaches.

  • 1. View the problem as a

satisfaction problem.

  • 2. Consider the constraints
  • n each set of k variables.

If they imply any simpler constraints, add them.

  • 3. Continue until equivalent

to some well-known problem. Unlike the P4-transverse matching problem, the clauses may be larger than two variables. Most simplified clauses are edges surrounding a vertex. The others are also expressible by matching covers.

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Introduction Small tree-transverse matchings Local Consistency Open Questions

CONSTRUCTING OBSTRUCTIONS

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Introduction Small tree-transverse matchings Local Consistency Open Questions

CONSTRUCTING OBSTRUCTIONS

Obstruction for simplified SAT problem Translation between graph and clause structures

Tm

AgraphhasaP4-transversematchingifandonlyifitdoes not have a specific configuration of “well-formed walks.”

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Introduction Small tree-transverse matchings Local Consistency Open Questions

OPEN QUESTIONS

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Introduction Small tree-transverse matchings Local Consistency Open Questions

OPEN QUESTIONS

Q

What is the complexity of the remaining tree-transverse matching problems?

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Introduction Small tree-transverse matchings Local Consistency Open Questions

OPEN QUESTIONS

Q

What is the complexity of the remaining tree-transverse matching problems? The T-transverse matching problem is...

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Introduction Small tree-transverse matchings Local Consistency Open Questions

OPEN QUESTIONS

Q

What is the complexity of the remaining tree-transverse matching problems? The T-transverse matching problem is...

◮ polynomial when T = P4 or T is the chair

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Introduction Small tree-transverse matchings Local Consistency Open Questions

OPEN QUESTIONS

Q

What is the complexity of the remaining tree-transverse matching problems? The T-transverse matching problem is...

◮ polynomial when T = P4 or T is the chair ◮ NP-complete when T is a tree of diameter ≥ 4

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Introduction Small tree-transverse matchings Local Consistency Open Questions

OPEN QUESTIONS

Q

What is the complexity of the remaining tree-transverse matching problems? The T-transverse matching problem is...

◮ polynomial when T = P4 or T is the chair ◮ NP-complete when T is a tree of diameter ≥ 4 ◮ polynomial for triangle-free inputs when T has diameter 3

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Introduction Small tree-transverse matchings Local Consistency Open Questions

OPEN QUESTIONS

Q

What is the complexity of the remaining tree-transverse matching problems? A cross-transverse matching of this graph contains 0, 1, 2, or 4 green edges. Impossible to express with matching covers: new ideas are needed.

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Introduction Small tree-transverse matchings Local Consistency Open Questions

OPEN QUESTIONS

The algebraic characterization for CSPs solvable by local implications doesn’t apply to H-transverse matching problems.

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Introduction Small tree-transverse matchings Local Consistency Open Questions

OPEN QUESTIONS

The algebraic characterization for CSPs solvable by local implications doesn’t apply to H-transverse matching problems.

Q

Find a sufficient condition for a restricted-domain CSP to be solvable by local implications.

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Introduction Small tree-transverse matchings Local Consistency Open Questions

TREE-TRANSVERSE MATCHINGS AND LOCAL CONSISTENCY

Ross Churchley

Simon Fraser University

13 June 2013