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

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 Consistency Open Questions H -TRANSVERSE MATCHINGS An H -TRANSVERSE MATCHING of a graph is a matching Df which covers all copies of H A C 4 -transverse matching.

Introduction Small tree-transverse matchings Local Consistency Open Questions H -TRANSVERSE MATCHINGS An H -TRANSVERSE MATCHING of a graph is a matching Df which covers all copies of H Not a C 4 -transverse matching.

Introduction Small tree-transverse matchings Local Consistency Open Questions MOTIVATION

Introduction Small tree-transverse matchings Local Consistency Open Questions M OTIVATION — GRAPH RAMSEY THEORY An H -TRANSVERSE MATCHING of a graph is a matching Df which covers all copies of H H -transverse matching � edge-colouring with no blue P 3 and no red H .

Introduction Small tree-transverse matchings Local Consistency Open Questions M OTIVATION — GRAPH RAMSEY THEORY An H -TRANSVERSE MATCHING of a graph is a matching Df which covers all copies of H H -transverse matching � edge-colouring with no blue P 3 and no red H . For every fixed H , only finitely many complete graphs Tm have an H -transverse matching.

Introduction Small tree-transverse matchings Local Consistency Open Questions M OTIVATION — TATAMI TILINGS An H -TRANSVERSE MATCHING of a graph is a matching Df which covers all copies of H a TATAMI TILING where no four tiles meet

Introduction Small tree-transverse matchings Local Consistency Open Questions M OTIVATION — TATAMI TILINGS An H -TRANSVERSE MATCHING of a graph is a matching Df which covers all copies of H a TATAMI TILING where no four tiles meet � C 4 -transverse matching in the dual graph

Introduction Small tree-transverse matchings Local Consistency Open Questions M OTIVATION — MONOPOLAR PARTITIONS An H -TRANSVERSE MATCHING of a graph is a matching Df which covers all copies of H Line graph L ( G ) has partition (ind. set, disjoint cliques) � G has a P 4 -transverse matching

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?

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 = C n + 4 NP-c

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 = C n + 4 3-connected NP-c NP-c

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 = C n + 4 3-connected diam 4+ trees NP-c NP-c NP-c

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 = C n + 4 3-connected diam 4+ trees small trees NP-c NP-c NP-c P / ???

Introduction Small tree-transverse matchings Local Consistency Open Questions H = P 4

Introduction Small tree-transverse matchings Local Consistency Open Questions P 4 -TRANSVERSE MATCHINGS — PROPAGATION Huang and Xu: certain paths “propagate” the inclusion of an edge:

Introduction Small tree-transverse matchings Local Consistency Open Questions P 4 -TRANSVERSE MATCHINGS — PROPAGATION Huang and Xu: certain paths “propagate” the inclusion of an edge:

Introduction Small tree-transverse matchings Local Consistency Open Questions P 4 -TRANSVERSE MATCHINGS — PROPAGATION Huang and Xu: certain paths “propagate” the inclusion of an edge:

Introduction Small tree-transverse matchings Local Consistency Open Questions P 4 -TRANSVERSE MATCHINGS — PROPAGATION Huang and Xu: certain paths “propagate” the inclusion of an edge:

Introduction Small tree-transverse matchings Local Consistency Open Questions P 4 -TRANSVERSE MATCHINGS — PROPAGATION Huang and Xu: certain paths “propagate” the inclusion of an edge: Some structures “force” an edge:

Introduction Small tree-transverse matchings Local Consistency Open Questions P 4 -TRANSVERSE MATCHINGS — PROPAGATION Huang and Xu: certain paths “propagate” the inclusion of an edge: Some structures “force” an edge:

Introduction Small tree-transverse matchings Local Consistency Open Questions P 4 -TRANSVERSE MATCHINGS — PROPAGATION Huang and Xu: certain paths “propagate” the inclusion of an edge: Some structures “force” an edge: Any maximal “compatible” set is a P 4 -transverse match- Tm ing (provided one exists) —Churchley + Huang

Introduction Small tree-transverse matchings Local Consistency Open Questions ANOTHER ALGORITHM

Introduction Small tree-transverse matchings Local Consistency Open Questions P 4 -TRANSVERSE MATCHINGS — REDUCTION Work on triangle-transverse matchings by Churchley, Huang, Xu inspired another solution.

Introduction Small tree-transverse matchings Local Consistency Open Questions P 4 -TRANSVERSE MATCHINGS — REDUCTION Work on triangle-transverse matchings by Churchley, Huang, Xu inspired another solution.

Introduction Small tree-transverse matchings Local Consistency Open Questions P 4 -TRANSVERSE MATCHINGS — REDUCTION Work on triangle-transverse matchings by Churchley, Huang, Xu inspired another solution. Finding a P 4 -transverse matching amounts to finding Tm one covering specific vertices.

Introduction Small tree-transverse matchings Local Consistency Open Questions CHAIRS

Introduction Small tree-transverse matchings Local Consistency Open Questions CHAIR-TRANSVERSE MATCHINGS The previous reduction works, but has many more cases.

Introduction Small tree-transverse matchings Local Consistency Open Questions CHAIR-TRANSVERSE MATCHINGS The previous reduction works, but has many more cases.

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?

Introduction Small tree-transverse matchings Local Consistency Open Questions LOCAL CONSISTENCY

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 edges ↔ variables matching ↔ true variables satisfaction problem.

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

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

Introduction Small tree-transverse matchings Local Consistency Open Questions The concept of LOCAL CONSISTENCY CHECKS from the study of CSPs unifies these approaches. g e 1 1. View the problem as a f satisfaction problem. e 2 2. Consider the constraints on each set of k variables. If they imply any simpler ( e 1 ∨ f ∨ g ) ∧ ( e 2 ∨ f ∨ g ) constraints, add them. ∧ ( e 1 ∨ e 2 )

Introduction Small tree-transverse matchings Local Consistency Open Questions The concept of LOCAL CONSISTENCY CHECKS from the study of CSPs unifies these approaches. g g e 1 e 1 f 1. View the problem as a f satisfaction problem. e 2 2. Consider the constraints on each set of k variables. If they imply any simpler ( e 1 ∨ f ∨ g ) ∧ ( e 2 ∨ f ∨ g ) constraints, add them. ∧ ( e 1 ∨ e 2 )

Introduction Small tree-transverse matchings Local Consistency Open Questions The concept of LOCAL CONSISTENCY CHECKS from the study of CSPs unifies these approaches. g g e 1 f 1. View the problem as a f satisfaction problem. e 2 e 2 2. Consider the constraints on each set of k variables. If they imply any simpler ( e 1 ∨ f ∨ g ) ∧ ( e 2 ∨ f ∨ g ) constraints, add them. ∧ ( e 1 ∨ e 2 )

Introduction Small tree-transverse matchings Local Consistency Open Questions The concept of LOCAL CONSISTENCY CHECKS from the study of CSPs unifies these approaches. g e 1 e 1 1. View the problem as a f satisfaction problem. e 2 e 2 2. Consider the constraints on each set of k variables. If they imply any simpler ( e 1 ∨ f ∨ g ) ∧ ( e 2 ∨ f ∨ g ) constraints, add them. ∧ ( e 1 ∨ e 2 )