New Algorithms for Planar Subgraph Isomorphism Jesper Nederlof

New Algorithms for Planar Subgraph Isomorphism Jesper Nederlof (Eindhoven University of Technology)

Bidimensionality • • • Feedback Vertex Set Cycle Packing Connected Vertex Cover etc. . We omit factors poly in input size

Proof Sketch Planar Grid Minor Theorem

Proof Sketch Planar Grid Minor Theorem N E W S

Proof Sketch Planar Grid Minor Theorem N E W S

Beyond Bidimensionality by the usual suspects Theorem(FLMPPS, FOCS’ 16) •

Beyond Bidimensionality by the usual suspects Theorem(FLMPPS, FOCS’ 16) • (in [FLMPPS]) (in several open problem sessions / talks)

Our Results: Planar Subgraph Isomorphism •

Our Results: Planar Subgraph Isomorphism • ETH-tight!

Count Patterns in Planar Graphs in sub-exponential time Jesper 1. 2. 3. 4. ‘Efficient’ Inclusion-Exclusion Lemma from [FLMPPS (FOCS’ 16)] to sparsify balanced separators Clever Divide&Conquer Technique from [BNvd. Z (ICALP’ 16)]

Outline •

Toy Example: #k-IS on Grids (inspired by FLMPPS’ 16)

Toy Example: #k-IS on Grids (inspired by FLMPPS’ 16) Proof Sketch 1 2 3 1 2

Toy Example: #k-IS on Grids (inspired by FLMPPS’ 16) Proof Sketch

Outline •

Menger-like Theorem Lemma[FLMPPS, FOCS’ 16] • s t or s t

Balanced Cycle Separators in Planar Graphs •

Lemma Weight-Oblivious Balanced Separator Sparsification (WOBSS) •

WOBSS algorithm N • Case 1: E s t W S

WOBSS algorithm • Case 1: s Case 2: t

WOBSS algorithm • Case 1: Case 2: Not necessarily balanced!! Fortunately, we can choose the part we continue with

Outline •

Obstacle 1: Boundary Increases •

Obstacle 2: Slow Run Time •

Obstacle 3: Only approximates • s t or s t

Obstacle 4: Other patterns? How to count occurrences of patterns? pattern

Conclusion • 2 main new ideas: • `Efficient Inclusion-Exclusion’ • Lemma to construct family of balanced cycle separator that have small intersection with any pattern Thanks a lot for your attention!