The Cinderella game on holes and anti-holes. Introduction - PowerPoint PPT Presentation

The Cinderella game on holes and anti-holes. Introduction Definitions Example game Marijke H.L. Bodlaender 1 Cor A.J. Hurkens 2 The game on general graphs Gerhard J. Woeginger 2 The game on holes Conjectures 22 June 2011 1 Department of Information and Computing Sciences, Universiteit Utrecht, The Netherlands 2 Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands 1

Introduction Introduction Definitions Definitions Example game The game on general graphs Example game The game on holes Conjectures The game on general graphs The game on holes Conjectures 2

Introduction to the game Introduction Definitions Example game The game on general graphs The game on holes Conjectures ◮ Proposed problem for the International Mathematical Olympiad ◮ We study variant where water arrives in rounds and the game board is an undirected graph 3

The game Introduction Definitions ◮ Game played on undirected simple graph G = ( V , E ) Example game ◮ Every vertex v contains a bucket The game on general graphs ◮ Every edge [ u , v ] ∈ E indicates an incompatibility The game on ◮ In every round the Stepmother distributes a liter of water holes Conjectures in the buckets ◮ Cinderella empties the buckets in an independent set ◮ Stepmother tries to reach an overflow ◮ Cinderella wants to avoid an overflow 4

Some definitions and notation Introduction Definitions ◮ bucket ′ ( G ): infimum of all bucket sizes Cinderella needs Example game to win The game on general graphs ◮ bucket ( G ) : bucket ′ ( G ) − 1 is the bucket number of G The game on holes ◮ GREEDY: empty maximum weight independent set every Conjectures turn ◮ g-bucket ( G ): bucket number of G when Cinderella uses a GREEDY strategy 5

Definitions Introduction ◮ x = ( x v ) v ∈ V where x v is the contents of bucket v at the Definitions Example game start of a round The game on ◮ x ( S ) = � v ∈ S x v general graphs The game on ◮ y v the contents of v after the Stepmother moved holes ◮ χ ( G ) the chromatic number of G Conjectures ◮ ω ( G ) the clique number of G ◮ For S ⊆ V we write χ ( S ) ◮ H � k � = 1 + 1 2 + 1 3 + 1 4 + . . . . . . + 1 k 6

Example with bucket size 1.6 Introduction Definitions ◮ x 1 = x 2 = x 3 = 0 Example game The game on general graphs The game on holes Conjectures 7

Example with bucket size 1.6 Introduction Definitions ◮ y 1 = y 2 = y 3 = 1 / 3 Example game The game on general graphs The game on holes Conjectures 8

Example with bucket size 1.6 Introduction Definitions ◮ x 1 = 1 / 3 Example game ◮ x 2 = x 3 = 0 The game on general graphs The game on holes Conjectures 9

Example with bucket size 1.6 Introduction Definitions ◮ y 1 = y 2 = 2 / 3 Example game ◮ y 3 = 0 The game on general graphs The game on holes Conjectures 10

Example with bucket size 1.6 Introduction Definitions ◮ x 1 = 2 / 3 Example game ◮ x 2 = x 3 = 0 The game on general graphs The game on holes Conjectures 11

Example with bucket size 1.6 Introduction Definitions ◮ y 1 = 5 / 3 > 1 . 6 Example game ◮ y 2 = y 3 = 0 The game on general graphs The game on holes Conjectures 12

Results in this paper Introduction Definitions ◮ g-bucket ( G ) ≤ H � χ ( G ) − 1 � Example game ◮ bucket ( G ) ≥ H � ω ( G ) − 1 � The game on general graphs ◮ bucket ( G ) = H � ω ( G ) − 1 � ∀ graphs on n ≤ 6 vertices The game on holes ◮ bucket ( C 2 m +1 ) = 1 Conjectures ◮ g-bucket ( C 2 m +1 ) = 1 + 1 m · 2 − m � � ◮ g-bucket C 2 m +1 ≤ H � m � − 1 / (2 m ) ≥ H � m − 1 � + m 2 − 3 m +1 � � ◮ g-bucket C 2 m +1 2 m 2 ( m − 1) 13

Upper bound on general graphs Introduction Definitions Theorem Example game Every graph G = ( V , E ) satisfies g-bucket ( G ) ≤ H � χ ( G ) − 1 � The game on general graphs Proof. The game on holes GREEDY maintains the following system of invariants Conjectures x ( S ) < χ ( S ) · ( 1 + H � χ ( G ) − 1 � − H � χ ( S ) � ) for all sets S ⊆ V (1) Apply (1) to S = { v } to show x v < H � χ ( G ) − 1 � 14

Upper bound for GREEDY on general graphs continued Introduction Definitions If χ ( S ) = χ ( G ), then Example game The game on y ( V − I ) ≤ χ ( G ) − 1 general graphs y ( S ) ≤ y ( V ) χ ( G ) The game on holes χ ( G ) − 1 Conjectures ≤ ( x ( V ) + 1) < χ ( G ) − 1 χ ( G ) Assume that χ ( S ) < χ ( G ) observe that y ( S ) ≤ χ ( S ) · y ( I ) (2) 15

Upper bound for GREEDY on general graphs continued Introduction Furthermore Definitions Example game x ( S ∪ I ) < ( χ ( S ) + 1 ) · ( 1 + H � χ ( G ) − 1 � − H � χ ( S ) + 1 � ) The game on (3) general graphs Applying (2) and (3) we derive The game on holes Conjectures χ ( S ) χ ( S ) y ( S ) ≤ χ ( S ) + 1 ( y ( S ) + y ( I )) ≤ χ ( S ) + 1 ( x ( S ∪ I ) + 1) � 1 � < χ ( S ) · 1 + H � χ ( G ) − 1 � − H � χ ( S ) + 1 � + χ ( S ) + 1 = χ ( S ) · (1 + H � χ ( G ) − 1 � − H � χ ( S ) � ) 16

Lower bound on general graphs Theorem Introduction Every graph G = ( V , E ) satisfies bucket ( G ) ≥ H � ω ( G ) − 1 � Definitions Example game Let ω ( G ) = n The game on general graphs Define a strategy for the Stepmother: The game on ◮ Play game on the the largest clique, K holes ◮ At the first phase: Conjectures • Fill repeatedly all buckets in K to the same level • This converges to 1 − ǫ ◮ In second phase • In r -th round fill n − r fullest buckets to the same level • At the end of round n − 2 at least one bucket contains H � n − 1 � − ǫ 17

Perfect graphs Introduction Definitions Example game The game on general graphs Theorem The game on Every perfect graph G has holes bucket ( G ) = g-bucket ( G ) = H � ω ( G ) − 1 � Conjectures 18

GREEDY odd holes: Upper bound Theorem The odd cycle C 2 m +1 has g-bucket ( C 2 m +1 ) ≤ 1 + 1 m · 2 − m Introduction Definitions Proof(Upper bound). Example game GREEDY maintains the following invariants The game on general graphs The game on 2 m +1 x i < m + 1 holes � Conjectures m i =1 k +2 t − 1 x i < 1 + 1 � m · 2 t − m for 1 ≤ k ≤ 2 m + 1, 1 ≤ t ≤ m i = k x k < 1 + 1 m · 2 − m for 1 ≤ k ≤ 2 m + 1 19

GREEDY odd holes: Lower bound Theorem Introduction The odd cycle C 2 m +1 has g-bucket ( C 2 m +1 ) ≥ 1 + 1 m · 2 − m Definitions Proof. Example game The game on First phase: Fill repeatedly all buckets to the same level general graphs The game on Second phase: holes B 1 B 2 B 3 B 2 m − 3 B 2 m − 2 B 2 m − 1 B 2 m . . . Conjectures 1 1 1 1 1 1 1 SL . . . m m m m m m m 1 1 1 1 CL 0 0 0 . . . m m m m 1 1 SL 0 . . . α 1 α 1 α 1 α 1 m m 1 CL 0 0 0 0 . . . α 1 α 1 m B k +1 B k +2 B 2 m − k − 1 B 2 m − k B 2 m − k +1 B 2 m − k +2 . . . . . . 1 1 CL 0 0 0 . . . α k − 1 α k − 1 m m 1 1 SL 0 . . . α k α k α k α k m m 1 CL 0 0 0 0 . . . α k α k m 20

GREEDY odd holes: Lower bound (continued) Third phase: Introduction Definitions . . . B m +1 B m +2 . . . Example game CL 0 α m − 1 α m − 1 0 The game on α m − 1 + 1 α m − 1 + 1 SL 0 0 general graphs 2 2 α m − 1 + 1 CL 0 0 0 The game on 2 holes Conjectures The alpha values solve to 1 � k + 1 + 2 − k � α k = ⇒ 2 m α m − 1 + 1 1 + 1 m · 2 − m = 2 21

Possible future research Introduction Definitions Conjecture Example game Every graph G satisfies bucket ( G ) = H � ω ( G ) − 1 � The game on general graphs Conjecture The game on holes A graph G is perfect, if and only if bucket ( G ) = g-bucket ( G ) Conjectures Conjecture The difference between g-bucket ( G ) and bucket ( G ) is bounded by an absolute constant (that does not depend on G) 22