The Firefighter Problem On the Grid

1. The Firefighter Problem On the Grid Joint work with RaniHod

2. The Firefighter Problem • A complete information solitaire positional game. • Played on a graph • Some vertices are “burning”. • Every turn: • a player protects some vertices • The fire spreads to neighboring vertices. Until the fire spreads no more.

3. Formally: • A graph the board. • A set of burning vertices. • , Set of fire-proof vertices. • A function , , the firefighter function. • Game step : Player picks a set of vertices in . .

4. Questions: • If is finite: • For every , how many vertices can we save? • If is infinite: • For which can we ever stop the fire? • Algorithms.

5. On grids: • Several grids to consider. Namely , , triangular and hexagonal. • For periodic , dimension greater then 2 is not relevant.

6. History: Finite : • Suggested by Hartnel (‘95)as a model for spreading phenomena. • Proven algorithmically hard for trees (FKMR ‘07), but approximable (CVY ‘08). Grids : • Wang and Moeller (‘02): , not enough for • Fogarty (’03): , enough for . • Ng and Raff(‘08): enough for .

7. Our results: • Formally:

8. Demonstration: Fire 2 3 4 5 2 1 2 6 Fire-Proof 4 3 1 1 6 5 1 2 2 5 1 0 1 3 6 4 7 1 1 3 6 8 4 1 7 5 2 7 2 2 4 6 8 2 2 3 5 3 3 3 3 6 5 3 4 7 8 3 3 4 4 5 6 4 4 8 7 5 5 5 7 6 1:4

9. Proof • We show that on if , satisfies t , then a square of fire cannot be stopped. • When we say time : • after the firefighters protected the vertices • before the fire spreads. • The main concept – Potential

10. Definitions • For Define • Define • We denote the fire fronts by(green) ｝ ｝

11. Potential function • endangered: on , not fireproof, and adjacent to a burning point. (if it belongs to two fronts – ½ endangered) • We define as: #endangered on (again corners count as half) • Observation ｝ ｝

12. Potential • We say the front is frozen at time if . Otherwise it is active. • We define to be 1 if is active, 0 otherwise. • We will show that at most one fire front is frozen at any given time. ｝ • Observation ｝

13. Conventions • When we omit fronts subscripts – we sum over all fronts. (example: ) • When we add * - we sum over all times (example: )

14. Dealing with firefighters • Whenever a fireproof vertex is on we say it becomes efficient. • We denote by the number of fireproof vertices which became efficient, on front , at time . • This treats inefficient fireproof vertices as movable. • Observation • A fireproof vertex never contributes to more then 1.

15. Proposition Proof: Let us examine the process: At turn we have burning vertices. These must have at least neighbors. Any of them which are fireproof increase and the rest increase .

16. Lemma 1 • Proof: • Summing over this we get :

17. Relation to length • Summing Lemma 1 over all fronts we get: • Summing over the length relation: • Key inequality

18. Lemma 2 • Suppose then: • Proof of 1: • Proof of 2: if : by 1. Else: • We apply: • To get:

19. End of the Proof • Suppose for all then for all as well and thus: • Proof: • Use induction. , thus by lemma 2No two fire fronts are frozen – that is and thus

20. Open Question