The Firefighter Problem On the Grid

Joint work with Rani Hod

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.

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 . .

• If is finite:– For every , how many vertices can we save?

• If is infinite:– For which can we ever stop the fire?

• Algorithms.

Questions:

On grids:

• Several grids to consider. Namely , , triangular and hexagonal.

• For periodic , dimension greater then 2 is not relevant.

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 .

History:

Our results:

• Formally:

0

Fire

Fire-Proof

1:4

1 11

1

1

11

112 2

2

2222222

33 3

3

3 3 3 33333

4 4 4

4

444444

5

5 5 5

555555

666

6

6666

77

7777

8888

4f

Demonstration:

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

Definitions

• For Define

• Define

• We denote the fire fronts by

(green) ｝

｝

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

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

Conventions

• When we omit fronts subscripts – we sum over all fronts. (example: )

• When we add * - we sum over all times (example: )

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.

ObservationA fireproof vertex never contributes to more then 1.

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 .

Lemma 1

Proof:

Summing over this we get :

Key inequality

Relation to length

Summing Lemma 1 over all fronts we get:

Summing over the length relation:

Lemma 2Suppose then:

Proof of 1: Proof of 2: if : by 1. Else:

We apply: To get:

1. 2.

End of the ProofSuppose for all then for all as

well and thus:Proof:Use induction. , thus by lemma 2

No two fire fronts are frozen – that is and thus

Open Question