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The Firefighter Problem On the Grid Joint work with Rani Hod

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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.

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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..

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

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

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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:

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Our results: Formally:

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0 Fire Fire- Proof 1:4 11 1 1 1 1 1 1 1 2 2 2 2 2 2 2 22 2 3 33 3 3 33 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 5 55 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 8 8 8 8 Demonstration:

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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

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

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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

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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

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

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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.

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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.

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Lemma 1 Proof: Summing over this we get :

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

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

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End of the Proof Suppose 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

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Open Question

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