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Containment Strategies in Network Models The Firefighter Problem and - - PowerPoint PPT Presentation

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Containment Strategies in Network Models The Firefighter Problem and Some Variations Lise E. Holte, Ryan M. Wagner, Daniel P . Biebighauser Concordia College, Moorhead, MN February 8th, 2011 1 Outline Introduction 1 Introduction to the

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Containment Strategies in Network Models

The Firefighter Problem and Some Variations Lise E. Holte, Ryan M. Wagner, Daniel P . Biebighauser

Concordia College, Moorhead, MN

February 8th, 2011

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Outline

1

Introduction Introduction to the Firefighter Problem Integer Programming

2

Flood Model

3

Untouchable Vertices

4

Directed Grids Hall-Like Theorem and a Quarter Plane Regular Directed Grids

5

Moveable Firefighters Integer Programming Results

6

Conclusion and Future Research

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Outline

1

Introduction Introduction to the Firefighter Problem Integer Programming

2

Flood Model

3

Untouchable Vertices

4

Directed Grids Hall-Like Theorem and a Quarter Plane Regular Directed Grids

5

Moveable Firefighters Integer Programming Results

6

Conclusion and Future Research

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Introduction to the Firefighter Problem

Graph theory problem introduced by Bert Hartnell in 1995 Focuses on finding containment strategies for an undesirable spread Examples:

Fire and Firefighters Flood and Sandbaggers Virus and Vaccine

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Introduction to the Firefighter Problem

Graph theory problem introduced by Bert Hartnell in 1995 Focuses on finding containment strategies for an undesirable spread Examples:

Fire and Firefighters Flood and Sandbaggers Virus and Vaccine

4

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Introduction to the Firefighter Problem

Graph theory problem introduced by Bert Hartnell in 1995 Focuses on finding containment strategies for an undesirable spread Examples:

Fire and Firefighters Flood and Sandbaggers Virus and Vaccine

4

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SLIDE 7

Introduction to the Firefighter Problem

Graph theory problem introduced by Bert Hartnell in 1995 Focuses on finding containment strategies for an undesirable spread Examples:

Fire and Firefighters Flood and Sandbaggers Virus and Vaccine

4

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SLIDE 8

Introduction to the Firefighter Problem

Graph theory problem introduced by Bert Hartnell in 1995 Focuses on finding containment strategies for an undesirable spread Examples:

Fire and Firefighters Flood and Sandbaggers Virus and Vaccine

4

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Introduction to the Firefighter Problem

Graph theory problem introduced by Bert Hartnell in 1995 Focuses on finding containment strategies for an undesirable spread Examples:

Fire and Firefighters Flood and Sandbaggers Virus and Vaccine

4

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Basic Graph Theory Definitions

Graph: A set of vertices and any edges connecting them, where:

a vertex is any point in a graph, and an edge is a “line” connecting two vertices of a graph.

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Description of the Firefighter Problem

Fire breaks out at one or more vertices at time t = 0 Firefighters are placed on some empty (non-burning) vertices at time t = 1 Fire spreads from burning vertices to undefended neighboring vertices Additional firefighters are placed on empty (non-burning and undefended) vertices at time t = 2

Previously defended vertices remain defended

Fire spreads again Our objective is to contain the spread of the fire by repeating this process until the fire can no longer spread

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Description of the Firefighter Problem

Fire breaks out at one or more vertices at time t = 0 Firefighters are placed on some empty (non-burning) vertices at time t = 1 Fire spreads from burning vertices to undefended neighboring vertices Additional firefighters are placed on empty (non-burning and undefended) vertices at time t = 2

Previously defended vertices remain defended

Fire spreads again Our objective is to contain the spread of the fire by repeating this process until the fire can no longer spread

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Description of the Firefighter Problem

Fire breaks out at one or more vertices at time t = 0 Firefighters are placed on some empty (non-burning) vertices at time t = 1 Fire spreads from burning vertices to undefended neighboring vertices Additional firefighters are placed on empty (non-burning and undefended) vertices at time t = 2

Previously defended vertices remain defended

Fire spreads again Our objective is to contain the spread of the fire by repeating this process until the fire can no longer spread

6

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Description of the Firefighter Problem

Fire breaks out at one or more vertices at time t = 0 Firefighters are placed on some empty (non-burning) vertices at time t = 1 Fire spreads from burning vertices to undefended neighboring vertices Additional firefighters are placed on empty (non-burning and undefended) vertices at time t = 2

Previously defended vertices remain defended

Fire spreads again Our objective is to contain the spread of the fire by repeating this process until the fire can no longer spread

6

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Description of the Firefighter Problem

Fire breaks out at one or more vertices at time t = 0 Firefighters are placed on some empty (non-burning) vertices at time t = 1 Fire spreads from burning vertices to undefended neighboring vertices Additional firefighters are placed on empty (non-burning and undefended) vertices at time t = 2

Previously defended vertices remain defended

Fire spreads again Our objective is to contain the spread of the fire by repeating this process until the fire can no longer spread

6

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Description of the Firefighter Problem

Fire breaks out at one or more vertices at time t = 0 Firefighters are placed on some empty (non-burning) vertices at time t = 1 Fire spreads from burning vertices to undefended neighboring vertices Additional firefighters are placed on empty (non-burning and undefended) vertices at time t = 2

Previously defended vertices remain defended

Fire spreads again Our objective is to contain the spread of the fire by repeating this process until the fire can no longer spread

6

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Description of the Firefighter Problem

Fire breaks out at one or more vertices at time t = 0 Firefighters are placed on some empty (non-burning) vertices at time t = 1 Fire spreads from burning vertices to undefended neighboring vertices Additional firefighters are placed on empty (non-burning and undefended) vertices at time t = 2

Previously defended vertices remain defended

Fire spreads again Our objective is to contain the spread of the fire by repeating this process until the fire can no longer spread

6

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Introduction to the Firefighter Problem

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Introduction to the Firefighter Problem

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Introduction to the Firefighter Problem

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Introduction to the Firefighter Problem

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Introduction to the Firefighter Problem

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Introduction to the Firefighter Problem

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Introduction to the Firefighter Problem

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Introduction to the Firefighter Problem

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Introduction to the Firefighter Problem

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Introduction to the Firefighter Problem

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Introduction to the Firefighter Problem

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Introduction to the Firefighter Problem

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Introduction to the Firefighter Problem

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Introduction to the Firefighter Problem

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Introduction to the Firefighter Problem

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Introduction to the Firefighter Problem

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Earlier Work

Wang and Moeller (2002) proved that two firefighters per time step is sufficient to contain any single outbreak of fire in a two dimensional infinite grid Hartke (2004) confirmed this result and also proved that this could be achieved in eight time steps, losing eighteen vertices (these numbers are both optimal) Fogarty (2003) proved that two firefighters per time step is sufficient to contain any finite outbreak of fire in these grids

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Earlier Work

Wang and Moeller (2002) proved that two firefighters per time step is sufficient to contain any single outbreak of fire in a two dimensional infinite grid Hartke (2004) confirmed this result and also proved that this could be achieved in eight time steps, losing eighteen vertices (these numbers are both optimal) Fogarty (2003) proved that two firefighters per time step is sufficient to contain any finite outbreak of fire in these grids

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Earlier Work

Wang and Moeller (2002) proved that two firefighters per time step is sufficient to contain any single outbreak of fire in a two dimensional infinite grid Hartke (2004) confirmed this result and also proved that this could be achieved in eight time steps, losing eighteen vertices (these numbers are both optimal) Fogarty (2003) proved that two firefighters per time step is sufficient to contain any finite outbreak of fire in these grids

8

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Integer Programs

Mathematical Programming (Optimization subject to constraints) Integer Programming Hartke gave an integer program for the Firefighter Problem Notation for the integer program:

bx,t is a Boolean variable that is 1 if vertex x is burning at time t and 0 otherwise dx,t is a Boolean variable that is 1 if vertex x is defended at time t and 0 otherwise T is the number of the final time step L is the graph under consideration (here a grid) For any set of vertices S, N(S) is the set of neighbors of S in L

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Integer Programs

Mathematical Programming (Optimization subject to constraints) Integer Programming Hartke gave an integer program for the Firefighter Problem Notation for the integer program:

bx,t is a Boolean variable that is 1 if vertex x is burning at time t and 0 otherwise dx,t is a Boolean variable that is 1 if vertex x is defended at time t and 0 otherwise T is the number of the final time step L is the graph under consideration (here a grid) For any set of vertices S, N(S) is the set of neighbors of S in L

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Integer Programs

Mathematical Programming (Optimization subject to constraints) Integer Programming Hartke gave an integer program for the Firefighter Problem Notation for the integer program:

bx,t is a Boolean variable that is 1 if vertex x is burning at time t and 0 otherwise dx,t is a Boolean variable that is 1 if vertex x is defended at time t and 0 otherwise T is the number of the final time step L is the graph under consideration (here a grid) For any set of vertices S, N(S) is the set of neighbors of S in L

9

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Integer Programs

Mathematical Programming (Optimization subject to constraints) Integer Programming Hartke gave an integer program for the Firefighter Problem Notation for the integer program:

bx,t is a Boolean variable that is 1 if vertex x is burning at time t and 0 otherwise dx,t is a Boolean variable that is 1 if vertex x is defended at time t and 0 otherwise T is the number of the final time step L is the graph under consideration (here a grid) For any set of vertices S, N(S) is the set of neighbors of S in L

9

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Integer Programs

Mathematical Programming (Optimization subject to constraints) Integer Programming Hartke gave an integer program for the Firefighter Problem Notation for the integer program:

bx,t is a Boolean variable that is 1 if vertex x is burning at time t and 0 otherwise dx,t is a Boolean variable that is 1 if vertex x is defended at time t and 0 otherwise T is the number of the final time step L is the graph under consideration (here a grid) For any set of vertices S, N(S) is the set of neighbors of S in L

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Integer Programs

Mathematical Programming (Optimization subject to constraints) Integer Programming Hartke gave an integer program for the Firefighter Problem Notation for the integer program:

bx,t is a Boolean variable that is 1 if vertex x is burning at time t and 0 otherwise dx,t is a Boolean variable that is 1 if vertex x is defended at time t and 0 otherwise T is the number of the final time step L is the graph under consideration (here a grid) For any set of vertices S, N(S) is the set of neighbors of S in L

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Integer Programs

Mathematical Programming (Optimization subject to constraints) Integer Programming Hartke gave an integer program for the Firefighter Problem Notation for the integer program:

bx,t is a Boolean variable that is 1 if vertex x is burning at time t and 0 otherwise dx,t is a Boolean variable that is 1 if vertex x is defended at time t and 0 otherwise T is the number of the final time step L is the graph under consideration (here a grid) For any set of vertices S, N(S) is the set of neighbors of S in L

9

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Integer Programs

Mathematical Programming (Optimization subject to constraints) Integer Programming Hartke gave an integer program for the Firefighter Problem Notation for the integer program:

bx,t is a Boolean variable that is 1 if vertex x is burning at time t and 0 otherwise dx,t is a Boolean variable that is 1 if vertex x is defended at time t and 0 otherwise T is the number of the final time step L is the graph under consideration (here a grid) For any set of vertices S, N(S) is the set of neighbors of S in L

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Integer Programs

Mathematical Programming (Optimization subject to constraints) Integer Programming Hartke gave an integer program for the Firefighter Problem Notation for the integer program:

bx,t is a Boolean variable that is 1 if vertex x is burning at time t and 0 otherwise dx,t is a Boolean variable that is 1 if vertex x is defended at time t and 0 otherwise T is the number of the final time step L is the graph under consideration (here a grid) For any set of vertices S, N(S) is the set of neighbors of S in L

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Integer Program

minimize

  • x∈L

bx,T subject to: bx,t + dx,t − by,t−1 ≥ 0, for all x ∈ L, for each y ∈ N(x), and 1 ≤ t ≤ T, (1) bx,t + dx,t ≤ 1, for all x ∈ L and 1 ≤ t ≤ T, (2) bx,t − bx,t−1 ≥ 0, for all x ∈ L and 1 ≤ t ≤ T, (3) dx,t − dx,t−1 ≥ 0, for all x ∈ L and 1 ≤ t ≤ T, (4)

  • x∈L

(dx,t − dx,t−1) ≤ 2, for 1 ≤ t ≤ T, (5) bx,0 =

  • 1 if x is the origin

0 otherwise , for all x ∈ L, (6) dx,0 = 0, for all x ∈ L, (7) bx,t, dx,t ∈ {0, 1}, for all x ∈ L and 0 ≤ t ≤ T. (8)

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Integer Program

Integer Program 2662 variables, 8040 constraints

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Integer Program

Integer Program 2662 variables, 8040 constraints

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An Optimal Solution from our Integer Program

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Outline

1

Introduction Introduction to the Firefighter Problem Integer Programming

2

Flood Model

3

Untouchable Vertices

4

Directed Grids Hall-Like Theorem and a Quarter Plane Regular Directed Grids

5

Moveable Firefighters Integer Programming Results

6

Conclusion and Future Research

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Weights for Moorhead

                

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Integer Program for the Flood Model

In order to create an integer program that would model the spread of water, we had to change our objective line and our spreading constraints: minimize

  • x∈L

wxbx,T subject to:            bx,t + dx,t − by,t−(elevation(x)−elevation(y)) ≥ 0, if (elevation(x) − elevation(y)) > 0 bx,t + dx,t − by,t ≥ 0, if (elevation(x) − elevation(y)) ≤ 0 , for all x ∈ L, for each y ∈ N(x), and 1 ≤ t ≤ T. (9)

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Optimal Containment of a Flood with Current Program

Figure: Flood model with F = 1, T = 13, and a starting river level of 37.5 ft.

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Updated Integer Program for Flood Model

We had to change the integer program again after noticing that the water was not spreading as desired: If elevation(x) ≤ t : bx,t + dx,t − by,t ≥ 0, for all x ∈ L, for each y ∈ N(x), and 1 ≤ t ≤ T, (10)

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Optimal Containment of Flood with New Integer Program

Figure: New flood model with F = 2, T = 7, and a starting river level of 37.5 ft.

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Outline

1

Introduction Introduction to the Firefighter Problem Integer Programming

2

Flood Model

3

Untouchable Vertices

4

Directed Grids Hall-Like Theorem and a Quarter Plane Regular Directed Grids

5

Moveable Firefighters Integer Programming Results

6

Conclusion and Future Research

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Introduction to Untouchable Vertices

The idea of untouchable vertices is a variation on the firefighter

  • problem. In this variation, there are declared vertices which must be

defended, but defenders cannot be placed on those vertices.

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A 2, 2 Success Case

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A 2, 2 Success Case

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A 2, 2 Success Case

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A 2, 2 Success Case

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A 2, 2 Success Case

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A 2, 2 Success Case

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A 2, 2 Success Case

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A 2, 2 Success Case

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A 2, 2 Success Case

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Alternate Defense using Fogarty

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Alternate Defense using Fogarty

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Alternate Defense using Fogarty

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Alternate Defense using Fogarty

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Alternate Defense using Fogarty

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Alternate Defense using Fogarty

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Alternate Defense using Fogarty

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The 2, 2 Cases That Fail

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The 2, 2 Cases That Fail

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The 2, 2 Cases That Fail

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The 2, 3, 4 Case That Fails

This is the only 2, 3, 4 case that fails, up to rotations and reflections.

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The 2, 3, 4 Case That Fails

This is the only 2, 3, 4 case that fails, up to rotations and reflections.

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All Failure Cases with Three Untouchable Vertices

2, 2, 2: All are failures except the following: (-2, 0), (-1, 1), (0, 2) 4 2, 2, 3: If a configuration in the 2, 2 case is impossible, then the corresponding configuration in the 2, 2, 3 case will fail. Only 2 configurations fail if the 2, 2 configuration is possible: (-2, 0), (2, 0), (0, 3) 4 (-2, 0), (2, 0), (-1, 2) 8 2, 2, 4: If a configuration in the 2, 2 case is impossible, then the corresponding configuration in the 2, 2, 4 case will fail. Only 3 configurations fail if the 2, 2 configuration is possible: (-2, 0), (2, 0), (0, -4) 4 (-2, 0), (2, 0), (1, -3) 8 (-2, 0), (2, 0), (2, -2) 8

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All Failure Cases with Three Untouchable Vertices

2, 2, 5+: If a configuration in the 2, 2 case is impossible, then the corresponding configurations in this case will fail. Otherwise, the con- figurations are possible. 2, 3, 3: The following 14 configurations fail: (-1, 1), (1, 2), (2, -1) 8 (-1, 1), (2, 1), (2, -1) 8 (-1, 1), (-1, -2), (2, -1) 8 (-1, 1), (-2, -1), (2, -1) 8 (-1, 1), (-1, -2), (1, 2) 8 (-1, 1), (1, 2), (1, -2) 8 (-1, 1), (-1, -2), (2, 1) 4 (-1, 1), (-1, -2), (2, 1) 8 (-1, 1), (1, -2), (2, 1) 8 (-2, 0), (1, 2), (1, -2) 4 (-2, 0), (1, 2), (2, -1) 8 (-2, 0), (1, -2), (2, 1) 8 (-1, 1), (-1, -2), (3, 0) 8 (-1, 1), (1, -2), (3, 0) 8

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All Failure Cases with Three Untouchable Vertices

2, 3, 4: The following configuration fails: (-2, 0), (1, 2), (2, -2) 8 2, 3+, 5+: All of the configurations are possible. 2, 4, 4: The following configuration fails: (-2, 0), (2, 2), (2, -2) 4 3, 3, 3: The following 3 configurations fail: (-3, 0), (1, 2), (1, -2) 4 (-2, -1), (-1, 2), (2, 1) 8 (-2, -1), (-1, 2), (2, -1) 8 3+, 3+, 4+: All of the configurations are possible.

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Outline

1

Introduction Introduction to the Firefighter Problem Integer Programming

2

Flood Model

3

Untouchable Vertices

4

Directed Grids Hall-Like Theorem and a Quarter Plane Regular Directed Grids

5

Moveable Firefighters Integer Programming Results

6

Conclusion and Future Research

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Definitions for Directed Grids

Arc: An arc is an edge with a direction assigned to it from one vertex to the other vertex. Directed Graph: A directed graph is a graph in which all edges are arcs.

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Definitions for Directed Grids

Arc: An arc is an edge with a direction assigned to it from one vertex to the other vertex. Directed Graph: A directed graph is a graph in which all edges are arcs.

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Hall-Like Theorem

In order to find a lower bound for the number of firefighters per time step needed to contain a fire in directed graphs, we proved a Hall-like theorem for directed graphs.

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Hall-Like Theorem

Assume that the fire begins at a single vertex of a directed graph (the origin). Let Dk be the set of vertices that are of distance k from the origin. Recall that, if S is a set of vertices, then N(S) is the set of neighbors of S in the underlying undirected graph. For any subset A in Dk, we will set N+(A) = N(A) ∩ Dk+1. Let Bk be the set of vertices in Dk that have been burned after time k.

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Hall-Like Theorem

Assume that the fire begins at a single vertex of a directed graph (the origin). Let Dk be the set of vertices that are of distance k from the origin. Recall that, if S is a set of vertices, then N(S) is the set of neighbors of S in the underlying undirected graph. For any subset A in Dk, we will set N+(A) = N(A) ∩ Dk+1. Let Bk be the set of vertices in Dk that have been burned after time k.

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SLIDE 88

Hall-Like Theorem

Assume that the fire begins at a single vertex of a directed graph (the origin). Let Dk be the set of vertices that are of distance k from the origin. Recall that, if S is a set of vertices, then N(S) is the set of neighbors of S in the underlying undirected graph. For any subset A in Dk, we will set N+(A) = N(A) ∩ Dk+1. Let Bk be the set of vertices in Dk that have been burned after time k.

31

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SLIDE 89

Hall-Like Theorem

Assume that the fire begins at a single vertex of a directed graph (the origin). Let Dk be the set of vertices that are of distance k from the origin. Recall that, if S is a set of vertices, then N(S) is the set of neighbors of S in the underlying undirected graph. For any subset A in Dk, we will set N+(A) = N(A) ∩ Dk+1. Let Bk be the set of vertices in Dk that have been burned after time k.

31

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SLIDE 90

Hall-Like Theorem

Assume that the fire begins at a single vertex of a directed graph (the origin). Let Dk be the set of vertices that are of distance k from the origin. Recall that, if S is a set of vertices, then N(S) is the set of neighbors of S in the underlying undirected graph. For any subset A in Dk, we will set N+(A) = N(A) ∩ Dk+1. Let Bk be the set of vertices in Dk that have been burned after time k.

31

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SLIDE 91

Hall-Like Theorem

Assume that the fire begins at a single vertex of a directed graph (the origin). Let Dk be the set of vertices that are of distance k from the origin. Recall that, if S is a set of vertices, then N(S) is the set of neighbors of S in the underlying undirected graph. For any subset A in Dk, we will set N+(A) = N(A) ∩ Dk+1. Let Bk be the set of vertices in Dk that have been burned after time k.

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Hall-Like Theorem

Let f be the number of firefighters we place each time step.

Theorem

Suppose we have a directed graph. For each k, if every A ⊆ Dk satisfies |N+(A)| ≥ |A| + f, then |Bn| ≥ 1 for all n. The proof is by induction on k.

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Hall-Like Theorem

Let f be the number of firefighters we place each time step.

Theorem

Suppose we have a directed graph. For each k, if every A ⊆ Dk satisfies |N+(A)| ≥ |A| + f, then |Bn| ≥ 1 for all n. The proof is by induction on k.

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Hall-Like Theorem

Let f be the number of firefighters we place each time step.

Theorem

Suppose we have a directed graph. For each k, if every A ⊆ Dk satisfies |N+(A)| ≥ |A| + f, then |Bn| ≥ 1 for all n. The proof is by induction on k.

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Hall-Like Theorem

One firefighter per time step is not enough, but a second firefighter at any time step would contain the fire.

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Hall-Like Theorem

One firefighter per time step is not enough, but a second firefighter at any time step would contain the fire.

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Regular Directed Grids

We now focus on regular directed grids, where every vertex has in-degree 2 and out-degree 2. Category A: One firefighter per time step is enough to contain the fire. Category B: One firefighter per time step is not enough to contain the fire, but a second firefighter at any time step will contain the fire.

Theorem

Every regular directed grid is either in Category A or in Category B.

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Regular Directed Grids

We now focus on regular directed grids, where every vertex has in-degree 2 and out-degree 2. Category A: One firefighter per time step is enough to contain the fire. Category B: One firefighter per time step is not enough to contain the fire, but a second firefighter at any time step will contain the fire.

Theorem

Every regular directed grid is either in Category A or in Category B.

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Regular Directed Grids

We now focus on regular directed grids, where every vertex has in-degree 2 and out-degree 2. Category A: One firefighter per time step is enough to contain the fire. Category B: One firefighter per time step is not enough to contain the fire, but a second firefighter at any time step will contain the fire.

Theorem

Every regular directed grid is either in Category A or in Category B.

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Regular Directed Grids

We now focus on regular directed grids, where every vertex has in-degree 2 and out-degree 2. Category A: One firefighter per time step is enough to contain the fire. Category B: One firefighter per time step is not enough to contain the fire, but a second firefighter at any time step will contain the fire.

Theorem

Every regular directed grid is either in Category A or in Category B.

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Regular Directed Grids

Without loss of generality, there are two cases for the arcs at the

  • rigin.

35

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SLIDE 102

Regular Directed Grids

Theorem (Case 1)

Let G be a regular infinite directed grid where the two arcs coming from the origin point along the positive x-axis and the positive y-axis. If each arc in the first quadrant (including the axes) is facing either right

  • r up, then G is a Category B grid. The arcs in the other quadrants are
  • irrelevant. If at least one arc in the first quadrant (including the axes)

faces down or left, then G is a Category A grid unless it is the exception shown below, which is a Category B grid.

36

slide-103
SLIDE 103

Regular Directed Grids

Theorem (Case 2)

Let G be a regular infinite directed grid where the vertical arc directly above the origin faces up and the vertical arc directly below the origin faces down. Then G is a Category A grid unless the grid is the exception shown below or a reflection of this figure across the y-axis. This exception is in Category B.

37

slide-104
SLIDE 104

Regular Directed Grids

The three types of Category B grids:

38

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SLIDE 105

Regular Directed Grids

We now show part of the proof that the rest of the regular directed grids are in Category A. Assume that the arcs pointing from the origin are along the positive x-axis and positive y-axis (Case 1).

39

slide-106
SLIDE 106

Regular Directed Grids

We now show part of the proof that the rest of the regular directed grids are in Category A. Assume that the arcs pointing from the origin are along the positive x-axis and positive y-axis (Case 1).

39

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SLIDE 107

Regular Directed Grids

Assume that some arc in the first quadrant points down or to the

  • left. Consider the closest such arc to the origin. Call it e. Without

loss of generality, e points down. We will only consider the case when e comes from a point in the first quadrant other than (1, 1).

40

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SLIDE 108

Regular Directed Grids

Assume that some arc in the first quadrant points down or to the

  • left. Consider the closest such arc to the origin. Call it e. Without

loss of generality, e points down. We will only consider the case when e comes from a point in the first quadrant other than (1, 1).

40

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SLIDE 109

Regular Directed Grids

The head of e must touch the x-axis.

41

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SLIDE 110

Regular Directed Grids

The head of e must touch the x-axis.

41

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SLIDE 111

Regular Directed Grids

The head of e must touch the x-axis.

41

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SLIDE 112

Regular Directed Grids

Then the arcs between e and the y-axis must look like this:

42

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SLIDE 113

Regular Directed Grids

Adding more arcs to our picture: In any case, we can contain the fire.

43

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SLIDE 114

Regular Directed Grids

Adding more arcs to our picture: In any case, we can contain the fire.

43

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SLIDE 115

Regular Directed Grids

Adding more arcs to our picture: In any case, we can contain the fire.

43

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SLIDE 116

Regular Directed Grids

Adding more arcs to our picture: In any case, we can contain the fire.

43

slide-117
SLIDE 117

Regular Directed Grids

Adding more arcs to our picture: In any case, we can contain the fire.

43

slide-118
SLIDE 118

Regular Directed Grids

Adding more arcs to our picture: In any case, we can contain the fire.

43

slide-119
SLIDE 119

Regular Directed Grids

Adding more arcs to our picture: In any case, we can contain the fire.

43

slide-120
SLIDE 120

Regular Directed Grids

Adding more arcs to our picture: In any case, we can contain the fire.

43

slide-121
SLIDE 121

Regular Directed Grids

Adding more arcs to our picture: In any case, we can contain the fire.

43

slide-122
SLIDE 122

Regular Directed Grids

Adding more arcs to our picture: In any case, we can contain the fire.

43

slide-123
SLIDE 123

Regular Directed Grids

Adding more arcs to our picture: In any case, we can contain the fire.

43

slide-124
SLIDE 124

Regular Directed Grids

Adding more arcs to our picture: In any case, we can contain the fire.

43

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SLIDE 125

Regular Directed Grids

Adding more arcs to our picture: In any case, we can contain the fire.

43

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SLIDE 126

Regular Directed Grids

Adding more arcs to our picture: In any case, we can contain the fire.

43

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SLIDE 127

Regular Directed Grids

Adding more arcs to our picture: In any case, we can contain the fire.

43

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SLIDE 128

Regular Directed Grids

Adding more arcs to our picture: In any case, we can contain the fire.

43

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SLIDE 129

Regular Directed Grids

Adding more arcs to our picture: In any case, we can contain the fire.

43

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SLIDE 130

Regular Directed Grids

We prove that the other types of grids in Case 1 and Case 2 are in Category A using similar methods, although most of the other subcases are more complicated. Our general strategy is to defend the fire along a directed cycle back to a vertex that has already been burning or defended. We also considered more general directed grids.

44

slide-131
SLIDE 131

Regular Directed Grids

We prove that the other types of grids in Case 1 and Case 2 are in Category A using similar methods, although most of the other subcases are more complicated. Our general strategy is to defend the fire along a directed cycle back to a vertex that has already been burning or defended. We also considered more general directed grids.

44

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SLIDE 132

Regular Directed Grids

We prove that the other types of grids in Case 1 and Case 2 are in Category A using similar methods, although most of the other subcases are more complicated. Our general strategy is to defend the fire along a directed cycle back to a vertex that has already been burning or defended. We also considered more general directed grids.

44

slide-133
SLIDE 133

Outline

1

Introduction Introduction to the Firefighter Problem Integer Programming

2

Flood Model

3

Untouchable Vertices

4

Directed Grids Hall-Like Theorem and a Quarter Plane Regular Directed Grids

5

Moveable Firefighters Integer Programming Results

6

Conclusion and Future Research

45

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SLIDE 134

New Variables

Stages of Burning: b1 : Vertex catches fire but cannot yet spread b2 : Vertex remains on fire and can spread to neighboring vertices b3 : Vertex remains on fire but cannot spread b4 : Vertex remains on fire and can spread to neighboring vertices s : Vertex has scorched; it is done burning and can no longer spread Defender Notation: dk signifies defender k

46

slide-135
SLIDE 135

Final Integer Program for Moveable Firefighters

minimize

  • x∈L

(b1x,T + b2x,T + b3x,T + b4x,T + sx,T) subject to: b1x,t + b2x,t + b3x,t + b4x,t +

K

  • k=1

dkx,t + sx,t − b2y,t−1 − b4y,t−1 ≥ 0, for all x ∈ L, for each y ∈ N(x), and 1 ≤ t ≤ T, (11) b1x,t −

  • y∈N(x)

b2y,t−1 −

  • y∈N(x)

b4y,t−1 ≤ 0, for all x ∈ L and 1 ≤ t ≤ T, (12) b1x,t + b2x,t + b3x,t + b4x,t + dkx,t + sx,t ≤ 1, for all x ∈ L, for each k, and 1 ≤ t ≤ T, (13)

  • x∈L

dkx,t = 1, for each k and 0 ≤ t ≤ T, (14) b2x,t − b1x,t−1 = 0, for all x ∈ L and 1 ≤ t ≤ T, (15)

47

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SLIDE 136

Final Integer Program for Moveable Firefighters

b3x,t − b2x,t−1 = 0, for all x ∈ L and 1 ≤ t ≤ T, (16) b4x,t − b3x,t−1 = 0, for all x ∈ L and 1 ≤ t ≤ T, (17) sx,t−1 + b4x,t−1 − sx,t = 0, for all x ∈ L and 1 ≤ t ≤ T, (18) dkx,t −

  • y∈N(x)

dky,t−1 ≤ 0, for all x ∈ L, for each k, and 1 ≤ t ≤ T, (19) b1x,0 =

  • 1 if x is the origin

0 otherwise , for all x ∈ L, (20) b2x,0 = b3x,0 = b4x,0 = sx,t = 0, for all x ∈ L, (21) dkx,0 =

  • 1 if x is dk’s initial position

0 otherwise , for each k, for all x ∈ L, (22) b1x,t, b2x,t, b3x,t, b4x,t, sx,t, dkx,t ∈ {0, 1}, for all x ∈ L, for each k, and 1 ≤ t ≤ T. (23)

48

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SLIDE 137

Example

       

49

slide-138
SLIDE 138

Example

       

49

slide-139
SLIDE 139

Example

           

49

slide-140
SLIDE 140

Example

           

49

slide-141
SLIDE 141

Example

                

49

slide-142
SLIDE 142

Example

                

49

slide-143
SLIDE 143

Example

                      

49

slide-144
SLIDE 144

Example

                      

49

slide-145
SLIDE 145

Example

                              

49

slide-146
SLIDE 146

Containment Strategy

Theorem

Suppose a finite outbreak of fire starts anywhere on an infinite grid and that vertices burn for m time steps before scorching. Assume that the defenders can move up to two vertices per time step. Then 2m + 1 defenders is sufficient to contain the fire, regardless of the starting positions of the 2m + 1 defenders.

50

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SLIDE 147

Final Containment Shape to Scale

51

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SLIDE 148

m Defenders Containing Upwards

52

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SLIDE 149

m Defenders Containing Upwards

52

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SLIDE 150

m Defenders Containing Upwards

52

slide-151
SLIDE 151

m Defenders Containing Upwards

52

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SLIDE 152

m Defenders Containing Upwards

52

slide-153
SLIDE 153

m Defenders Containing Upwards

52

slide-154
SLIDE 154

m Defenders Containing Upwards

52

slide-155
SLIDE 155

Final Containment Shape to Scale

53

slide-156
SLIDE 156

m + 1 Defenders Containing Downward

54

slide-157
SLIDE 157

m + 1 Defenders Containing Downward

54

slide-158
SLIDE 158

m + 1 Defenders Containing Downward

54

slide-159
SLIDE 159

m + 1 Defenders Containing Downward

54

slide-160
SLIDE 160

m + 1 Defenders Containing Downward

54

slide-161
SLIDE 161

m + 1 Defenders Containing Downward

54

slide-162
SLIDE 162

m + 1 Defenders Containing Downward

54

slide-163
SLIDE 163

m + 1 Defenders Containing Downward

54

slide-164
SLIDE 164

m + 1 Defenders Containing Downward

54

slide-165
SLIDE 165

Final Containment Shape to Scale

55

slide-166
SLIDE 166

Outline

1

Introduction Introduction to the Firefighter Problem Integer Programming

2

Flood Model

3

Untouchable Vertices

4

Directed Grids Hall-Like Theorem and a Quarter Plane Regular Directed Grids

5

Moveable Firefighters Integer Programming Results

6

Conclusion and Future Research

56

slide-167
SLIDE 167

Further Research

The Flooding Problem

Time steps represent incremental increases in the volume of water rather than the elevation of water

Untouchable Vertices

We considered configurations of 4 or more untouchable vertices, including proving a theorem about how to add more untouchable vertices sufficiently far away from the origin For 4 untouchable vertices, we classified many of the configurations — some of the most difficult remaining cases are the 2, 3, 5, 7 case, the 2, 3, 5, 6 case, and the 2, 3, 5, 5 case The number of different configurations increases substantially with more untouchable vertices

57

slide-168
SLIDE 168

Further Research

The Flooding Problem

Time steps represent incremental increases in the volume of water rather than the elevation of water

Untouchable Vertices

We considered configurations of 4 or more untouchable vertices, including proving a theorem about how to add more untouchable vertices sufficiently far away from the origin For 4 untouchable vertices, we classified many of the configurations — some of the most difficult remaining cases are the 2, 3, 5, 7 case, the 2, 3, 5, 6 case, and the 2, 3, 5, 5 case The number of different configurations increases substantially with more untouchable vertices

57

slide-169
SLIDE 169

Further Research

The Flooding Problem

Time steps represent incremental increases in the volume of water rather than the elevation of water

Untouchable Vertices

We considered configurations of 4 or more untouchable vertices, including proving a theorem about how to add more untouchable vertices sufficiently far away from the origin For 4 untouchable vertices, we classified many of the configurations — some of the most difficult remaining cases are the 2, 3, 5, 7 case, the 2, 3, 5, 6 case, and the 2, 3, 5, 5 case The number of different configurations increases substantially with more untouchable vertices

57

slide-170
SLIDE 170

Further Research

The Flooding Problem

Time steps represent incremental increases in the volume of water rather than the elevation of water

Untouchable Vertices

We considered configurations of 4 or more untouchable vertices, including proving a theorem about how to add more untouchable vertices sufficiently far away from the origin For 4 untouchable vertices, we classified many of the configurations — some of the most difficult remaining cases are the 2, 3, 5, 7 case, the 2, 3, 5, 6 case, and the 2, 3, 5, 5 case The number of different configurations increases substantially with more untouchable vertices

57

slide-171
SLIDE 171

Further Research

The Flooding Problem

Time steps represent incremental increases in the volume of water rather than the elevation of water

Untouchable Vertices

We considered configurations of 4 or more untouchable vertices, including proving a theorem about how to add more untouchable vertices sufficiently far away from the origin For 4 untouchable vertices, we classified many of the configurations — some of the most difficult remaining cases are the 2, 3, 5, 7 case, the 2, 3, 5, 6 case, and the 2, 3, 5, 5 case The number of different configurations increases substantially with more untouchable vertices

57

slide-172
SLIDE 172

Further Research

The Flooding Problem

Time steps represent incremental increases in the volume of water rather than the elevation of water

Untouchable Vertices

We considered configurations of 4 or more untouchable vertices, including proving a theorem about how to add more untouchable vertices sufficiently far away from the origin For 4 untouchable vertices, we classified many of the configurations — some of the most difficult remaining cases are the 2, 3, 5, 7 case, the 2, 3, 5, 6 case, and the 2, 3, 5, 5 case The number of different configurations increases substantially with more untouchable vertices

57

slide-173
SLIDE 173

Further Research

More general directed grids

Conjecture

Let G be an infinite directed grid, and assume that the fire begins at the origin. If we are given one firefighter per time step and an

  • ccasional second firefighter is given on some finite number of time

steps (the number may depend on the grid), the fire can be contained.

58

slide-174
SLIDE 174

Further Research

More general directed grids

Conjecture

Let G be an infinite directed grid, and assume that the fire begins at the origin. If we are given one firefighter per time step and an

  • ccasional second firefighter is given on some finite number of time

steps (the number may depend on the grid), the fire can be contained.

58

slide-175
SLIDE 175

Future Research

59

slide-176
SLIDE 176

Further Research

Moveable Firefighters

We conjecture that 2m + 1 defenders are necessary to contain some finite outbreaks of fire if it takes m time steps for the fire to scorch Applying the concept of moveable defenders to the flood model Applying the concept of stages to the virus model (SIR models) Allowing more than one firefighter on a vertex, forcing all burning neighbors to burn (and hence to scorch) more quickly

60

slide-177
SLIDE 177

Further Research

Moveable Firefighters

We conjecture that 2m + 1 defenders are necessary to contain some finite outbreaks of fire if it takes m time steps for the fire to scorch Applying the concept of moveable defenders to the flood model Applying the concept of stages to the virus model (SIR models) Allowing more than one firefighter on a vertex, forcing all burning neighbors to burn (and hence to scorch) more quickly

60

slide-178
SLIDE 178

Further Research

Moveable Firefighters

We conjecture that 2m + 1 defenders are necessary to contain some finite outbreaks of fire if it takes m time steps for the fire to scorch Applying the concept of moveable defenders to the flood model Applying the concept of stages to the virus model (SIR models) Allowing more than one firefighter on a vertex, forcing all burning neighbors to burn (and hence to scorch) more quickly

60

slide-179
SLIDE 179

Further Research

Moveable Firefighters

We conjecture that 2m + 1 defenders are necessary to contain some finite outbreaks of fire if it takes m time steps for the fire to scorch Applying the concept of moveable defenders to the flood model Applying the concept of stages to the virus model (SIR models) Allowing more than one firefighter on a vertex, forcing all burning neighbors to burn (and hence to scorch) more quickly

60

slide-180
SLIDE 180

Further Research

Moveable Firefighters

We conjecture that 2m + 1 defenders are necessary to contain some finite outbreaks of fire if it takes m time steps for the fire to scorch Applying the concept of moveable defenders to the flood model Applying the concept of stages to the virus model (SIR models) Allowing more than one firefighter on a vertex, forcing all burning neighbors to burn (and hence to scorch) more quickly

60

slide-181
SLIDE 181

Acknowledgement

We would like to thank the Concordia College Centennial Scholars Research Program for supporting this research.

61

slide-182
SLIDE 182

Bibliography

City of Moorhead website, http://www.ci.moorhead.mn.us/, accessed June 2010.

  • S. Finbow and G. MacGillivray, The Firefighter Problem: A survey of

results, directions and questions, Australasian Journal of Combinatorics, 43 (2009), 57-77. P . Fogarty, Catching the fire on grids, M.Sc. Thesis, Department of Mathematics, University of Vermont (2003). Stephen G. Hartke, Graph-Theoretic Models of Spread and Competition, Ph.D. dissertation, Rutgers University (2004).

  • B. Hartnell, Firefighter! An application of domination, presentation,

Twentieth Conference on Numerical Mathematics and Computing, University of Manitoba in Winnipeg, Canada (1995).

  • R. Pedersen, Strategic Containment of Fires on Networks, presentation,

Pi Mu Epsilon Conference, Saint John’s University (2009). P . Wang and S. A. Moeller, Fire control on graphs, J. Combin. Math.

  • Combin. Comput., 41 (2002), 19-34.

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