Integer Programs Mathematical Programming (Optimization subject to constraints) Integer Programming Hartke gave an integer program for the Firefighter Problem Notation for the integer program: b x , t is a Boolean variable that is 1 if vertex x is burning at time t and 0 otherwise d x , 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
Integer Programs Mathematical Programming (Optimization subject to constraints) Integer Programming Hartke gave an integer program for the Firefighter Problem Notation for the integer program: b x , t is a Boolean variable that is 1 if vertex x is burning at time t and 0 otherwise d x , 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
Integer Programs Mathematical Programming (Optimization subject to constraints) Integer Programming Hartke gave an integer program for the Firefighter Problem Notation for the integer program: b x , t is a Boolean variable that is 1 if vertex x is burning at time t and 0 otherwise d x , 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
Integer Programs Mathematical Programming (Optimization subject to constraints) Integer Programming Hartke gave an integer program for the Firefighter Problem Notation for the integer program: b x , t is a Boolean variable that is 1 if vertex x is burning at time t and 0 otherwise d x , 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
Integer Programs Mathematical Programming (Optimization subject to constraints) Integer Programming Hartke gave an integer program for the Firefighter Problem Notation for the integer program: b x , t is a Boolean variable that is 1 if vertex x is burning at time t and 0 otherwise d x , 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
Integer Programs Mathematical Programming (Optimization subject to constraints) Integer Programming Hartke gave an integer program for the Firefighter Problem Notation for the integer program: b x , t is a Boolean variable that is 1 if vertex x is burning at time t and 0 otherwise d x , 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
Integer Programs Mathematical Programming (Optimization subject to constraints) Integer Programming Hartke gave an integer program for the Firefighter Problem Notation for the integer program: b x , t is a Boolean variable that is 1 if vertex x is burning at time t and 0 otherwise d x , 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
Integer Program � minimize b x , T x ∈ L subject to: b x , t + d x , t − b y , t − 1 ≥ 0 , for all x ∈ L , for each y ∈ N ( x ) , and 1 ≤ t ≤ T , (1) b x , t + d x , t ≤ 1 , for all x ∈ L and 1 ≤ t ≤ T , (2) b x , t − b x , t − 1 ≥ 0 , for all x ∈ L and 1 ≤ t ≤ T , (3) d x , t − d x , t − 1 ≥ 0 , for all x ∈ L and 1 ≤ t ≤ T , (4) � ( d x , t − d x , t − 1 ) ≤ 2 , for 1 ≤ t ≤ T , (5) x ∈ L � 1 if x is the origin b x , 0 = , for all x ∈ L , (6) 0 otherwise d x , 0 = 0 , for all x ∈ L , (7) b x , t , d x , t ∈ { 0 , 1 } , for all x ∈ L and 0 ≤ t ≤ T . (8) 10
Integer Program Integer Program 2662 variables, 8040 constraints 11
Integer Program Integer Program 2662 variables, 8040 constraints 11
An Optimal Solution from our Integer Program 12
Outline Introduction 1 Introduction to the Firefighter Problem Integer Programming Flood Model 2 Untouchable Vertices 3 Directed Grids 4 Hall-Like Theorem and a Quarter Plane Regular Directed Grids Moveable Firefighters 5 Integer Programming Results Conclusion and Future Research 6 13
Weights for Moorhead 14
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 w x b x , T x ∈ L b x , t + d x , t − b y , t − ( elevation ( x ) − elevation ( y )) ≥ 0 , if ( elevation ( x ) − elevation ( y )) > 0 subject to: , b x , t + d x , t − b y , t ≥ 0 , if ( elevation ( x ) − elevation ( y )) ≤ 0 for all x ∈ L , for each y ∈ N ( x ) , and 1 ≤ t ≤ T . (9) 15
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. 16
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 : b x , t + d x , t − b y , t ≥ 0 , for all x ∈ L , for each y ∈ N ( x ) , and 1 ≤ t ≤ T , (10) 17
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. 18
Outline Introduction 1 Introduction to the Firefighter Problem Integer Programming Flood Model 2 Untouchable Vertices 3 Directed Grids 4 Hall-Like Theorem and a Quarter Plane Regular Directed Grids Moveable Firefighters 5 Integer Programming Results Conclusion and Future Research 6 19
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. 20
A 2, 2 Success Case 21
A 2, 2 Success Case 21
A 2, 2 Success Case 21
A 2, 2 Success Case 21
A 2, 2 Success Case 21
A 2, 2 Success Case 21
A 2, 2 Success Case 21
A 2, 2 Success Case 21
A 2, 2 Success Case 21
Alternate Defense using Fogarty 22
Alternate Defense using Fogarty 22
Alternate Defense using Fogarty 22
Alternate Defense using Fogarty 22
Alternate Defense using Fogarty 22
Alternate Defense using Fogarty 22
Alternate Defense using Fogarty 22
The 2, 2 Cases That Fail 23
The 2, 2 Cases That Fail 23
The 2, 2 Cases That Fail 23
The 2, 3, 4 Case That Fails This is the only 2, 3, 4 case that fails, up to rotations and reflections. 24
The 2, 3, 4 Case That Fails This is the only 2, 3, 4 case that fails, up to rotations and reflections. 24
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 25
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 26
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. 27
Outline Introduction 1 Introduction to the Firefighter Problem Integer Programming Flood Model 2 Untouchable Vertices 3 Directed Grids 4 Hall-Like Theorem and a Quarter Plane Regular Directed Grids Moveable Firefighters 5 Integer Programming Results Conclusion and Future Research 6 28
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. 29
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. 29
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. 30
Hall-Like Theorem Assume that the fire begins at a single vertex of a directed graph (the origin ). Let D k 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 D k , we will set N + ( A ) = N ( A ) ∩ D k + 1 . Let B k be the set of vertices in D k that have been burned after time k . 31
Hall-Like Theorem Assume that the fire begins at a single vertex of a directed graph (the origin ). Let D k 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 D k , we will set N + ( A ) = N ( A ) ∩ D k + 1 . Let B k be the set of vertices in D k that have been burned after time k . 31
Hall-Like Theorem Assume that the fire begins at a single vertex of a directed graph (the origin ). Let D k 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 D k , we will set N + ( A ) = N ( A ) ∩ D k + 1 . Let B k be the set of vertices in D k that have been burned after time k . 31
Hall-Like Theorem Assume that the fire begins at a single vertex of a directed graph (the origin ). Let D k 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 D k , we will set N + ( A ) = N ( A ) ∩ D k + 1 . Let B k be the set of vertices in D k that have been burned after time k . 31
Hall-Like Theorem Assume that the fire begins at a single vertex of a directed graph (the origin ). Let D k 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 D k , we will set N + ( A ) = N ( A ) ∩ D k + 1 . Let B k be the set of vertices in D k that have been burned after time k . 31
Hall-Like Theorem Assume that the fire begins at a single vertex of a directed graph (the origin ). Let D k 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 D k , we will set N + ( A ) = N ( A ) ∩ D k + 1 . Let B k be the set of vertices in D k that have been burned after time k . 31
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 ⊆ D k satisfies | N + ( A ) | ≥ | A | + f, then | B n | ≥ 1 for all n. The proof is by induction on k . 32
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 ⊆ D k satisfies | N + ( A ) | ≥ | A | + f, then | B n | ≥ 1 for all n. The proof is by induction on k . 32
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 ⊆ D k satisfies | N + ( A ) | ≥ | A | + f, then | B n | ≥ 1 for all n. The proof is by induction on k . 32
Hall-Like Theorem One firefighter per time step is not enough, but a second firefighter at any time step would contain the fire. 33
Hall-Like Theorem One firefighter per time step is not enough, but a second firefighter at any time step would contain the fire. 33
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. 34
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. 34
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. 34
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. 34
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