SLIDE 1
Solving Evacuation Problems in Polynomial Space
Solving Evacuation Problems in Polynomial Space
Miriam Schlöter & Martin Skutella
√
Solving Evacuation Problems in Polynomial Space
Flows over Time
Definition
Solving Evacuation Problems in Polynomial Space Miriam Schlter & - - PowerPoint PPT Presentation
Solving Evacuation Problems in Polynomial Space Miriam Schlter & - - PowerPoint PPT Presentation
Solving Evacuation Problems in Polynomial Space Miriam Schlter & Martin Skutella Solving Evacuation Problems in Polynomial Space Flows over Time Definition Solving Evacuation Problems in Polynomial Space Flows over Time Definition
SLIDE 2
SLIDE 3
Solving Evacuation Problems in Polynomial Space
Flows over Time
Definition
Many real life problems crucially depend on time:
- logistic - public transport - evacuation problems
SLIDE 4
Solving Evacuation Problems in Polynomial Space
Flows over Time
Definition
Flows over time are like… …like classical (static) network flows + time component:
- flow needs time to travel through an arc a :
arc a has a transit time 𝜐a (length)
- a bounded amount of flow can enter an arc a per time unit:
arc a has a capacity ua (width) Many real life problems crucially depend on time:
- logistic - public transport - evacuation problems
SLIDE 5
Solving Evacuation Problems in Polynomial Space
Flows over Time
Definition
SLIDE 6
Solving Evacuation Problems in Polynomial Space
Flows over Time
Definition
dynamic network 𝒪= (G =(V,A ),u,𝜐,S+,S- ): digraph G = (V,A ) with capacities u, transit times 𝜐, sources S+ ⊂V and sinks S- ⊂V
SLIDE 7
Solving Evacuation Problems in Polynomial Space
Flows over Time
Definition
dynamic network 𝒪= (G =(V,A ),u,𝜐,S+,S- ): digraph G = (V,A ) with capacities u, transit times 𝜐, sources S+ ⊂V and sinks S- ⊂V
1 1 2 2 2 2
t = 0
𝒪
SLIDE 8
Solving Evacuation Problems in Polynomial Space
Flows over Time
Definition
dynamic network 𝒪= (G =(V,A ),u,𝜐,S+,S- ): digraph G = (V,A ) with capacities u, transit times 𝜐, sources S+ ⊂V and sinks S- ⊂V
1 1 2 2 2 2
t = 1
𝒪
SLIDE 9
Solving Evacuation Problems in Polynomial Space
Flows over Time
Definition
dynamic network 𝒪= (G =(V,A ),u,𝜐,S+,S- ): digraph G = (V,A ) with capacities u, transit times 𝜐, sources S+ ⊂V and sinks S- ⊂V
1 1 2 2 2 2
t = 2
𝒪
SLIDE 10
Solving Evacuation Problems in Polynomial Space
Flows over Time
Definition
dynamic network 𝒪= (G =(V,A ),u,𝜐,S+,S- ): digraph G = (V,A ) with capacities u, transit times 𝜐, sources S+ ⊂V and sinks S- ⊂V
1 1 2 2 2 2
t = 3
𝒪
SLIDE 11
Solving Evacuation Problems in Polynomial Space
Flows over Time
Definition
dynamic network 𝒪= (G =(V,A ),u,𝜐,S+,S- ): digraph G = (V,A ) with capacities u, transit times 𝜐, sources S+ ⊂V and sinks S- ⊂V
1 1 2 2 2 2
t = 4
𝒪
SLIDE 12
Solving Evacuation Problems in Polynomial Space
Flows over Time
Definition
dynamic network 𝒪= (G =(V,A ),u,𝜐,S+,S- ): digraph G = (V,A ) with capacities u, transit times 𝜐, sources S+ ⊂V and sinks S- ⊂V
1 1 2 2 2 2
t = 5
𝒪
SLIDE 13
Solving Evacuation Problems in Polynomial Space
Flows over Time
Definition
dynamic network 𝒪= (G =(V,A ),u,𝜐,S+,S- ): digraph G = (V,A ) with capacities u, transit times 𝜐, sources S+ ⊂V and sinks S- ⊂V
1 1 2 2 2 2
t = 6
𝒪
SLIDE 14
Solving Evacuation Problems in Polynomial Space
Flows over Time
Definition
dynamic network 𝒪= (G =(V,A ),u,𝜐,S+,S- ): digraph G = (V,A ) with capacities u, transit times 𝜐, sources S+ ⊂V and sinks S- ⊂V
1 1 2 2 2 2
t = 7
𝒪
SLIDE 15
Solving Evacuation Problems in Polynomial Space
Flows over Time
Definition
dynamic network 𝒪= (G =(V,A ),u,𝜐,S+,S- ): digraph G = (V,A ) with capacities u, transit times 𝜐, sources S+ ⊂V and sinks S- ⊂V
1 1 2 2 2 2
t = 8
𝒪
SLIDE 16
Solving Evacuation Problems in Polynomial Space
Earliest Arrival Flows
single source / single sink
Given: 𝒪 = (G = (V,A), u, 𝜐, s, t ) and time horizon T Aim: Flow over time f in 𝒪 s.t. at each point in time θ ≤ T as much flow as possible has reached the sink
SLIDE 17
Solving Evacuation Problems in Polynomial Space
Earliest Arrival Flows
single source / single sink
s T = 8 t
Given: 𝒪 = (G = (V,A), u, 𝜐, s, t ) and time horizon T Aim: Flow over time f in 𝒪 s.t. at each point in time θ ≤ T as much flow as possible has reached the sink
2 1
Solution [Minieka ’71, Wilkinson ’71]: Send flow along paths P with 𝜐(P )≤ T occurring in successive shortest path algorithm from s to t
SLIDE 18
Solving Evacuation Problems in Polynomial Space
Earliest Arrival Flows
single source / single sink
s T = 8 t
Given: 𝒪 = (G = (V,A), u, 𝜐, s, t ) and time horizon T Aim: Flow over time f in 𝒪 s.t. at each point in time θ ≤ T as much flow as possible has reached the sink
2 1
Solution [Minieka ’71, Wilkinson ’71]: Send flow along paths P with 𝜐(P )≤ T occurring in successive shortest path algorithm from s to t
SLIDE 19
Solving Evacuation Problems in Polynomial Space
Earliest Arrival Flows
single source / single sink
s T = 8 t
Given: 𝒪 = (G = (V,A), u, 𝜐, s, t ) and time horizon T Aim: Flow over time f in 𝒪 s.t. at each point in time θ ≤ T as much flow as possible has reached the sink
2 1
Solution [Minieka ’71, Wilkinson ’71]: Send flow along paths P with 𝜐(P )≤ T occurring in successive shortest path algorithm from s to t
SLIDE 20
Solving Evacuation Problems in Polynomial Space
Earliest Arrival Flows
single source / single sink
s T = 8 t
Given: 𝒪 = (G = (V,A), u, 𝜐, s, t ) and time horizon T Aim: Flow over time f in 𝒪 s.t. at each point in time θ ≤ T as much flow as possible has reached the sink
2 1
Solution [Minieka ’71, Wilkinson ’71]: Send flow along paths P with 𝜐(P )≤ T occurring in successive shortest path algorithm from s to t
SLIDE 21
Solving Evacuation Problems in Polynomial Space
Earliest Arrival Flows
single source / single sink
s T = 8 t
Given: 𝒪 = (G = (V,A), u, 𝜐, s, t ) and time horizon T Aim: Flow over time f in 𝒪 s.t. at each point in time θ ≤ T as much flow as possible has reached the sink
2 1
Solution [Minieka ’71, Wilkinson ’71]: Send flow along paths P with 𝜐(P )≤ T occurring in successive shortest path algorithm from s to t
SLIDE 22
Solving Evacuation Problems in Polynomial Space
Earliest Arrival Flows
single source / single sink
s T = 8 t
Given: 𝒪 = (G = (V,A), u, 𝜐, s, t ) and time horizon T Aim: Flow over time f in 𝒪 s.t. at each point in time θ ≤ T as much flow as possible has reached the sink
2 1
Solution [Minieka ’71, Wilkinson ’71]: Send flow along paths P with 𝜐(P )≤ T occurring in successive shortest path algorithm from s to t
SLIDE 23
Solving Evacuation Problems in Polynomial Space
Earliest Arrival Flows
single source / single sink
s T = 8 t
Given: 𝒪 = (G = (V,A), u, 𝜐, s, t ) and time horizon T Aim: Flow over time f in 𝒪 s.t. at each point in time θ ≤ T as much flow as possible has reached the sink
2 1
Solution [Minieka ’71, Wilkinson ’71]: Send flow along paths P with 𝜐(P )≤ T occurring in successive shortest path algorithm from s to t
SLIDE 24
Solving Evacuation Problems in Polynomial Space
Earliest Arrival Flows
single source / single sink
s T = 8 t
Given: 𝒪 = (G = (V,A), u, 𝜐, s, t ) and time horizon T Aim: Flow over time f in 𝒪 s.t. at each point in time θ ≤ T as much flow as possible has reached the sink
2 1
Solution [Minieka ’71, Wilkinson ’71]: Send flow along paths P with 𝜐(P )≤ T occurring in successive shortest path algorithm from s to t
SLIDE 25
Solving Evacuation Problems in Polynomial Space
Earliest Arrival Flows
single source / single sink
s T = 8 t
Given: 𝒪 = (G = (V,A), u, 𝜐, s, t ) and time horizon T Aim: Flow over time f in 𝒪 s.t. at each point in time θ ≤ T as much flow as possible has reached the sink
2 1
Solution [Minieka ’71, Wilkinson ’71]: Send flow along paths P with 𝜐(P )≤ T occurring in successive shortest path algorithm from s to t
SLIDE 26
Solving Evacuation Problems in Polynomial Space
Earliest Arrival Flows
single source / single sink
s T = 8 t
Given: 𝒪 = (G = (V,A), u, 𝜐, s, t ) and time horizon T Aim: Flow over time f in 𝒪 s.t. at each point in time θ ≤ T as much flow as possible has reached the sink
2 1
Solution [Minieka ’71, Wilkinson ’71]: Send flow along paths P with 𝜐(P )≤ T occurring in successive shortest path algorithm from s to t
SLIDE 27
Solving Evacuation Problems in Polynomial Space
Earliest Arrival Flows
single source / single sink
s T = 8 t
Given: 𝒪 = (G = (V,A), u, 𝜐, s, t ) and time horizon T Aim: Flow over time f in 𝒪 s.t. at each point in time θ ≤ T as much flow as possible has reached the sink
2 1
Solution [Minieka ’71, Wilkinson ’71]: Send flow along paths P with 𝜐(P )≤ T occurring in successive shortest path algorithm from s to t
SLIDE 28
Solving Evacuation Problems in Polynomial Space
Earliest Arrival Flows
single source / single sink
s T = 8 t
Given: 𝒪 = (G = (V,A), u, 𝜐, s, t ) and time horizon T Aim: Flow over time f in 𝒪 s.t. at each point in time θ ≤ T as much flow as possible has reached the sink
2 1
Solution [Minieka ’71, Wilkinson ’71]: Send flow along paths P with 𝜐(P )≤ T occurring in successive shortest path algorithm from s to t
SLIDE 29
Solving Evacuation Problems in Polynomial Space
Earliest Arrival Flows
multiple sources / single sink
Given: 𝒪 = (G = (V,A), u, 𝜐, S+, t ) with supplies v on the sources Aim: Flow over time f in 𝒪 that respects the supplies such that at each point in time as much flow as possible has reached the sink
SLIDE 30
Solving Evacuation Problems in Polynomial Space
Earliest Arrival Flows
multiple sources / single sink
Given: 𝒪 = (G = (V,A), u, 𝜐, S+, t ) with supplies v on the sources Aim: Flow over time f in 𝒪 that respects the supplies such that at each point in time as much flow as possible has reached the sink → Earliest Arrival Flows can be used to model evacuation scenarios
SLIDE 31
Solving Evacuation Problems in Polynomial Space
Earliest Arrival Flows
multiple sources / single sink
Given: 𝒪 = (G = (V,A), u, 𝜐, S+, t ) with supplies v on the sources Aim: Flow over time f in 𝒪 that respects the supplies such that at each point in time as much flow as possible has reached the sink Baumann & Skutella, (2006), No explicit time-expansion:
- 1. Construct earliest arrival pattern p
- 2. Compute the earliest arrival flow f using p
→ Earliest Arrival Flows can be used to model evacuation scenarios
SLIDE 32
Solving Evacuation Problems in Polynomial Space
Earliest Arrival Flows
multiple sources / single sink
Given: 𝒪 = (G = (V,A), u, 𝜐, S+, t ) with supplies v on the sources Aim: Flow over time f in 𝒪 that respects the supplies such that at each point in time as much flow as possible has reached the sink Baumann & Skutella, (2006), No explicit time-expansion:
- 1. Construct earliest arrival pattern p
- 2. Compute the earliest arrival flow f using p
→ Earliest Arrival Flows can be used to model evacuation scenarios
attaches exponentially many super sources to the network
SLIDE 33
Solving Evacuation Problems in Polynomial Space
Earliest Arrival Flows
multiple sources / single sink - our results, SODA 2017
SLIDE 34
Solving Evacuation Problems in Polynomial Space
New strongly polynomial time algorithm for the Quickest Transshipment Problem
Earliest Arrival Flows
multiple sources / single sink - our results, SODA 2017
SLIDE 35
Solving Evacuation Problems in Polynomial Space
New strongly polynomial time algorithm for the Quickest Transshipment Problem
- A quickest transshipment problem can be solved
as a convex combination of lex-max flows over time.
Earliest Arrival Flows
multiple sources / single sink - our results, SODA 2017
SLIDE 36
Solving Evacuation Problems in Polynomial Space
New strongly polynomial time algorithm for the Quickest Transshipment Problem
- A quickest transshipment problem can be solved
as a convex combination of lex-max flows over time.
- The convex combination can be computed via
- ne submodular function minimization
Earliest Arrival Flows
multiple sources / single sink - our results, SODA 2017
SLIDE 37
Solving Evacuation Problems in Polynomial Space
New strongly polynomial time algorithm for the Quickest Transshipment Problem
- A quickest transshipment problem can be solved
as a convex combination of lex-max flows over time.
- The convex combination can be computed via
- ne submodular function minimization
A polynomial space algorithm for a generalization of lex-max flows over time
Earliest Arrival Flows
multiple sources / single sink - our results, SODA 2017
SLIDE 38
Solving Evacuation Problems in Polynomial Space
New strongly polynomial time algorithm for the Quickest Transshipment Problem A polynomial space algorithm for a generalization of lex-max flows over time
+
Earliest Arrival Flows
multiple sources / single sink - our results, SODA 2017
SLIDE 39
Solving Evacuation Problems in Polynomial Space
New strongly polynomial time algorithm for the Quickest Transshipment Problem A polynomial space algorithm for a generalization of lex-max flows over time
+
Earliest arrival flow problems can be solved as convex combinations of generalization of lex- max flows over time which can be computed using only polynomial space
Earliest Arrival Flows
multiple sources / single sink - our results, SODA 2017
SLIDE 40
Solving Evacuation Problems in Polynomial Space