Nash Flows Over Time Leon Sering and Martin Skutella COGA - PowerPoint PPT Presentation

Nash Flows over Time (with one sink) no particle can be faster by changing its route flow over time ( f + , f − ) & inflow distribution ( f i ): • Entering a source directly is always a fastest option. F i ( φ ) / r i = ℓ s i ( φ ) waiting time in front of s i earliest (possible) arrival time at s i • Flow only travels along current shortest paths.

Nash Flows over Time (with one sink) no particle can be faster by changing its route flow over time ( f + , f − ) & inflow distribution ( f i ): • Entering a source directly is always a fastest option. F i ( φ ) / r i = ℓ s i ( φ ) waiting time in front of s i earliest (possible) arrival time at s i • Flow only travels along current shortest paths. f + uv ( ℓ u ( φ )) > 0 ⇒ uv active for φ

Nash Flows over Time (with one sink) no particle can be faster by changing its route flow over time ( f + , f − ) & inflow distribution ( f i ): • Entering a source directly is always a fastest option. F i ( φ ) / r i = ℓ s i ( φ ) waiting time in front of s i earliest (possible) arrival time at s i • Flow only travels along current shortest paths. f + uv ( ℓ u ( φ )) > 0 ⇒ uv active for φ earliest arrival time at u part of a current shortest path

Nash Flows over Time (with one sink) no particle can be faster by changing its route flow over time ( f + , f − ) & inflow distribution ( f i ): • Entering a source directly is always a fastest option. F i ( φ ) / r i = ℓ s i ( φ ) waiting time in front of s i earliest (possible) arrival time at s i • Flow only travels along current shortest paths. f + uv ( ℓ u ( φ )) > 0 ⇒ uv active for φ earliest arrival time at u part of a current shortest path ⇒ flow always takes fastest routes to the sink

Thin Flows with Resetting [Koch & Skutella ’09] Given: • Nash flow, and a fixed particle φ . • current shortest path network G ′ = ( V , E ′ ) • edges with queue (resetting edges) E ∗ ⊆ E ′ . • earliest arrival time derivatives ℓ ′ Find: v • static flow x ′ e and x ′ i

Thin Flows with Resetting [Koch & Skutella ’09] Given: • Nash flow, and a fixed particle φ . • current shortest path network G ′ = ( V , E ′ ) • edges with queue (resetting edges) E ∗ ⊆ E ′ . • earliest arrival time derivatives ℓ ′ Find: v • static flow x ′ e and x ′ i l ′ s i = x ′ i / r i l ′ e = us i ρ e ( l ′ u , x ′ s i ≤ min e ) l ′ e = uv ρ e ( l ′ u , x ′ v = min e ) for v �∈ S l ′ v = ρ e ( l ′ u , x ′ if x ′ e ) e > 0 where � { x ′ if e ∈ E ∗ e /ν e } ρ e ( l ′ u , x ′ e ) := max { l ′ u , x ′ if e ∈ E ′ \ E ∗ . e /ν e }

Thin Flows with Resetting [Koch & Skutella ’09] Given: • Nash flow, and a fixed particle φ . • current shortest path network G ′ = ( V , E ′ ) • edges with queue (resetting edges) E ∗ ⊆ E ′ . • earliest arrival time derivatives ℓ ′ Find: v • static flow x ′ e and x ′ i 2 1 2 φ 1 s 1 t 1 2 2 3 s 2 1

Thin Flows with Resetting [Koch & Skutella ’09] Given: • Nash flow, and a fixed particle φ . • current shortest path network G ′ = ( V , E ′ ) • edges with queue (resetting edges) E ∗ ⊆ E ′ . • earliest arrival time derivatives ℓ ′ Find: v • static flow x ′ e and x ′ i 2 2 φ 1 s 1 t 1 2 2 3 s 2 1

Thin Flows with Resetting [Koch & Skutella ’09] Given: • Nash flow, and a fixed particle φ . • current shortest path network G ′ = ( V , E ′ ) • edges with queue (resetting edges) E ∗ ⊆ E ′ . • earliest arrival time derivatives ℓ ′ Find: v • static flow x ′ e and x ′ 3 i 4 2 3 0 4 3 1 4 2 2 3 φ 1 3 4 s 1 4 t 1 2 1 1 2 1 4 3 4 s 2 1 1 8 1 4

Thin Flows with Resetting [Koch & Skutella ’09] Given: • Nash flow, and a fixed particle φ . • current shortest path network G ′ = ( V , E ′ ) • edges with queue (resetting edges) E ∗ ⊆ E ′ . • earliest arrival time derivatives ℓ ′ Find: v • static flow x ′ e and x ′ 3 i 4 2 3 0 4 3 1 4 2 2 3 φ 1 3 4 s 1 4 t 1 2 1 1 2 1 4 3 4 s 2 1 1 8 1 4 Thm: [Cominetti et. all. ’11] There always exists a thin flow with resetting.

Thin Flows with Resetting [Koch & Skutella ’09] Given: • Nash flow, and a fixed particle φ . • current shortest path network G ′ = ( V , E ′ ) • edges with queue (resetting edges) E ∗ ⊆ E ′ . • earliest arrival time derivatives ℓ ′ Find: v • static flow x ′ e and x ′ 3 i 4 2 3 0 4 3 1 4 2 2 3 φ 1 3 4 s 1 4 t 1 2 1 1 2 1 4 3 4 s 2 1 1 8 1 4 Thm: [Cominetti et. all. ’11] There always exists a thin flow with resetting. MIP

Thin Flows with Resetting [Koch & Skutella ’09] Given: • Nash flow, and a fixed particle φ . • current shortest path network G ′ = ( V , E ′ ) • edges with queue (resetting edges) E ∗ ⊆ E ′ . • earliest arrival time derivatives ℓ ′ Find: v • static flow x ′ e and x ′ 3 i 4 2 3 0 4 3 1 4 2 2 3 φ 1 3 4 s 1 4 t 1 2 1 1 2 1 4 3 4 s 2 1 1 8 1 4 Thm: [Cominetti et. all. ’11] Theorem: [Koch & Skutella ’09] There always exists a thin The derivatives of a Nash flow are almost flow with resetting. everywhere a thin flow with resetting. MIP

Constructing Nash Flows from Thin Flows 1 2 6 1 1 t 1 2 1 1 s 1 1 1 3 6 4 1 1 3

Constructing Nash Flows from Thin Flows 1 2 10 6 1 1 t 1 2 ℓ v ( φ ) 1 5 1 s 1 0 1 1 0 1 2 3 4 3 6 4 φ 1 1 1 f + e ( θ ) 3 0 0 15 θ θ 5 4 3 2 1 0

Constructing Nash Flows from Thin Flows 1 2 10 6 1 1 t 1 2 ℓ v ( φ ) 1 5 1 s 1 0 1 1 0 1 2 3 4 3 6 4 φ 1 1 1 f + e ( θ ) 3 0 0 15 θ θ 5 4 3 2 1 0

Constructing Nash Flows from Thin Flows 1 2 10 6 1 1 t 1 2 ℓ v ( φ ) 1 5 1 s 1 0 1 1 0 1 2 3 4 3 6 4 φ 1 1 1 f + e ( θ ) 3 0 0 15 θ θ 5 4 3 2 1 0 Thin Flow: 2 1 1 1 6 1 1 3

Constructing Nash Flows from Thin Flows 1 2 10 6 1 1 t 1 2 ℓ v ( φ ) 1 5 1 3 s 1 0 1 1 0 1 2 3 4 3 6 4 φ 1 1 3 1 f + 9 e ( θ ) 3 0 0 15 θ θ 5 4 3 2 1 0 3 Thin Flow: 1 2 2 1 1 1 6 5 1 1 3

Constructing Nash Flows from Thin Flows 1 2 10 6 1 1 t 1 2 ℓ v ( φ ) 1 5 1 3 s 1 0 1 1 0 1 2 3 4 3 6 4 φ 1 1 3 1 f + 9 e ( θ ) 3 0 0 15 θ θ 5 4 3 2 1 0 3 Thin Flow: 1 2 2 2 1 3 1 1 2 3 5 1 1 0 3 1

Constructing Nash Flows from Thin Flows 1 2 10 6 1 1 t 1 2 ℓ v ( φ ) 5 1 5 1 s 1 0 1 1 0 1 2 3 4 3 6 4 φ 1 1 11 6 1 f + e ( θ ) 3 0 0 15 θ θ 5 4 3 2 1 0 4 Thin Flow: 2 2 2 1 3 1 1 2 3 6 1 1 0 3 1

Constructing Nash Flows from Thin Flows 1 2 10 6 1 1 t 1 2 ℓ v ( φ ) 5 1 5 1 s 1 0 1 1 0 1 2 3 4 3 6 4 φ 1 1 11 6 1 f + e ( θ ) 3 0 0 15 θ θ 5 4 3 2 1 0 4 Thin Flow: 2 1.5 3 3 4 4 1 0 1.5 6 1 1 1 4 4 1

Constructing Nash Flows from Thin Flows 1 2 10 6 1 1 t 1 2 ℓ v ( φ ) 1 5 1 s 1 0 1 1 0 1 2 3 4 3 6 4 φ 1 1 1 f + e ( θ ) 3 0 0 15 θ θ 5 4 3 2 1 0 Thin Flow: 1.5 3 3 4 4 1 0 1.5 1 1 1 4 4 1

Demands 1 2 t 1 s 1 t 2 s 2 1 3 1 t 3 6

Demands 1 2 t 1 s 1 t 2 super sink t s 2 1 3 1 t 3 6

Demands 1 2 t 1 s 1 1 ε · 2 ε · 1 3 t 2 super sink t s 2 1 1 3 ε · 6 1 t 3 6