Nadine k-Institut Maren Skutella, - TU Berlin › coga › pub › Jahres...s-t-cuts k-splitt. o...

DFG Jahres-Kolloquium, July 2004 slide 1 of 39

Efficient Algorithms for Path-Based and Dynamic Flow Problems

in Large Networks

Rolf H. Möhring, Heiko Schilling

Technische Universität Berlin

Martin Skutella, Nadine Baumann

Max-Planck-Institut Saarbrücken

Georg Baier, Ekkehard Köhler, Ines Spenke, Maren Martens


Shortest Path Acceleration Methods

SP acceleration methods

• Input

•Separator heuristic

•Bit-labels

•Conlusion

•Cooperation

Length-bounded s-t-cuts

k-splitt. flows - complexity


Our separator heuristic

1. find graph separator of small size, balanced region size[Lipton and Tarjan ’79],[Goodrich ’95]

2. determine apsp matrix for separator vertices of each region3. compute sps using hierarchy on these separator vertices

[Buchholz and Riedhofer ’97]⇒ speedup factor 22-46


• Input


•Bit-labels

•Conlusion

•Cooperation




Computing SPs using separators

PSfrag replacementss

t


• Input


•Bit-labels

•Conlusion

•Cooperation






t


• Input


•Bit-labels

•Conlusion

•Cooperation




Bit-labels at arcs for target-regions

1

0000

11

1000

0

1

11

0

• Running time: speedup factor 200-350 compared to dijkstra• Preprocessing time: | grid crossing arcs | ∗O(n log n)

[Ulrich Lauther, Siemens: Juli ’04 GeoInformatik Workshop Münster]


• Input


•Bit-labels

•Conlusion

•Cooperation




Bit-labels - visited arcs

instance #graph arcs #dijkstra arcs #bit-label arcs

AA 920.464 783.599 (85.1%) 788 (0.09%)HS 1.696.054 1.566.296 (92.3%) 3.077 (0.18%)NO 1.611.148 1.398.721 (86.8%) 6.509 (0.40%)NW 1.410.076 1.251.424 (88.7%) 1.601 (0.11%)OS 1.169.224 1.115.437 (95.4%) 2.088 (0.18%)TH 1.030.148 947.274 (92.0%) 3.037 (0.29%)

number of visited arcs (30x30 raster)


• Input


•Bit-labels

•Conlusion

•Cooperation




Separator/bit-labels - running time

graph dijkstra separator bit-labels

AA 8.16 0.37 (x22) 0.04 (x204)TH 12.45 0.46 (x27) 0.06 (x208)

NW 13.02 0.47 (x28) 0.06 (x217)NO 26.63 0.75 (x36) 0.1 (x266)OS 18.68 0.44 (x42) 0.06 (x311)HS 28.60 0.61 (x46) 0.08 (x358)

running time in seconds (30x30 raster)


• Input


•Bit-labels

•Conlusion

•Cooperation




Bit-labels - raster size

grid size init time [sec] #visited arcs time [sec]

5 × 5 12057.83 71540 0.2310 × 10 24213.69 18816 0.0415 × 15 26926.64 10445 0.02

instance OS (474431 nodes, 1.169.224 arcs)


• Input


•Bit-labels

•Conlusion

•Cooperation




Memory consumption

memory

graph datastructure 2 GBbit-labels computation 7 GB

separator heuristic computation 22 GB

germany instance


• Input


•Bit-labels

•Conlusion

•Cooperation




Bit-labels - preprocessing (# dijkstra computations)

#regions raster sep raster-bi sep-bi

25 3028 1597 6056 3194100 7632 4009 15264 8018225 11468 6384 22936 12768400 15882 9100 31764 18200625 20040 12057 40080 24114900 23128 15097 46256 30194

instance OS (474431 nodes, 1.169.224 arcs)

[George Karypis, University of Minnesota,METIS: A Family of Multilevel Partitioning Algorithms]

Running time: speedup factor > 2 compared to rastering approach→ overall speedup factor 400-700


• Input


•Bit-labels

•Conlusion

•Cooperation




Bit-labels - number of visited arcs in bi-directional case

|d| `d |sp-arcs| raster sep raster-bi sep-bi

1 long 831 4255 4362 832 832100 short 12011 813480 232547 160409 47528100 av. 44012 652804 334401 84949 72036100 long 81507 524052 393403 102360 101957

instance OS (474431 nodes, 1.169.224 arcs, 400 regions)


• Input


•Bit-labels

•Conlusion

•Cooperation





• DFG-SPP Workshop on Shortest Paths in Karlsruhe ’04

• Acceleration Methods for constrained shortest path(Standard acceleration Methods for SP (do, bi, do-bi) can beapplied for labeling dijkstra algorithm.)

• [joint work with Ekkehard Köhler and Rolf Möhring ’04]


• Input


•Bit-labels

•Conlusion

•Cooperation





• DFG-SPP Workshop on Shortest Paths in Karlsruhe ’04• Acceleration Methods for constrained shortest path

(Standard acceleration Methods for SP (do, bi, do-bi) can beapplied for labeling dijkstra algorithm.)

• [joint work with Ekkehard Köhler and Rolf Möhring ’04]


• Input


•Bit-labels

•Conlusion

•Cooperation




Cooperation with University Karlsruhe

KD-Tree subdivision of AA

started cooperation with AG Dorothea Wagner on accelerationsmethods for shortest paths

◦ What are good subdivisions of graphs for use with bit-labels?◦ How can bit-labels be combined with geometric containers?

[Wagner and Willhalm ’03]


• Input


•Bit-labels

•Conlusion

•Cooperation




Cooperation with ETH Zürich/TU Berlin

Portland, 9am

started cooperation with AG Kai Nagel (ETH/TU) on applications ofshortest path acceleration methods in traffic simulation

matsim [www.matsim.org], Transims [http://www.acl.lanl.gov]


• Input


•Bit-labels

•Conlusion

•Cooperation





housingland use/ activities

(demand)mode choice/

routestraffic

micro−sim

scenario data (road network, demographics); behavioral data

compute routes◦ dijkstra with time-dependent travel times

apply acceleration methods reduce sp computation time(aim: 40min↘ 15min)

◦ dynamic flow models


• Input


•Bit-labels

•Conlusion

•Cooperation





HOME

WORKLUNCH

WORK

DOCTOR

SHOP

HOME

Activity plan for one day

HOME

WORKLUNCH

WORK

DOCTOR

SHOP

HOME

Routes for one day

rating of computed routes by simulation◦ number of turnings, street type changes, etc.◦ quality of service (different travel times on different days?)


Length-bounded s− t−cuts



• s-t-cuts and length-bounds

•Complexity

•dist(s,t)-LBC

•Approximation algorithm



Length-bounded s− t−cuts

• Definition and Elementary Properties• Complexity Issues• Approximation Algorithm




•Complexity

•dist(s,t)-LBC




s− t−cuts and length-bounds

• graph G = (V,E) (directed or undirected)• vertices s, t ∈ V

• edge-capacities u : E → Q≥0

• edge-length d : E → Q≥0

• length bound L ≥ 0

s-t-cut

S ⊂ E: S hits each s-t-path




•Complexity

•dist(s,t)-LBC





• graph G = (V,E) (directed or undirected)• vertices s, t ∈ V• edge-capacities u : E → Q≥0



s-t-cut

S ⊂ E: S hits each s-t-path

cut capacity

|S| = ∑e∈S ue




•Complexity

•dist(s,t)-LBC








L-length-bounded s-t-cut:

S ⊂ E: S hits each s-t-path of length ≤ L




•Complexity

•dist(s,t)-LBC








L-length-bounded s-t-cut:

S ⊂ E: S hits each s-t-path of length ≤ Lcut capacity:

|S| = ∑e∈S ue




•Complexity

•dist(s,t)-LBC




Elementary properties of length-bounded s− t−cuts

• L-length-bounded cut need not tobe a (standard) s-t-cut

• i.e. s and t can be part of thesame connected component ofG \ S

• minimum s-t-cut capacitiesstandard↔ length-bounded→ arbitrarily apart

PSfrag replacementss t

G

de ≈ ∞




•Complexity

•dist(s,t)-LBC








PSfrag replacements

s tG

de ≈ ∞




•Complexity

•dist(s,t)-LBC








PSfrag replacements

s tGde ≈ ∞

ue, de ≈ ∞




•Complexity

•dist(s,t)-LBC




Complexity of length-bounded s− t−cuts

L Vertex-cut Edge-cut

1...




•Complexity

•dist(s,t)-LBC





vertex-cut: L = 1

no inner vertices for L = 1 L Vertex Edge

1 –




•Complexity

•dist(s,t)-LBC





edge-cut: L = 1

delete all s− t− edges

→ polynomial (O(m))

L Vertex Edge

1 – poly




•Complexity

•dist(s,t)-LBC





vertex-cut: L = 2

delete all vertices in N(s)⋂N(t)

→ polynomial (O(n))

L Vertex Edge

1 – poly

2 poly




•Complexity

•dist(s,t)-LBC





edge-cut: L = 2

s t

N(s)⋂N(t) L Vertex Edge

1 – poly

2 poly poly




•Complexity

•dist(s,t)-LBC





edge-cut: L = 2

s t

N(s)⋂N(t)

→ polynomial (O(m))

L Vertex Edge

1 – poly

2 poly poly




•Complexity

•dist(s,t)-LBC





vertex-cut: L = 3

s t

N(s)⋂N(t)

N(s)−N(s)⋂N(t) N(t)−N(s)

⋂N(t) L Vertex Edge

1 – poly

2 poly poly

3 poly




•Complexity

•dist(s,t)-LBC





vertex-cut: L = 3

s t

N(s)⋂N(t)


⋂N(t)

all vertices ∈ N(s)⋂N(t) are in min cut

L Vertex Edge

1 – poly

2 poly poly

3 poly




•Complexity

•dist(s,t)-LBC





vertex-cut: L = 3

s t

N(s)⋂N(t)


⋂N(t)

bipartite graph: min vertex cover =max matching [König ’31]→ (O(

√nm))

L Vertex Edge

1 – poly

2 poly poly

3 poly




•Complexity

•dist(s,t)-LBC





edge-cut: L = 3

polynomial:[McCormick and Mahjoub, ’03]

(simpler constructionwith length expanded graph)

L Vertex Edge

1 – poly

2 poly poly

3 poly poly




•Complexity

•dist(s,t)-LBC





vertex-cut: L = 4

polynomial:[Lovász, Neumann-Lara und Plummer ’78]

L Vertex Edge

1 – poly

2 poly poly

3 poly poly

4 poly




•Complexity

•dist(s,t)-LBC





edge-cut: L = 4

reduction from Vertex Cover(gap-preserving)

L Vertex Edge

1 – poly

2 poly poly

3 poly poly

4 poly APX-hard




•Complexity

•dist(s,t)-LBC





edge-cut: L = 4

u

v

s t

gadget represents edge u− v

L Vertex Edge

1 – poly

2 poly poly

3 poly poly

4 poly APX-hard




•Complexity

•dist(s,t)-LBC





edge-cut: L = 4

u

v

s t

this edge choice for min cut

L Vertex Edge

1 – poly

2 poly poly

3 poly poly

4 poly APX-hard




•Complexity

•dist(s,t)-LBC





edge-cut: L = 4

u

v

s t

force these edges to be in the cut as well

L Vertex Edge

1 – poly

2 poly poly

3 poly poly

4 poly APX-hard




•Complexity

•dist(s,t)-LBC





edge-cut: L = 4

u

v

s t

define that situation as vertex v is choosenfor VC (2 edges are choosen for min cut)

L Vertex Edge

1 – poly

2 poly poly

3 poly poly

4 poly APX-hard




•Complexity

•dist(s,t)-LBC





edge-cut: L = 4

u

v

s t

vice versa: start with this edge choice(defined as vertex u in VC)

L Vertex Edge

1 – poly

2 poly poly

3 poly poly

4 poly APX-hard




•Complexity

•dist(s,t)-LBC





edge-cut: L = 4

u

v

s t

forces this edge to be in the cut

→ APX-hard

L Vertex Edge

1 – poly

2 poly poly

3 poly poly

4 poly APX-hard




•Complexity

•dist(s,t)-LBC





vertex-cut: L = 5

reduction from Vertex Cover(different gadget, gap-preserving)

→ APX-hard

L Vertex Edge

1 – poly

2 poly poly

3 poly poly

4 poly APX-hard

5 APX-hard




•Complexity

•dist(s,t)-LBC





Lv,e Vertex Edge

1 poly poly

2 poly poly

3 poly poly

4 APX-hard APX-hard

[joint work with Georg Baier, Thomas Erlebach, and Alex Hall ’04]




•Complexity

•dist(s,t)-LBC




dist(s,t)-Length-bounded s− t−cuts

given• (un)directed graph G = (V,E)

• edge-capacities: u : E → Q≥0

• edge-lengths: d : E → Q≥0

directed graph G := G[Es] induced by all edges on shortests-t-paths• all s-t-paths in G are shortest paths• G can by computed using a modified Dijkstra-Algorithm

in G: determine a standard minimum s-t-cut S→ S is a minimum dist(s, t)-length-bounded s-t-cut in G

[Lovász, Neumann-Lara und Plummer ’78]




•Complexity

•dist(s,t)-LBC








directed graph G := G[Es] induced by all edges on shortests-t-paths

• all s-t-paths in G are shortest paths• G can by computed using a modified Dijkstra-Algorithm






•Complexity

•dist(s,t)-LBC








directed graph G := G[Es] induced by all edges on shortests-t-paths• all s-t-paths in G are shortest paths

• G can by computed using a modified Dijkstra-Algorithm






•Complexity

•dist(s,t)-LBC














•Complexity

•dist(s,t)-LBC









in G: determine a standard minimum s-t-cut S

→ S is a minimum dist(s, t)-length-bounded s-t-cut in G





•Complexity

•dist(s,t)-LBC














•Complexity

•dist(s,t)-LBC




Length-bounded s− t−cuts — approximation algorithm

given: G = (V,E), L > 0 and unit edge-lengths

determine the following cuts:• S0 = min. D-length-bnd s-t-cut in G (D := distG(s, t))• S1 = min. (D + 1)-length-bnd s-t-cut in G \ S0

. . .

• SL−D = min. L-length-bnd s-t-cut in G \ ∪L−D−1i=0 Si

Let S :=⋃L−Di=0 Si

→ S is a L-length-bounded s-t-cut in G

→ S is a (L+ 1− dist(s, t))-approximationsince: ∀i : |S∗| ≥ |Si|

[Georg Baier ’03]




•Complexity

•dist(s,t)-LBC






determine the following cuts:• S0 = min. D-length-bnd s-t-cut in G (D := distG(s, t))

• S1 = min. (D + 1)-length-bnd s-t-cut in G \ S0

. . .





[Georg Baier ’03]




•Complexity

•dist(s,t)-LBC







. . .





[Georg Baier ’03]




•Complexity

•dist(s,t)-LBC







. . .



→ S is a L-length-bounded s-t-cut in G→ S is a (L+ 1− dist(s, t))-approximation

since: ∀i : |S∗| ≥ |Si|

[Georg Baier ’03]


k−splittable s− t−flows




•Definition

• k const

• k(m and n)

• k(m)

• k(m) in simple graphs


Definition — k-splittable s− t−Flows

given:• Graph G = (V,E)

• Terminals s, t ∈ V• Edge capacities u : E → Q>0

• Bound k ∈ Z>0

k-splittable s-t-Flow

f : Ps,t → R≥0

|{P ∈ Ps,t|f(P ) > 0}| ≤ k

PSfrag replacements

s

t1

1

2

3

3

345 7

PSfrag replacements

s

t1

2

3457

[Baier, Köhler and Skutella ’02]




•Definition

• k const

• k(m and n)

• k(m)



Maximum k−splittable flows — k const

1k :

k const

2 3 ...

polynomial solvable NP-hard

• strongly NP-hard for fixed k ≥ 2 in undirected and directedgraphs (reduction from 3SAT)




•Definition

• k const

• k(m and n)

• k(m)




1k :

k const

2 3 ...



• not to approximate better than with a guarantee of 56




•Definition

• k const

• k(m and n)

• k(m)




1k :

k const

2 3 ...



• not to approximate better than with a guarantee of 56

(best result so far strongly NP-hard for directed graphs andnon-approximability bound k

k+1 )




•Definition

• k const

• k(m and n)

• k(m)



Maximum k−splittable flows — k(m,n)

1k :

k const

2 3 ...

1k :

k depending on m and n

2 m− n+ 1 m− n+ 2...


• each s− t−flow can be decomposed in at most m− n+ 2paths (follows from Ford and Fulkerson ’62)⇒ polynomial for k ≥ m− n+ 2 (standard max s− t−flow)




•Definition

• k const

• k(m and n)

• k(m)



Maximum k−splittable flows — k(m,n)

1k :

k const

2 3 ...

1k :


2 m− n+ 1 m− n+ 2...


• each s− t−flow can be decomposed in at most m− n+ 2paths (follows from Ford and Fulkerson ’62)⇒ polynomial for k ≥ m− n+ 2 (standard max s− t−flow)

• strongly NP-hard for 2 ≤ k ≤ m− n+ 1(reductions from 3-PARTITION)




•Definition

• k const

• k(m and n)

• k(m)



Maximum k−splittable flows — k(m)

1k :

k const

2 3 ...

1k :


2 m− n+ 1 m− n+ 2...

1k :

k depending on m

2 ... m− 1m− 2


• polynomial for n = 2⇒ strongly NP-hard for 2 ≤ k ≤ m− 2




•Definition

• k const

• k(m and n)

• k(m)



Maximum k−splittable flows — k(m) in simple graphs

1k :

k const

2 3 ...

1k :


2 m− n+ 1 m− n+ 2...

1k :

gap

m− (log log p(m,n))εk depending on m

2 m−mε...in simple graphs

1k :

k depending on m

2 ... m− 1m− 2


• polynomial for k = m− c, const c ∈ N: (prev. results for big n)case n < const⇒ m < const (simple graph)




•Definition

• k const

• k(m and n)

• k(m)




1k :

k const

2 3 ...

1k :


2 m− n+ 1 m− n+ 2...

1k :

gap



1k :

k depending on m

2 ... m− 1m− 2


• polynomial for k = m− c, const c ∈ N: (prev. results for big n)case n < const⇒ m < const (simple graph)

• number of simple paths in G is bounded by a constant(polynomial sovable with LP)




•Definition

• k const

• k(m and n)

• k(m)




1k :

k const

2 3 ...

1k :


2 m− n+ 1 m− n+ 2...

1k :

gap



1k :

k depending on m

2 ... m− 1m− 2


• better estimating: polynomial for k ≥ m− (log log p(m,n))ε

∀ polynomials p(n,m), ε > 0




•Definition

• k const

• k(m and n)

• k(m)




1k :

k const

2 3 ...

1k :


2 m− n+ 1 m− n+ 2...

1k :

gap



1k :

k depending on m

2 ... m− 1m− 2


• better estimating: polynomial for k ≥ m− (log log p(m,n))ε

∀ polynomials p(n,m), ε > 0

• strongly NP-hard for 2 ≤ k ≤ m−mε

(reduction from 3-PARTITION)




•Definition

• k const

• k(m and n)

• k(m)



Maximum k−splittable flows — complexity

1k :

k const

2 3 ...

1k :


2 m− n+ 1 m− n+ 2...

1k :

gap



1k :

k depending on m

2 ... m− 1m− 2


[Georg Baier, Ekkehard Köhler, Ronald Koch, and Ines Spenke ’04]




•Definition

• k const

• k(m and n)

• k(m)



Overview

Part IShortest path acceleration methods (A1, B1, D)Complexity of length-bounded s− t−cuts (A1)Complexity of k-splittable flows (A2)

Part IIOn k-splittable flows with path capacities (A2/A3)Flows over time and earliest arrival flows (B/C)

Part IIIEvacuation with flow-dependent transit times (C2)

Nadine k-Institut Maren Skutella, - TU Berlin › coga › pub › Jahres...s-t-cuts k-splitt. o...

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