Energy-efficient Delivery by Heterogeneous Mobile...

Energy-efficient Delivery by Heterogeneous Mobile Agents

Andreas Bärtschi

Jérémie Chalopin, Shantanu Das, Yann Disser, Daniel Graf, Jan Hackfeld, Paolo Penna

Department of Computer Science

Motivation / Toy model

s t

10`/100 km

7`/100 km 6`/100 km 5`/100 km

100 km 100 km 100 km 100 km

100 km 100 km 100 km

100km

100km

Department of Computer Science Andreas Bärtschi Maastricht Visit Nederland June 15, 2017 2 / 10


s t

10`/100 km

7`/100 km 6`/100 km 5`/100 km

2 · 6`



s t

10`/100 km

7`/100 km 6`/100 km 5`/100 km

2 · 6`+ 1 · 6`



s t

10`/100 km

7`/100 km 6`/100 km 5`/100 km

2 · 6`+ 4 · 6`



s t

10`/100 km

7`/100 km 6`/100 km 5`/100 km

2 · 6`+ 4 · 6` = 36`



s t

10`/100 km

7`/100 km 6`/100 km 5`/100 km

100 km 100 km 100 km 100 km

100 km 100 km 100 km

100km

100km

We extend this with:

multiple items to be delivered (messages)

varying road lengths (edge lengths)

many vehicles (mobile agents)

handovers at cities (nodes)



s t

10`/100 km

7`/100 km 6`/100 km 5`/100 km








s t

10`/100 km

7`/100 km 6`/100 km 5`/100 km

10`








s t

10`/100 km

7`/100 km 6`/100 km 5`/100 km

10`+ 7`








s t

10`/100 km

7`/100 km 6`/100 km 5`/100 km

10`+ 7`+ 2 · 6`








s t

10`/100 km

7`/100 km 6`/100 km 5`/100 km

10`+ 7`+ 2 · 6`+ 5`








s t

10`/100 km

7`/100 km 6`/100 km 5`/100 km

10`+ 7`+ 2 · 6`+ 5` = 34`







Model

Setting

undirected graph G = (V ,E )with edges E having lengths

m messages, given bysource-target node pairs (si , ti )

anyone can use any edge

Agents

k agents each with capacity κ and

starting position pi ∈ Vrate of energy consumption wialso called weights

Assumptions

global coordination

handovers possible at nodes V

Task

Find a delivery schedule which minimizesoverall energy cost, given by the weightedsum of each agent’s travel distance di :

k∑i=1

wi · di


1 IntroductionMotivationModel

2 Collaboration, Planning and CoordinationCollaborationPlanningCoordination

3 Conclusion


Collaboration, Planning and CoordinationC

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How should the agents work together on each message?

Defines all handover points of a message and their order.

An agent then carries it between consecutive handover points.

→ This includes the case of a single message.

Which route should each agent take?

Gives an order of all pick-ups and all drop-offs of each agent.

→ This includes the case of a single agent.

How should the agents be assigned to the messages?

Assigns a subset of the messages to each agent.

Depends on the starting position of an agent, and on its weight.

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Collaboration, Planning and CoordinationC

olla

bo

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Pla

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ord

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n

How should the agents work together on each message?

Defines all handover points of a message and their order.

An agent then carries it between consecutive handover points.

→ This includes the case of a single message.Which route should each agent take?

Gives an order of all pick-ups and all drop-offs of each agent.

→ This includes the case of a single agent.How should the agents be assigned to the messages?

Assigns a subset of the messages to each agent.

Depends on the starting position of an agent, and on its weight.

sim

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aneo

usl

y+

mor

ed

etai

ls!


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How should the agents work together on each message?m = 1: Agent weights are decreasing → dynamic programming

m >1: No characterization.

How far off is a schedule in which agents do not collaborate?

Theorem (Benefit of Collaboration BoC)

The benefit of collaboration is at most 2.


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How should the agents work together on each message?m > 1: No characterization.

s1 s2

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t1

t2 t3

4 4

3 3

2

2

w1=2

w2=3

1

2

3





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s1 s2

s3

t1

t2 t3

4 4

3 3

2

2

w1=2

w2=3

1

2

3 energy cost= 2 · (4 + 3 + 3 + 4 + 3)

+ 3 · (2 + 2) = 46





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s1 s2

s3

t1

t2 t3

4 4

3 3

2

2

w1=2

w2=3

1

2

3 energy cost= 2 · (4 + 3 + 3 + 4 + 2)

+ 3 · (2 + 3) = 47





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s1 s2

s3

t1

t2 t3

4 4

3 3

2

2

w1=2

w2=3

1

2

3 energy cost= 2 · (4 + 3 + 3 + 4 + 2)

+ 3 · (2 + 3) = 47




Proof Idea: Build non-collaborative solution froman arbitrary optimum (with collaboration).

1 Trajectory graph + backward edges

2 Generalization of Euler tours


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s1 s2

s3

t1

t2 t3

4 4

3 3

2

2

w1=2

w2=3

1

2

3 energy cost= 2 · (4 + 3 + 3 + 4 + 2)

+ 3 · (2 + 3) = 47




For κ = 1, this holds even if in the non-collaboration scenario(i) messages are directly delivered, and(ii) agents return to their starting position in the end.


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s1 s2

s3

t1

t2 t3

4 4

3 3

2

2

w1=2

w2=3

1

2

3 energy cost= 2 · 2 · (4 + 3 + 4 + 2 + 2 + 3)= 72






Co

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s1 s2

s3

t1

t2 t3

4 4

3 3

2

2

w1=2

w2=3

1

2

3 κ = 1: no collaboration+ direct delivery + return:

BoC ≤ 2.How far off is a schedule in which agents do not collaborate?





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Which route should each agent take?NP-hard on planar graphs even for a single agent:

Hp1

G

1 1 1

1 1 1

1 1 1

1

1 1 1 1

1 1 1 1

1 1

0 0 0

0 0

0 0 0 0

x

0 0 0

d1 = |V |+ x

p′1

0

Similarly: NP-hard to approximate better than 1 + 1122 · 13 = 367366 .Theorem (Planning restricted to direct delivery)

For κ = 1, restricted planning can be 2−approximated.


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Hp1

G

1 1 1

1 1 1

1 1 1

1

1 1 1 1

1 1 1 1

1 1

0 0 0

0 0

0 0 0 0

x

0 0 0

d1 = |V |+ x

p′1

0

Similarly: NP-hard to approximate better than 1 + 1122 · 13 = 367366 .

Theorem (Planning restricted to direct delivery)

For κ = 1, restricted planning can be 2−approximated.


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Hp1

G

1 1 1

1 1 1

1 1 1

1

1 1 1 1

1 1 1 1

1 1

0 0 0

0 0

0 0 0 0

x

0 0 0

d1 = |V |+ x

p′1

0

Similarly: NP-hard to approximate better than 1 + 1122 · 13 = 367366 .Theorem (Planning restricted to direct delivery)

For κ = 1, restricted planning can be 2−approximated.Proof Idea:

1 Build a minimum spanning tree that contains all (si , ti )-edges,adding the other edges in a Kruskal-like fashion.

2 Traverse the minimum spanning tree twice.


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How should the agents be assigned to the messages?NP-hard on planar graphs even in simple cases, where there isno collaboration and a total message order:

v1false

true

true

falsetrue

false true

false

v1 v2 v3 v4

v1 ∨ v2 ∨ v4 v2 ∨ v3 ∨ v4

v1 ∨ v2 v2 ∨ v3 ∨ v4 v4

v2 v3 v4

G(F )H

Theorem

For κ = 1 and uniform weights,and given complete information about collaboration and planning,coordination can be solved in polynomial time.

Increasing each weight to uniform weight wmax thus gives awmaxwmin

-approximation of coordination.


Conclusion

(for capacity κ = 1) Open: κ > 1

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s1 s2

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t1

t2 t3

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w1=2

w2=3

1

2


BoC ≤ 2.

κ = 1: For each agent, compute traversal of a minimum spanningtree that connects its starting position to its subset of messages;direct delivery of each message → 2−approximation.κ = 1: Increasing all weights to uniform weight wmax results ina loss of at most a factor of wmaxwmin .

Assign agents to messages (use no collaboration + direct delivery)paying attention only to their starting position.

(wmax

wmin·2·2

)-ap

prox

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ion


Conclusion

(for capacity κ = 1) Open: κ > 1

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s1 s2

s3

t1

t2 t3

4 4

3 3

2

2

w1=2

w2=3

1

2


BoC ≤ 2.

κ = 1: For each agent, compute traversal of a minimum spanningtree that connects its starting position to its subset of messages;direct delivery of each message → 2−approximation.

κ = 1: Increasing all weights to uniform weight wmax results ina loss of at most a factor of wmaxwmin .


(

wmax

wmin

·2·2

)-ap

prox

imat

ion


Conclusion (for capacity κ = 1)

Open: κ > 1

Co

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s1 s2

s3

t1

t2 t3

4 4

3 3

2

2

w1=2

w2=3

1

2


BoC ≤ 2.

κ = 1: For each agent, compute traversal of a minimum spanningtree that connects its starting position to its subset of messages;direct delivery of each message → 2−approximation.

κ = 1: Increasing all weights to uniform weight wmax results ina loss of at most a factor of wmaxwmin .


(

wmax

wmin·2

·2)-

appr

oxim

atio

n


Conclusion (for capacity κ = 1)

Open: κ > 1

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s1 s2

s3

t1

t2 t3

4 4

3 3

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2

w1=2

w2=3

1

2


BoC ≤ 2.



(wmax

wmin·2·2

)-ap

prox

imat

ion


Conclusion (for capacity κ = 1) Open: κ > 1C

olla

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s1 s2

s3

t1

t2 t3

4 4

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w1=2

w2=3

1

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BoC ≤ 2.



(wmax

wmin·2·2

)-ap

prox

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ion


IntroductionMotivationModel

Collaboration, Planning and CoordinationCollaborationPlanningCoordination

Conclusion

Energy-efficient Delivery by Heterogeneous Mobile...

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