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Los Alamos National Laboratory LA-UR-19-24756 Efficient Delivery with Mobile Agents Andreas B artschi NSEC/CNLS, baertschi@lanl.gov CNLS Postdoc Seminar April 18, 2019 Managed by Triad National Security, LLC for the U.S. Department of

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Los Alamos National Laboratory LA-UR-19-24756

Efficient Delivery with Mobile Agents

Andreas B¨ artschi NSEC/CNLS, baertschi@lanl.gov CNLS Postdoc Seminar April 18, 2019

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

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Los Alamos National Laboratory

Model of delivery

Setting undirected graph G = (V , E) with edges e ∈ E having lengths ℓe m packages: source si and target ti k agents, each starting at node pi, budget βi, weight ωi & velocity υi, able to transport 1 package at a time. s t

400 km 300 km 600 km 400 km 400 km

Assumptions agents cooperate by global, centralized coordination handovers possible at nodes V as well as inside edges Task: Find an efficient delivery in terms of energy consumption E terms of form ωi · ℓe delivery time T terms of form 1

υi · ℓe

constrained resources βi range of agents

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 2

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Los Alamos National Laboratory

Outline

Resource-efficiency Energy-efficiency Time-efficiency

F6 F5 F4 F3 F2 F1 ∼ n size: DAG structure

M[1, 1] M[2, 2] M[3, 3] M[4, 4] M[1, 2] M[2, 3] M[3, 4] M[1, 3] M[2, 4] M[1, 4] 10′000′000 100′000 10′000′000 101′000 110′000 201′000

s t

400 km 300 km 600 km ω1 = 12ℓ/100km υ1 = 10km/h ω2 = 6ℓ/100km υ2 = 30km/h ω3 = 5ℓ/100km υ3 = 50km/h ω4 = 7ℓ/100km υ4 = 100km/h ω5 = 5ℓ/100km υ5 = 80km/h 400 km 400 km

1 Examples and Results

Resource-efficiency: Budgets only Energy-efficiency: Weights only Time-efficiency: Velocities only

2 Dynamic programming

Technique Matrix chain multiplication

3 A detailed discussion

Combining energy- and time-efficiency OPT characterization Polynomial-time algorithm

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 3

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Los Alamos National Laboratory

Examples and Results

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 4

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Los Alamos National Laboratory

Agents with budgets

The agents’ resources constrain how far each agent can go (budgets βi). Can we decide whether there is a package delivery which respects all battery constraints?

s t

400 km 300 km 600 km β1 = 200km β2 = 300km β3 = 1100km β4 = 200km β5 = 600km 400 km 400 km

not on shortest path in-edge-handovers no clear characterization

[1] B., Chalopin, Das, Disser, Geissmann, Graf, Labourel, Mihal´ ak: Collaborative delivery with energy-constrained mobile robots. Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 5

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Los Alamos National Laboratory

Agents with budgets

The agents’ resources constrain how far each agent can go (budgets βi). Can we decide whether there is a package delivery which respects all battery constraints?

s t

400 km 300 km 600 km β1 = 200km β2 = 300km β3 = 1100km β4 = 200km β5 = 600km 400 km 400 km

not on shortest path in-edge-handovers no clear characterization

[1] B., Chalopin, Das, Disser, Geissmann, Graf, Labourel, Mihal´ ak: Collaborative delivery with energy-constrained mobile robots. Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 5

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Los Alamos National Laboratory

Resource-efficiency NP-hard, even for a single package (m = 1) on simple graphs. Energy-efficiency Time-efficiency

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 6

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Los Alamos National Laboratory

Agents with weights

Each agent has its individual energy consumption (weight) ωi. Can we optimize the total energy consumption E needed to deliver the package?

s t

400 km 300 km 600 km ω1 = 12ℓ/100km ω2 = 6ℓ/100km ω3 = 5ℓ/100km ω4 = 7ℓ/100km ω5 = 5ℓ/100km 400 km 400 km

not on shortest path vertex handovers decreasing weights

[2] B., Chalopin, Das, Disser, Graf, Hackfeld, Penna: Energy-Efficient Delivery by Heterogeneous Mobile Agents. [3] B., Graf, Penna: Truthful Mechanisms for Delivery with Agents. Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 7

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Los Alamos National Laboratory

Resource-efficiency NP-hard, even for a single package (m = 1) on simple graphs. Energy-efficiency Agents face 3 major challenges: Time-efficiency Collaboration: How to work together on a package? – 2-approximation without collaboration. Planning: Which route to take? – NP-hard, polynomial-time 2-approximation. Coordination: How to assign agents to packages? – NP-hard, polynomial-time max ωi

ωj -approximation.

⇒ Polynomial-time 4 max ωi

ωj -approximation.

⇒ Polynomial-time algorithm for one package.

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 8

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Los Alamos National Laboratory

Agents with velocities

Each agent has its individual velocity υi. Can we optimize the total time T needed to deliver the package?

s t

400 km 300 km 600 km υ1 = 20km/h υ2 = 20km/h υ3 = 76km/h υ4 = 40km/h υ5 = 60km/h 400 km 400 km

not on shortest path (multiple) in-edge-handovers increasing velocities

[4] B., Graf, Mihal´ ak: Collective fast delivery by energy-efficient agents. Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 9

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Los Alamos National Laboratory

Resource-efficiency NP-hard, even for a single package (m = 1) on simple graphs. Energy-efficiency Agents face 3 major challenges: Time-efficiency Collaboration: How to work together on a package? – 2-approximation without collaboration. Existence of optima is unclear, maybe only infima exist. Planning: Which route to take? – NP-hard, polynomial-time 2-approximation. − → NP-hard. Coordination: How to assign agents to packages? – NP-hard, polynomial-time max ωi

ωj -approximation.

⇒ Polynomial-time 4 max ωi

ωj -approximation.

⇒ Polynomial-time algorithm for one package. Poly-time algo. for one package. Can we combine energy- and time-efficiency for m = 1?

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 10

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Los Alamos National Laboratory

Combining energy- and time-efficiency

Each agent has its individual weight ωi and velocity υi. We want a delivery that is both efficient in its energy consumption and its delivery time: Fastest delivery among all energy-optimum ones. Task: lexicographically minimize (E, T ). Energy-optimum delivery among all fastest ones. Task: lexicographically minimize (T , E). Tradeoff between the two measures. Task: minimize ε · T + (1 − ε) · E, ε ∈ (0, 1). ⇒ Part 3. [5]        ⇒ NP-hard. [4]

[4] B., Graf, Mihal´ ak: Collective fast delivery by energy-efficient agents. [5] B., Tschager: Energy-Efficient Fast Delivery by Mobile Agents. Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 11

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Los Alamos National Laboratory

Dynamic programming

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 12

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Los Alamos National Laboratory

Dynamic programming: Technique

Programming technique which can be used if an optimal solution of a problem can be found by combining optimal solutions of subproblems: optimal substructure. Applications: Graph: Shortest Paths & Dijkstra & Floyd-Warshall Bioinformatics: De Novo Peptide Sequencing Economics: Optimal Saving Control Theory Matrix chain multiplication Toy Example: Fibonacci Sequence Fn = Fn−1 + Fn−2 F2 = 1 F1 = 1 Task: Compute n-th Fibonacci number.

F6 F5 F4 F4 F3 F3 F3 F2 F2 F2 F2 F2 F1 F1 F1 ∼ 1.6n size:

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 13

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Los Alamos National Laboratory

Dynamic programming: Technique

Programming technique which can be used if an optimal solution of a problem can be found by combining optimal solutions of subproblems: optimal substructure. Applications: Graph: Shortest Paths & Dijkstra & Floyd-Warshall Bioinformatics: De Novo Peptide Sequencing Economics: Optimal Saving Control Theory Matrix chain multiplication Toy Example: Fibonacci Sequence Fn = Fn−1 + Fn−2 F2 = 1 F1 = 1 Task: Compute n-th Fibonacci number.

F6 F5 F4 F3 F2 F1 ∼ n size: DAG structure

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 13

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Los Alamos National Laboratory

Dynamic programming: Matrix chain multiplication

A1 A2 A3 A4 n

q m n

m = 100, n = 10, o = 10′000, p = 1, q = 1′000.

m·o·q=109

m·n·o=107
·p·q=107
(A1 × A2) × (A3 × A4)

(A1 × A2) × (A3 × A4) ⇒ 1′020′000′000

Find best multiplication order by: Testing all ways to insert parentheses: ∼ 4(#Matrices) many!

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 14

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Los Alamos National Laboratory

Dynamic programming: Matrix chain multiplication

A1 A2 A3 A4 n

q m n

m = 100, n = 10, o = 10′000, p = 1, q = 1′000.

m·p·q=105

m·o·p=106
m·n·o=107
((A1 × A2) × A3) × A4

((A1 × A2) × A3) × A4 ⇒ 11′100′000 (A1 × A2) × (A3 × A4) ⇒ 1′020′000′000

Find best multiplication order by: Testing all ways to insert parentheses: ∼ 4(#Matrices) many!

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 14

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Los Alamos National Laboratory

Dynamic programming: Matrix chain multiplication

A1 A2 A3 A4 n

q m n

m = 100, n = 10, o = 10′000, p = 1, q = 1′000.

A1 × A2 × A3 × A4

(A1 × (A2 × A3)) × A4 ⇒ 201′000 ((A1 × A2) × A3) × A4 ⇒ 11′100′000 (A1 × A2) × (A3 × A4) ⇒ 1′020′000′000

Find best multiplication order by: Testing all ways to insert parentheses: ∼ 4(#Matrices) many! Dynamic Programming: ∼ (#Matrices)2 subproblems: M[x, y] = Cost of best parentheses for Ax × · · · × Ay = min

x≤i<y

M[x,i] + M[i+1,y]

+ cost of multiplying two subproblems

M[1, 1]

M[2, 2] M[3, 3] M[4, 4] M[1, 2] M[2, 3] M[3, 4] M[1, 3] M[2, 4] M[1, 4]

10′000′000 100′000 10′000′000 101′000 110′000 201′000

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 14

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Los Alamos National Laboratory

A detailed discussion

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 15

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Los Alamos National Laboratory

Agents with weights and velocities

Each agent has its individual weight ωi and velocity υi. Among all energy-optimum deliveries, can we find the fastest?

s t

400 km 300 km 600 km ω1 = 12ℓ/100km υ1 = 10km/h ω2 = 6ℓ/100km υ2 = 30km/h ω3 = 5ℓ/100km υ3 = 50km/h ω4 = 7ℓ/100km υ4 = 100km/h ω5 = 5ℓ/100km υ5 = 80km/h 400 km 400 km

(E, T ) = (84, 38) (E, T ) = (84, 40)

not on shortest path 0 or 1 in-edge-handovers decreasing tuples (ωi, υ−1

)

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 16

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Los Alamos National Laboratory

Characterization of an optimum delivery

Theorem (OPT characterization)

There is an optimum delivery, with the involved agents denoted by 1, . . . , i, i + 1, . . . , k, in which the following holds for each consecutive pair of agents: Decreasing weights: ωi ≥ ωi+1. If ωi = ωi+1, then υi < υi+1. If ωi = ωi+1, then agent i + 1 does not move without the package. υi < υi+1 < . . . ω1 = ω2 = . . . > ωi = ωi+1 = . . . > . . . = ωk

x

y

⇒ First look at the problem for each weight class Wj separately.

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 17

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Los Alamos National Laboratory

Uniform energy consumption weights

s t

4 1 2 5 2 4 2 2 1 7 6 9 2 9 8 6 6 4

t s 4 1 4 9 9 5

10 11 19 14 (3, 1) (3, 2) (3, 5) (3, 4)

Example: 4 agents, weight 3, velocities υ1 = 1, υ2 = 2, υ3 = 4, υ4 = 5. Approach:

1 Move closest agent to source s. Costs (E, T )[p1] = (3 · 3, 3/1) = (9, 3). 2 Order agents by increasing velocity. Transform graph to DAG. Compute

(E, T )[pi] = the energy consumption E[pi] and delivery time T [pi] of an optimum delivery of the package from s to pi, using only agents 1, . . . , i − 1.

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 18

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Los Alamos National Laboratory

Uniform energy consumption weights

s t

4 1 2 5 2 4 2 2 1 7 6 9 2 9 8 6 6 4

s 4 1 4 9 9 5

(3, 1) (3, 2) (3, 5) (3, 4)

t

10 11 19 14

Example: 4 agents, weight 3, velocities υ1 = 1, υ2 = 2, υ3 = 4, υ4 = 5. Approach:

1 Move closest agent to source s. Costs (E, T )[p1] = (3 · 3, 3/1) = (9, 3). 2 Order agents by increasing velocity. Transform graph to DAG. Compute

(E, T )[p2] = (E, T )[p1] + (5 · 3, 5/1) = (24, 8). (E, T )[p3] = min{(E, T )[p1] + (9 · 3, 9/1), (E, T )[p2] + (4 · 3, 4/2)} = (36, 10). (E, T )[p4] = min{. . . , (E, T )[p2] + (4 · 3, 4/2), . . .} = (36, 10).

3 Compute optimum delivery time among energy-optimum deliveries: (E, T ) = (66, 12).

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 18

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Los Alamos National Laboratory

Vertex handovers

u i v i + 1 u i v i + 1 u i v i + 1 wi > 2wi+1 wi < 2wi+1 If ωi = 2ωi+1, then agent i does not handover the package to i + 1 inside an edge. Assume we have ωi = 2ωj ∀i, j. ⇒ We can use the precomputed weight class solutions. Define the subproblems (E, T )[j, y] = the energy consumption E[j, y] and delivery time T [j, y] of an

ptimum delivery of the package from s up to node y,

using only agents from the first j weight classes W1, . . . , Wj. ⇒ We can compute (E, T )[j, y] from all smaller subproblems (E, T )[j − 1, x]!

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 19

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Los Alamos National Laboratory

In-edge handovers

u i v i + 1 wi = 2wi+1 = 4wi+2 i + 2 u v ? ?

arrival time at u velocity

i + 1

N O

We can have at most one in-edge-handover per edge. But: Which agents are involved in an in-edge-handover? Pareto frontier! Adapt previous methods as follows:

1 Incorporate the Pareto frontier into the weight class computations. 2 Incorporate in-edge-handovers into the main dynamic program.

Running time: Preprocessing poly(APSP + k + |V |) Per weight class: poly(|V | · k2 + |V |2 · k) Main dynamic program: poly(k · |V | · |V |)        poly(k + |V |3)

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 20

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Los Alamos National Laboratory

Resource-efficiency, m = 1 NP-hard on simple graphs. Energy-efficiency Time-efficiency Collaboration: 2-apx without collaboration Existence of optima is un- clear (maybe only infima) Planning: NP-hard, poly-time 2-apx NP-hard Coordination: NP-hard, poly-time max ωi

ωj -apx

Poly-time algorithm for m = 1 Poly-time algo. for m = 1 Combining energy- and time-efficiency, m = 1 Some versions NP-hard, but lexicographically minimizing (E, T ) can be done in polynomial time.

https://xkcd.com/1925/ Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 21