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
Los Alamos National Laboratory Model of delivery Setting undirected graph G = ( V , E ) with edges e ∈ E having lengths ℓ e 400 km m packages: source s i and target t i 300 km 400 km s t 400 km 600 km k agents, each starting at node p i , budget β i , weight ω i & velocity υ i , able to transport 1 package at a time. Assumptions Task : Find an efficient delivery in terms of agents cooperate by energy consumption E � terms of form ω i · ℓ e global, centralized coordination � terms of form 1 delivery time T υ i · ℓ e handovers possible at nodes V constrained resources β i � range of agents as well as inside edges Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 2
Los Alamos National Laboratory Outline 1 Examples and Results Resource-efficiency Resource-efficiency: Budgets only Energy-efficiency Time-efficiency Energy-efficiency: Weights only Time-efficiency: Velocities only F 6 M [1 , 4] 201 ′ 000 F 5 2 Dynamic programming M [1 , 3] M [2 , 4] F 4 101 ′ 000 110 ′ 000 Technique F 3 M [1 , 2] M [2 , 3] M [3 , 4] 10 ′ 000 ′ 000 100 ′ 000 10 ′ 000 ′ 000 Matrix chain multiplication F 2 M [1 , 1] M [2 , 2] M [3 , 3] M [4 , 4] DAG size: 0 0 0 0 structure F 1 ∼ n 3 A detailed discussion ω 4 = 7 ℓ/ 100km ω 5 = 5 ℓ/ 100km Combining energy- and time-efficiency υ 4 = 100km /h υ 5 = 80km /h 400 km 300 km 400 km OPT characterization s t 400 km 600 km ω 1 = 12 ℓ/ 100km ω 2 = 6 ℓ/ 100km ω 3 = 5 ℓ/ 100km Polynomial-time algorithm υ 1 = 10km /h υ 2 = 30km /h υ 3 = 50km /h Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 3
Los Alamos National Laboratory Examples and Results Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 4
Los Alamos National Laboratory Agents with budgets The agents’ resources constrain how far each agent can go ( budgets β i ). Can we decide whether there is a package delivery which respects all battery constraints? β 4 = 200km β 5 = 600km 400 km 300 km 400 km s t 400 km 600 km β 1 = 200km β 2 = 300km β 3 = 1100km 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
Los Alamos National Laboratory Agents with budgets The agents’ resources constrain how far each agent can go ( budgets β i ). Can we decide whether there is a package delivery which respects all battery constraints? β 4 = 200km β 5 = 600km 400 km 300 km 400 km s t 400 km 600 km β 1 = 200km β 2 = 300km β 3 = 1100km 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
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
Los Alamos National Laboratory Agents with weights Each agent has its individual energy consumption (weight) ω i . Can we optimize the total energy consumption E needed to deliver the package? ω 4 = 7 ℓ/ 100km ω 5 = 5 ℓ/ 100km 400 km 300 km 400 km s t 400 km 600 km ω 1 = 12 ℓ/ 100km ω 2 = 6 ℓ/ 100km ω 3 = 5 ℓ/ 100km 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
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 ω i ω j -approximation. ⇒ Polynomial-time 4 max ω i ω j -approximation. ⇒ Polynomial-time algorithm for one package. Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 8
Los Alamos National Laboratory Agents with velocities Each agent has its individual velocity υ i . Can we optimize the total time T needed to deliver the package? υ 4 = 40km /h υ 5 = 60km /h 400 km 300 km 400 km s t 400 km 600 km υ 1 = 20km /h υ 2 = 20km /h υ 3 = 76km /h 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
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? Existence of optima is unclear, – 2-approximation without collaboration. 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 ω i ω j -approximation. ⇒ Polynomial-time 4 max ω i ω j -approximation. Poly-time algo. for one package. ⇒ Polynomial-time algorithm 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
Los Alamos National Laboratory Combining energy- and time-efficiency Each agent has its individual weight ω i and velocity υ i . We want a delivery that is both efficient in its energy consumption and its delivery time: Fastest delivery among all energy-optimum ones. ⇒ Part 3 . [5] Task: lexicographically minimize ( E , T ). Energy-optimum delivery among all fastest ones. Task: lexicographically minimize ( T , E ). ⇒ NP-hard. [4] Tradeoff between the two measures. Task: minimize ε · T + (1 − ε ) · E , ε ∈ (0 , 1). [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
Los Alamos National Laboratory Dynamic programming Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 12
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: Toy Example: F 6 Fibonacci Sequence Graph: Shortest Paths & Dijkstra & Floyd-Warshall F n = F n − 1 + F n − 2 F 5 Bioinformatics: F 2 = 1 F 4 F 4 De Novo Peptide Sequencing F 1 = 1 Economics: F 3 F 3 F 3 Task: Compute n -th Optimal Saving Fibonacci number. F 2 F 2 F 2 F 2 F 2 Control Theory Matrix chain multiplication size: F 1 F 1 ∼ 1 . 6 n F 1 Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 13
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: Toy Example: F 6 Fibonacci Sequence Graph: Shortest Paths & Dijkstra & Floyd-Warshall F n = F n − 1 + F n − 2 F 5 Bioinformatics: F 2 = 1 F 4 De Novo Peptide Sequencing F 1 = 1 Economics: F 3 Task: Compute n -th Optimal Saving Fibonacci number. F 2 Control Theory Matrix chain multiplication DAG size: structure F 1 ∼ n Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov 4/18/2019 | 13
Los Alamos National Laboratory Dynamic programming: Matrix chain multiplication p q o A 4 p A 2 n n A 3 o m · o · q =10 9 m A 1 � �� � m · n · o =10 7 o · p · q =10 7 ( A 1 × A 2 ) × ( A 3 × A 4 ) � �� � � �� � m = 100 , n = 10 , o = 10 ′ 000 , p = 1 , q = 1 ′ 000. ( A 1 × A 2 ) × ( A 3 × A 4 ) ⇒ 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
Los Alamos National Laboratory Dynamic programming: Matrix chain multiplication p q o A 4 p A 2 n n m · p · q =10 5 A 3 � �� � (( A 1 × A 2 ) × A 3 ) × A 4 o m · o · p =10 6 m A 1 11 ′ 100 ′ 000 ⇒ � �� � m · n · o =10 7 ( A 1 × A 2 ) × ( A 3 × A 4 ) � �� � m = 100 , n = 10 , o = 10 ′ 000 , p = 1 , q = 1 ′ 000. (( A 1 × A 2 ) × A 3 ) × A 4 ⇒ 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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