Planning and Optimization D7. M&S: Generic Algorithm and Heuristic Properties Gabriele R¨ oger and Thomas Keller Universit¨ at Basel November 7, 2018
Generic Algorithm Heuristic Properties Summary Content of this Course Tasks Progression/ Regression Classical Complexity Heuristics Planning MDPs Uninformed Search Probabilistic Heuristic Search Monte-Carlo Methods
Generic Algorithm Heuristic Properties Summary Content of this Course: Heuristics Abstractions Delete Relaxation in General Pattern Abstraction Databases Merge & Heuristics Landmarks Shrink Potential Heuristics Cost Partitioning
Generic Algorithm Heuristic Properties Summary Generic Algorithm
Generic Algorithm Heuristic Properties Summary Content of this Course: Merge & Shrink Synchronized Product Merge & Shrink Algorithm Merge & Shrink Heuristic Properties Strategies Label Reduction
Generic Algorithm Heuristic Properties Summary Generic Merge-and-shrink Abstractions: Outline Using the results from the previous chapter, we can develop the ideas of a generic abstraction computation procedure that takes all state variables into account: Initialization step: Compute all abstract transition systems for atomic projections to form the initial abstraction collection. Merge steps: Combine two abstract systems in the collection by replacing them with their synchronized product. (Stop once only one transition system is left.) Shrink steps: If the abstractions in the collection are too large to compute their synchronized product, make them smaller by abstracting them further (applying an arbitrary abstraction to them). We explain these steps with our running example.
Generic Algorithm Heuristic Properties Summary Back to the Running Example ALR ARL LLR RRL ALL ARR LRR LLL RRR RLL BLL BRR LRL RLR BRL BLR Logistics problem with one package, two trucks, two locations: state variable package: { L , R , A , B } state variable truck A: { L , R } state variable truck B: { L , R }
Generic Algorithm Heuristic Properties Summary Initialization Step: Atomic Projection for Package T π { package } : M ⋆⋆⋆ M ⋆⋆⋆ A M ⋆⋆⋆ DAR PAL DAL PAR L R PBL DBR DBL PBR B M ⋆⋆⋆
Generic Algorithm Heuristic Properties Summary Initialization Step: Atomic Projection for Truck A T π { truck A } : PAL,DAL,MB ⋆⋆ , PAR,DAR,MB ⋆⋆ , PB ⋆ ,DB ⋆ PB ⋆ ,DB ⋆ MALR L R MARL
Generic Algorithm Heuristic Properties Summary Initialization Step: Atomic Projection for Truck B T π { truck B } : PBL,DBL,MA ⋆⋆ , PBR,DBR,MA ⋆⋆ , PA ⋆ ,DA ⋆ PA ⋆ ,DA ⋆ MBLR L R MBRL current collection: {T π { package } , T π { truck A } , T π { truck B } }
Generic Algorithm Heuristic Properties Summary First Merge Step T 1 := T π { package } ⊗ T π { truck A } : MB ⋆⋆ MB ⋆⋆ MALR AL AR PAL DAR MARL L P A A D R MB ⋆⋆ MB ⋆⋆ MB ⋆⋆ MB ⋆⋆ MALR MALR P R B B L D LL LR RL RR DBL R MARL MARL B P R P B B D L DBL PBR MALR BL BR MARL MB ⋆⋆ MB ⋆⋆ current collection: {T 1 , T π { truck B } }
Generic Algorithm Heuristic Properties Summary Need to Simplify? If we have sufficient memory available, we can now compute T 1 ⊗ T π { truck B } , which would recover the complete transition system of the task. However, to illustrate the general idea, let us assume that we do not have sufficient memory for this product. More specifically, we will assume that after each product operation we need to reduce the result transition system to four states to obey memory constraints. So we need to reduce T 1 to four states. We have a lot of leeway in deciding how exactly to abstract T 1 . In this example, we simply use an abstraction that leads to a good result in the end.
Generic Algorithm Heuristic Properties Summary First Shrink Step T 2 := some abstraction of T 1 MB ⋆⋆ MB ⋆⋆ MALR AL AR PAL DAR MARL L P A A D R MB ⋆⋆ MB ⋆⋆ MB ⋆⋆ MB ⋆⋆ MALR MALR P R B B L D LL LR RL RR DBL R MARL MARL B P R P B B D L DBL PBR MALR BL BR MARL MB ⋆⋆ MB ⋆⋆
Generic Algorithm Heuristic Properties Summary First Shrink Step T 2 := some abstraction of T 1 MB ⋆⋆ MB ⋆⋆ MALR AL AR PAL DAR MARL L P A A D R MB ⋆⋆ MB ⋆⋆ MB ⋆⋆ MB ⋆⋆ MALR MALR P R B B L D LL LR RL RR DBL R MARL MARL B P R P B B D L DBL PBR MALR BL BR MARL MB ⋆⋆ MB ⋆⋆
Generic Algorithm Heuristic Properties Summary First Shrink Step T 2 := some abstraction of T 1 MB ⋆⋆ MB ⋆⋆ MALR AL AR PAL D MARL A L P A R A D R MB ⋆⋆ MB ⋆⋆ MALR DBR M ⋆⋆⋆ P B L LL LR R R DBL MARL B P R P B B R D L B DBL P MALR BL BR MARL MB ⋆⋆ MB ⋆⋆
Generic Algorithm Heuristic Properties Summary First Shrink Step T 2 := some abstraction of T 1 MB ⋆⋆ MB ⋆⋆ MALR AL AL AR AR PAL D MARL A L P A R A D R MB ⋆⋆ MB ⋆⋆ MALR DBR M ⋆⋆⋆ P B L LL LR R R DBL MARL B P R P B B R D L B DBL P MALR BL BR MARL MB ⋆⋆ MB ⋆⋆
Generic Algorithm Heuristic Properties Summary First Shrink Step T 2 := some abstraction of T 1 M ⋆⋆⋆ A PAL DAR DAL P A R MB ⋆⋆ MB ⋆⋆ MALR DBR M ⋆⋆⋆ P B L LL LR R R DBL MARL B P R P B B R D L B DBL P MALR BL BR MARL MB ⋆⋆ MB ⋆⋆
Generic Algorithm Heuristic Properties Summary First Shrink Step T 2 := some abstraction of T 1 M ⋆⋆⋆ A PAL DAR DAL P A R MB ⋆⋆ MB ⋆⋆ MALR DBR M ⋆⋆⋆ P B L LL LR R R DBL MARL B P R P B B R D L B DBL P MALR BL BL BR BR MARL MB ⋆⋆ MB ⋆⋆
Generic Algorithm Heuristic Properties Summary First Shrink Step T 2 := some abstraction of T 1 M ⋆⋆⋆ A PAL DAR DAL P A R MB ⋆⋆ MB ⋆⋆ MALR M ⋆⋆⋆ PBL LL LR R MARL DBL R PBL B D PBR DBL B M ⋆⋆⋆
Generic Algorithm Heuristic Properties Summary First Shrink Step T 2 := some abstraction of T 1 M ⋆⋆⋆ A PAL DAR DAL P A R MB ⋆⋆ MB ⋆⋆ MALR M ⋆⋆⋆ PBL LL LR R MARL DBL R PBL B D PBR DBL B M ⋆⋆⋆
Generic Algorithm Heuristic Properties Summary First Shrink Step T 2 := some abstraction of T 1 MB ⋆⋆ M ⋆⋆⋆ MB ⋆⋆ D ⋆ R MALR PBL M ⋆⋆⋆ LL LR I R MARL DBL P ⋆ R P ⋆ L D ⋆ L
Generic Algorithm Heuristic Properties Summary First Shrink Step T 2 := some abstraction of T 1 MB ⋆⋆ M ⋆⋆⋆ MB ⋆⋆ D ⋆ R MALR PBL M ⋆⋆⋆ LL LR I R MARL DBL P ⋆ R P ⋆ L D ⋆ L current collection: {T 2 , T π { truck B } }
Generic Algorithm Heuristic Properties Summary Second Merge Step T 3 := T 2 ⊗ T π { truck B } : LRL P B M L DBL B L M R B R MA ⋆⋆ L LRR DAR MALR MARL R L PAR L R IL RL M M A A B B L L M M P ⋆ L M M R R B B R R L L L ⋆ D IR RR D ⋆ R MA ⋆⋆ P ⋆ R LLL PAL MA ⋆⋆ MA ⋆⋆ M B L DAL M R B R L LLR current collection: {T 3 }
Generic Algorithm Heuristic Properties Summary Another Shrink Step? Normally we could stop now and use the distances in the final abstract transition system as our heuristic function. However, if there were further state variables to integrate, we would simplify further, e.g. leading to the following abstraction (again with four states): M ⋆⋆⋆ M ⋆⋆⋆ M ⋆⋆⋆ M ⋆ RL P ⋆ L D ⋆ R LLL LRR I R LRL LLR M ⋆ LR D ⋆ L P ⋆ R We get a heuristic value of 3 for the initial state, better than any PDB heuristic that is a proper abstraction. The example generalizes to more locations and trucks, even if we stick to the size limit of 4 (after merging).
Generic Algorithm Heuristic Properties Summary Generic Algorithm Template Generic Merge & Shrink Algorithm abs := {T π { v } | v ∈ V } while abs contains more than one abstract transition system: select A 1 , A 2 from abs shrink A 1 and/or A 2 until size ( A 1 ) · size ( A 2 ) ≤ N abs := abs \ {A 1 , A 2 } ∪ {A 1 ⊗ A 2 } return the remaining abstract transition system in abs N : parameter bounding number of abstract states Questions for practical implementation: Which abstractions to select? � merging strategy How to shrink an abstraction? � shrinking strategy How to choose N ? � usually: as high as memory allows
Generic Algorithm Heuristic Properties Summary Heuristic Properties
Generic Algorithm Heuristic Properties Summary Content of this Course: Merge & Shrink Synchronized Product Merge & Shrink Algorithm Merge & Shrink Heuristic Properties Strategies Label Reduction
Generic Algorithm Heuristic Properties Summary Heuristic Properties Each iteration of the algorithm corresponds to a transformation of the collection abs of transition systems. The exact transformation depends on the specific instantiation of the generic algorithm (e.g. of the merging and the shrinking strategy). For analyzing the properties of the resulting heuristic, we analyze properties of the individual transformations.
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