Subset-Saturated Cost Partitioning for Optimal Classical Planning - PowerPoint PPT Presentation

Subset-Saturated Cost Partitioning for Optimal Classical Planning Jendrik Seipp, Malte Helmert July 14, 2019 University of Basel, Switzerland

In a nutshell • optimal classical planning • multiple heuristics • cost partitioning 1/14 • A ∗ search + admissible heuristic

In a nutshell • optimal classical planning • multiple heuristics 1/14 • A ∗ search + admissible heuristic • saturated cost partitioning

In a nutshell • optimal classical planning • multiple heuristics 1/14 • A ∗ search + admissible heuristic • subset-saturated cost partitioning

h 1 h 2 s 2 Saturated cost partitioning Saturated cost partitioning algorithm 8 3 5 h SCP 1 1 4 4 s 5 s 2 ,s 3 ,s 4 s 1 1 1 4 4 s 4 ,s 5 s 3 s 1 ,s 2 • use remaining costs for subsequent heuristics • use minimum costs preserving all estimates of h • order heuristics, then for each heuristic h: 2/14 max( h 1 ( s 2 ) , h 2 ( s 2 )) = max( 5 , 4 ) = 5

h 1 h 2 s 2 Saturated cost partitioning 1 8 3 5 h SCP s 5 s 2 ,s 3 ,s 4 s 1 1 Saturated cost partitioning algorithm 4 4 s 4 ,s 5 s 3 s 1 ,s 2 • use remaining costs for subsequent heuristics • use minimum costs preserving all estimates of h • order heuristics, then for each heuristic h: 2/14

h 1 h 2 s 2 Saturated cost partitioning 1 8 3 5 h SCP s 5 s 2 ,s 3 ,s 4 s 1 0 Saturated cost partitioning algorithm 1 4 s 4 ,s 5 s 3 s 1 ,s 2 • use remaining costs for subsequent heuristics • use minimum costs preserving all estimates of h • order heuristics, then for each heuristic h: 2/14

h 1 h 2 s 2 Saturated cost partitioning s 2 ,s 3 ,s 4 8 3 5 h SCP 0 1 3 0 s 5 s 1 Saturated cost partitioning algorithm 1 0 1 4 s 4 ,s 5 s 3 s 1 ,s 2 • use remaining costs for subsequent heuristics • use minimum costs preserving all estimates of h • order heuristics, then for each heuristic h: 2/14

h 1 h 2 s 2 Saturated cost partitioning s 2 ,s 3 ,s 4 8 3 5 h SCP 0 0 3 0 s 5 s 1 Saturated cost partitioning algorithm 1 0 1 4 s 4 ,s 5 s 3 s 1 ,s 2 • use remaining costs for subsequent heuristics • use minimum costs preserving all estimates of h • order heuristics, then for each heuristic h: 2/14

Saturated cost partitioning Saturated cost partitioning algorithm h SCP 0 0 3 0 s 5 s 2 ,s 3 ,s 4 s 1 1 0 1 4 s 4 ,s 5 s 3 s 1 ,s 2 • use remaining costs for subsequent heuristics • use minimum costs preserving all estimates of h • order heuristics, then for each heuristic h: 2/14 ⟨ h 1 , h 2 ⟩ ( s 2 ) = 5 + 3 = 8

4 saturators Saturated cost function 3/14 1. scf ( o ) ≤ cost ( o ) for all operators o ∈ O 2. h ( scf , s ) = h ( cost , s ) for all states s ∈ S

Saturated cost function 4 saturators 3/14 1. scf ( o ) ≤ cost ( o ) for all operators o ∈ O 2. h ( scf , s ) = h ( cost , s ) for all states s ∈ S ′ ⊆ S

Saturated cost function 3/14 1. scf ( o ) ≤ cost ( o ) for all operators o ∈ O 2. h ( scf , s ) = h ( cost , s ) for all states s ∈ S ′ ⊆ S → 4 saturators

Saturate for all states Saturator all o h=1 h=2 h=2 h=0 1 1 1 1 1 4/14 scf ( o ) = max a → b ∈ T ( h ( a ) − h ( b )) −

Saturate for reachable states Saturator reach • remove unreachable states and transitions h=1 h=2 h=2 h=0 1 1 1 1 1 5/14 • use all saturator for remaining states

Saturate for reachable states Saturator reach • remove unreachable states and transitions h=1 h=2 h=0 1 1 1 1 5/14 • use all saturator for remaining states

Saturate for reachable states Saturator reach • remove unreachable states and transitions h=1 h=2 h=0 1 0 1 1 5/14 • use all saturator for remaining states

Saturate for a perimeter Saturator perim • cap all distances at k • use all saturator h=1 h=2 h=2 h=0 1 1 1 1 1 6/14 • use perimeter k around goal states (k = h ( s ) )

Saturate for a perimeter Saturator perim • cap all distances at k • use all saturator h=1 h=1 h=1 h=0 1 1 1 1 1 6/14 • use perimeter k around goal states (k = h ( s ) )

Saturate for a perimeter Saturator perim • cap all distances at k • use all saturator h=1 h=1 h=1 h=0 1 0 0 0 0 6/14 • use perimeter k around goal states (k = h ( s ) )

Saturate for a single state h=1 1 1 1 1 1 h=0 h=2 h=2 7/14 Saturator lp H a ≤ 0 for all a ∈ S ⋆ H a ≤ C o + H b → b ∈ T − for all a o C o ≤ cost ( o ) for all o ∈ O H α ( s ) = h ( s )

Saturate for a single state h=1 1 1 1 1 h=0 h=2 7/14 Saturator lp H a ≤ 0 for all a ∈ S ⋆ H a ≤ C o + H b → b ∈ T − for all a o C o ≤ cost ( o ) for all o ∈ O H α ( s ) = h ( s )

Saturate for a single state h=1 0 0 0 1 h=0 h=1 7/14 Saturator lp H a ≤ 0 for all a ∈ S ⋆ H a ≤ C o + H b → b ∈ T − for all a o C o ≤ cost ( o ) for all o ∈ O H α ( s ) = h ( s )

No unique minimum saturated cost function s 1 s 2 s 3 1 1 1 • large Pareto frontier • chain saturators 8/14

No unique minimum saturated cost function s 1 s 2 s 3 1 1 1 • large Pareto frontier 8/14 • → chain saturators

Chaining saturators reach, perim: h=1 h=2 h=2 h=0 1 1 1 1 1 9/14

Chaining saturators reach, perim: h=1 h=2 h=0 1 0 1 1 9/14

Chaining saturators reach, perim: h=1 h=1 h=0 1 0 0 0 9/14

Experiments

Setup • 30 minutes, 3.5 GiB • hill climbing and systematic PDBs, Cartesian abstractions • online and offline • non-negative and general costs 10/14

Heuristic value for initial state – non-negative costs (1506 tasks) – 79 79 84 2 perim 433 430 – 142 15 15 6 lp 493 490 229 all, lp 2 507 505 219 179 – 0 12 reach, lp 507 505 219 179 0 – 12 perim, lp 512 510 217 185 16 16 – 2 88 all – reach perim lp all, lp reach, lp perim, lp all 2 5 2 88 2 2 2 reach 15 – 11/14

Heuristic value for initial state – non-negative costs (1506 tasks) – 79 79 84 2 perim 433 430 – 142 15 15 6 lp 493 490 229 all, lp 2 507 505 219 179 – 0 12 reach, lp 507 505 219 179 0 – 12 perim, lp 512 510 217 185 16 16 – 2 88 all – reach perim lp all, lp reach, lp perim, lp all 2 5 2 88 2 2 2 reach 15 – 11/14

Heuristic value for initial state – non-negative costs (1506 tasks) – 79 79 84 2 perim 433 430 – 142 15 15 6 lp 493 490 229 all, lp 2 507 505 219 179 – 0 12 reach, lp 507 505 219 179 0 – 12 perim, lp 512 510 217 185 16 16 – 2 88 all – reach perim lp all, lp reach, lp perim, lp all 2 5 2 88 2 2 2 reach 15 – 11/14

Heuristic value for initial state – non-negative costs (1506 tasks) all, lp 2 perim 433 430 6 lp 493 490 229 – 79 79 84 507 505 219 179 2 – 0 12 reach, lp 507 505 219 179 0 – 12 perim, lp 512 510 217 185 16 16 – 2 88 all 2 reach perim lp all, lp reach, lp perim, lp all – 2 5 88 2 2 2 reach 15 – 11/14 – 142 15 15

Heuristic value for initial state – non-negative costs (1506 tasks) lp 2 2 perim 433 430 – 142 15 15 6 493 490 229 all all, lp 507 505 219 179 – 0 12 0 – 12 – 2 88 5 – reach perim lp all, lp reach, lp perim, lp all 11/14 – 2 2 88 2 2 2 reach 15 – 79 79 84 reach, lp 507 505 219 179 perim, lp 512 510 217 185 16 16

Heuristic value for initial state – non-negative costs (1506 tasks) – 79 79 84 2 perim 433 430 – 142 15 15 6 lp 493 490 229 all, lp 2 507 505 219 179 – 0 12 reach, lp 507 505 219 179 0 – 12 – 2 88 all – reach perim lp all, lp reach, lp perim, lp all 2 5 2 88 2 2 2 reach 15 – 11/14 perim, lp 512 510 217 185 16 16

Online – non-negative (nn) vs. general costs (gen) reach, lp 320 8.7 6.9 493 144 all, lp 385 354 10.0 8.1 138 180 384 lp 355 9.9 8.1 120 181 perim, lp 392 367 11.0 8.8 31 134 360 99 Coverage 4 Evals/sec nn gen nn gen nn gen all 680 703 940.9 946.2 7 195 reach 657 679 501.4 514.1 5 222 perim 722 726 897.7 906.6 12/14 h ( s 0 ) higher

Offline subset-saturated cost partitioning perim 1144 – 4 4 11 36 perim 694 0 – 0 0 0 lp 1117 0 – 36 1 2 1134 all 1 8 36 0 – reach 1136 1 9 36 – 1 all Coverage perim lp perim reach 13/14

Offline subset-saturated cost partitioning all 1144 – 694 0 – 0 0 0 lp 1117 0 – 36 1 2 perim 1134 1 8 36 0 – reach 1136 1 9 36 – 1 all Coverage lp perim reach 13/14 perim ⋆ perim ⋆ 4 4 11 36

Summary • generalization of saturated cost partitioning • three new saturators • stronger heuristics 14/14