1. Tractable Class of a Problem of Finding Supports Pavel Surynek Roman Barták Charles University, Prague Czech Republic

2. Problem of finding supports Problem: select a set of non-mutex (non- conflicting) actions supporting a goal Problem: select a set of non-mutex (non- conflicting) actions supporting a goal Must be solved many times within GraphPlan Must be solved many times within GraphPlan CP Doctoral Consortium 2007Pavel Surynek, Roman Barták supports load(box_2, transporter_C) load(box_3, transporter_C) supports load(box_1, transporter_B) load(box_2, transporter_B) load(box_1, transporter_A) load(box_2, transporter_A) load(box_3, transporter_A) free-capacity(transporter_B)=0 free-capacity(transporter_C)=0 free-capacity(transporter_A)=0

3. Mutex graph - clique cover CP Doctoral Consortium 2007 3 2 1 A B 4 5 X Y Z Pavel Surynek, Roman Barták

4. Clique counting arguments CP Doctoral Consortium 2007 Clique decomposition C 1, C 2,..., C k Clique decomposition C 1, C 2,..., C k Contribution of an action a Contribution of an action a c(a) = number of supported atoms Contribution of a clique C Contribution of a clique C c(C) = max a  C c(a) Counting argument (simplest form) Counting argument (simplest form) if ∑ i=1...k c(C i ) < size of the goal ►►► ►►► the goal is unsatisfiable Pavel Surynek, Roman Barták

5. Tractable class (diagram of supported atoms) CP Doctoral Consortium 2007Pavel Surynek, Roman Barták Atoms in positive effects C1C1 C3C3 C4C4 C 5 C6C6 C7C7 C8C8 C3C3 C4C4 C 10 C 11 C9C9 Action cliques C2C2 C5C5 C 12

6. Tractable class (clique intersection graph) CP Doctoral Consortium 2007Pavel Surynek, Roman Barták C1C1 C3C3 C6C6 C5C5 C7C7 C4C4 C2C2 C 12 C8C8 C 10 C 11 C9C9

7. Difficult (artificial) problems CP Doctoral Consortium 2007Pavel Surynek, Roman Barták Encodes Dirichlet’s box principle (pigeon holes) Encodes Dirichlet’s box principle (pigeon holes) Difficult for today’s state-of-the-art planners Difficult for today’s state-of-the-art planners Three types of problems Three types of problems –standard pigeon holes (unsolvable) –pigeon holes as bottleneck (solvable) –pigeon holes as two stage bottleneck (unsolvable)

8. Experiments CP Doctoral Consortium 2007Pavel Surynek, Roman Barták Instance Our planner (seconds) Speedup ratio w.r.t SGPlan 5.1 Speedup ratio w.r.t IPP 4.1 Speedup ratio w.r.t MaxPlan Speedup ratio w.r.t SATPlan Speedup ratio w.r.t CPT Speedup ratio w.r.t LPG-td ujam-02_01 0.06N/A 1.00N/A ujam-03_02 0.54N/A0.020.04N/A 9.25 ujam-04_03 3.39N/A0.190.30N/A 1.76 ujam-05_04 23.88N/A3.500.35N/A 0.37 ujam-06_05 177.9N/A> 3.37 N/A > 3.37 jam-06_05 1.29N/A20.002.140.71177.32N/A jam-07_06 2.48N/A> 241.9312.461.21> 241.93N/A jam-08_07 4.60N/A> 130.4349.563.19> 130.43N/A jam-09_08 8.77N/A> 68.42 17.33> 68.42N/A jam-10_09 17.05N/A> 35.19 N/A holes-06_05 0.27N/A0.44+ 1000.00N/A 18.51 holes-07_06 0.55N/A3.44> 1090.00N/A 10.90 holes-08_07 1.08N/A28.09> 555.55N/A 12.96 holes-09_08 2.08N/A276.25> 288.46N/A > 288.46 holes-10_09 4.06N/A> 147.78 N/A > 147.78

9. Conclusions CP Doctoral Consortium 2007Pavel Surynek, Roman Barták Acyclic clique intersection graph Acyclic clique intersection graph –problem of supports solvable using counting arguments (in polynomial time) Successful on difficult planning problems Successful on difficult planning problems –difficult for today’s state-of-the-art planners (however, problems are artificial) Significant improvements of time to find a solution Significant improvements of time to find a solution