Tractable Class of a Problem of Goal Satisfaction in Mutual Exclusion Network

1. Tractable Class of a Problem of Goal Satisfaction inMutual Exclusion Network Pavel SurynekFaculty of Mathematics and PhysicsCharles University, PragueCzech Republic

2. Outline of the talk • Problem definition - goal satisfaction in mutex network • Motivation by concurrent AI planning • A special consistency technique • A very special consistency technique • polynomial time (backtrack free) solving method • Experimental evaluation • random problems • concurrent planning problems Pavel Surynek FLAIRS 2008

3. S(2)={c} 2 7 S(7)={d,g,h,i} S(6)={e,f} 3 S(3)={d} 6 4 1 8 S(8)={g,h} S(4)={h} S(5)={a,b,j} S(1)={a,b} 5 Goal g = a b c d e f g h Problem definition - goal satisfaction in mutex network • A finite set of symbols S, a graph G=(V,E), wherevV S(v)S, and a goal gS • Find a stable set of vertices UV, such that uUS(u)g • An NP-complete problem, unfortunately Solution U={2,5,6,7} (S(2)S(5)S(6)S(7)={c}{a,b,j}{e,f}{d,g,h,i}={a,b,c,d,e,f,g,h,i,j}g) Pavel Surynek FLAIRS 2008

4. Why to deal with such an artificial problem? • It is a problem that arises in artificial intelligence • Consider a concurrent planning problem • multiple agents, agents interfere with each other, parallel action execution Goal state Initial state Pavel Surynek FLAIRS 2008

5. B A 1 2 X Y 3 5 4 Z Structure of goal satisfaction problem • Concurrent planning problems solved using planning-graphs  sequence of goal satisfaction problems • goal satisfaction problems are highly structured Graph of the problem small number of large complete sub-graphs Pavel Surynek FLAIRS 2008

6. A special consistency technique • Cliquedecomposition V=C1C2 ... Ck, i Ci is a complete sub-graph • at most one vertex from each clique can be selected • Contribution of a vertex v ... c(v) = |S(v)| • Contribution of a clique C ... c(C) = maxvC c(v) • Counting argument (simplest form) if∑i=1...k c(Ci) < size of the goal ►►► the goal is unsatisfiable Pavel Surynek FLAIRS 2008

7. C1 C2 C3 C3 C4 C4 C5 C5 C6 C7 C8 C9 C10 C11 C12 symbols A very special consistency technique (1) • The interference among symbols of cliques of the cliquedecompositionC1, C2,..., Ckis limited Pavel Surynek FLAIRS 2008

8. C8 C5 C12 C4 C6 C3 C2 C1 C7 C9 C11 C10 A very special consistency technique (2) • Intersection graph of clique symbols is almost acyclic  the problem is highly structured • If the clique intersection graph is acyclic the goal satisfaction problem can be solvedin polynomial time (backtrack free) Pavel Surynek FLAIRS 2008

9. Solving time Time (seconds) Probability of random edges (m) m=0.04 m=0.00 m=0.08 Experimental evaluation onrandom problems • As structure is more dominant the proposed consistency technique becomes more efficient Pavel Surynek FLAIRS 2008

10. Generalized Hanoi towers Dock worker robots Refueling planes Experimental evaluation withconcurrent planning • Consistency integrated in GraphPlan planning algorithm • For all the problems consistency performs significantly better Pavel Surynek FLAIRS 2008

11. Conclusions and future work • We proposed a (very) special consistency technique that can solve problems with acyclic clique intersection graphs in polynomial time • We evaluated the proposed technique experimentally on random problems and on problems arising in concurrent planning • For future work we want to identify more general structures and properties within problems than cliques and acyclicity of graph Pavel Surynek FLAIRS 2008