Tractable Class of a Problem of Goal Satisfaction in Mutual Exclusion Network Pavel Surynek Faculty of Mathematics and Physics Charles University, Prague Czech Republic

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 FLAIRS 2008 Pavel Surynek

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  U S(u)  g An NP-complete problem, unfortunately FLAIRS 2008 Pavel Surynek 1 S(1)={a,b} S(2)={c} S(4)={h} S(3)={d} S(5)={a,b,j} S(6)={e,f} S(7)={d,g,h,i} S(8)={g,h} a b c d e f g h Goal g = 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)

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 FLAIRS 2008 Pavel Surynek Initial state Goal state

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

A special consistency technique Clique decomposition V=C 1  C 2 ...  C k,  i C i 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) = max v  C c(v) Counting argument (simplest form) if ∑ i=1...k c(C i ) < size of the goal ►►► the goal is unsatisfiable FLAIRS 2008 Pavel Surynek

A very special consistency technique (1) The interference among symbols of cliques of the clique decomposition C 1, C 2,..., C k is limited FLAIRS 2008 Pavel Surynek C1C1 C3C3 C4C4 C 5 C6C6 C7C7 C8C8 C3C3 C4C4 C 10 C 11 C9C9 C2C2 C5C5 C 12 symbols

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 solved in polynomial time (backtrack free) FLAIRS 2008 Pavel Surynek C1C1 C7C7 C 10 C 11 C9C9 C3C3 C6C6 C5C5 C4C4 C2C2 C 12 C8C8

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

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

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 FLAIRS 2008 Pavel Surynek