Tractable Class of a Problem of Goal Satisfaction inMutual Exclusion Network

Pavel SurynekFaculty of Mathematics and Physics

Charles University, PragueCzech 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), wherevV S(v)S, and a goal gS

Find a stable set of vertices UV, such that uUS(u)g An NP-complete problem, unfortunately

FLAIRS 2008 Pavel Surynek

1

S(1)={a,b}

2

3

4

5

6

7

8

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

321

A B

45

X Y

Z

A special consistency technique

Clique decomposition V=C1C2 ... 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) = maxvC c(v) Counting argument (simplest form)

if ∑i=1...k c(Ci) < 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 C1, C2,..., Ck is limited

FLAIRS 2008 Pavel Surynek

C1

C3

C4

C5

C6

C7

C8

C3

C4

C10

C11

C9

C2

C5

C12

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

C1

C7C10C11C9

C3

C6

C5C4

C2

C12C8

Experimental evaluation onrandom problems

FLAIRS 2008 Pavel Surynek

0.1

1

10

100

00.010.020.030.040.050.060.070.080.090.1

Arc-consistency

Projection consistencyTractable projection

Probability of random edges (m)

Solving time

Tim

e (

se

co

nd

s)

As structure is more dominant the proposed consistency technique becomes more efficient

m=0.08

m=0.04

m=0.00

Experimental evaluation withconcurrent planning

Consistency integrated in GraphPlan planning algorithm

For all the problems consistency performs significantly better

FLAIRS 2008 Pavel Surynek

Plan Extraction Time

0.01

0.10

1.00

10.00

100.00

1000.00

10000.00

ha

n0

1

dw

r03

dw

r04

ha

n0

2

pln

04

dw

r02

dw

r01

ha

n0

4

ha

n0

3

pln

01

pln

10

ha

n0

7

pln

05

dw

r05

pln

06

pln

11

dw

r07

pln

13

ha

n0

8

dw

r16

dw

r17

Tim

e (

seco

nd

s)

StandardArc-consistencyProjection consistencyTractable projection

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