Transcript ptac2002 1640

Dynamics of DPLL algorithm
S. Cocco (Strasbourg), R. Monasson (Paris)
Articles available on http://www.lpt.ens.fr/~monasson
Rigorous analysis of search heuristics
Quantitative study
of search trees
(with backtracking)
Distribution of
resolution times
(left tail)
probability
2 N
?
resolution
time
Analysis of the GUC heuristic
p,
t
p’,’
p0   1 , α 0   α 0
dp
 Fp p,α , t 
dt
dα
 Fα p,α , t 
dt
(ODE)
(unsat)
sat
Chao,
Franco ‘90
Frieze,
Suen ‘96
Complete search trees ( > 4.3)
DPLL induces a non Markovian
evolution of the search tree
Imaginary, and parallel building up
of the search tree
one branch: p(t) , (t)
ODE
many branches:
 (p,,t)
PDE
Analysis of the search tree growth (I)
Average number
of branches with
clause populations
C1, C2, C3
Branching matrix
• B(C1, C2, C3;T)  exp[ N (c2, c3;t) ] where ci = Ci/N , t = T/N
• Distribution of C1 becomes stationary over O(1) time scale
Analysis of the search tree growth (II)
t+dt
t
(PDE)
+ moving frontier between alive and dead branches
Analysis of the search tree growth (III)
t = 0.01
Halt line
(Delocalization transition
in C1 space)
t = 0.05
unsat
(sat)
t = 0.09
0  10
Comparison to numerical experiments
Q2
N
unsat
sat
(nodes)
(leaves)
2
3  5   1  5  1 0.292



ln 

6 ln 2   2  

Beame, Karp, Pitassi, Saks ‘98
The polynomial/exponentiel crossover
sat
(exp)
G
sat
(poly)
unsat
(exp)
“dynamical” transition
(depends on the heuristic)
sat
unsat
Satisfiable, hard instances 3.003<  < 4.3
Vardi et al. ’00
Cocco, R.M. ‘01
Achlioptas, Beame, Molloy ‘02
Fluctuations of complexity for finite instance size
Histograms of
solving times
3.5
Exponential
regime
Complexity
= 2 0.035 N
Linear regime
Very rare! frequence = 2-0.011 N
Application to Stop & Restart resolution
Resolution through systematic stop-and-restart of the search:
- stop algorithm after time N;
- restart until a solution is found.
Time of resolution :
2 0.035 N
Cocco, R.M. ‘02
2 0.011 N
Halt line for first branch = accumulation of unitary clauses
Contradictory
region
Easy resolution
trajectories
manage to survive in
the contradictory region!
Analysis of the probability of survival (I)
Probability of survival
of the first branch with
clause populations
C1, C2, C3
Transition matrix
*(1-C1/2/(N-T))C1-1
• B(C1, C2, C3;T)  exp[ - N (c1, c2, c3;t) ] where ci = Ci/N , t = T/N
• two cases: C1=O(1) (safe regime), C1=O(N) (dangerous regime).
Analysis of the probability of survival (II)
• Safe regime:
c1= 0
 = log(probability)/N
t
=0
t+dt
(PDE)
• Dangerous regime: c1=O(1) ,  < 0