Dynamics of DPLL algorithm

1
Dynamics of DPLL algorithm
S. Cocco (Strasbourg), R. Monasson
(Paris) Articles available on http//www.lpt.ens.f
r/monasson
Rigorous analysis of search heuristics
Distribution of resolution times (left tail)
Quantitative study of search trees (with
backtracking)
probability
2 N ?
?
resolution time
2
Analysis of the GUC heuristic
p,?
Chao, Franco 90 Frieze, Suen 96
t
p,?
(ODE)
(unsat)
sat
3
Complete search trees (? gt 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)
many branches ? (p,?,t)
ODE
PDE
4
Analysis of the search tree growth (I)
Average number of branches with clause
populations C1, C2, C3
Branching matrix

• B(C1, C2, C3T) ? exp N ?(c2, c3t)
  where ci Ci/N , t T/N
• Distribution of C1 becomes stationary over O(1)
  time scale

5
Analysis of the search tree growth (II)
tdt
t
(PDE)
moving frontier between alive and dead branches
6
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
7
Comparison to numerical experiments
unsat sat
(nodes)
(leaves)
Beame, Karp, Pitassi, Saks 98
8
The polynomial/exponentiel crossover
sat (exp)
unsat (exp)
sat (poly)
G
dynamical transition (depends on the heuristic)
sat
unsat
Vardi et al. 00 Cocco, R.M. 01 Achlioptas,
Beame, Molloy 02
Satisfiable, hard instances 3.003lt ? lt 4.3
9
Fluctuations of complexity for finite instance
size
Histograms of solving times a3.5
Exponential regime Complexity 2 0.035 N
Linear regime Very rare! frequence 2-0.011 N
10
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.
Cocco, R.M. 02
Time of resolution
2 0.035 N
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!
11
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, C3T) ? exp - N ?(c1, c2, c3t)
  where ci Ci/N , t T/N
• two cases C1O(1) (safe regime), C1O(N)
  (dangerous regime).

12
Analysis of the probability of survival (II)
? log(probability)/N

• Safe regime
• c1 0

? 0
t
tdt
(PDE)

• Dangerous regime c1O(1) , ? lt 0