Dynamics of DPLL algorithm

Rigorous analysis of search heuristics

Quantitative studyof search trees

(with backtracking)

Distribution of resolution times

(left tail)

resolutiontime

probability

2 N

?

S. Cocco (Strasbourg), R. Monasson (Paris)Articles available on http://www.lpt.ens.fr/~monasson

Analysis of the GUC heuristic

t

t,αp,Fdt

αd

t,αp,Fdt

dp

α0α,10p

α

p

0

(ODE)

(unsat)

sat

Chao,Franco ‘90

Frieze,Suen ‘96

p,

p’,’

Complete search trees ( > 4.3)

DPLL induces a non Markovianevolution of the search tree

Imaginary, and parallel building upof the search tree

one branch: p(t) , (t) many branches: (p,,t) ODE PDE

Analysis of the search tree growth (I)

Branching matrix

Average number of branches with

clause populations C1, C2, C3

• 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)

(PDE)

t

t+dt

+ moving frontier between alive and dead branches

Analysis of the search tree growth (III)

(sat)

unsat

Halt line(Delocalization transition

in C1 space)

t = 0.01

t = 0.05

t = 0.09

100

Comparison to numerical experiments

(nodes) (leaves)

292.01

2

51ln

2ln6

532

Beame, Karp, Pitassi, Saks ‘98

unsat

sat

N2Q

The polynomial/exponentiel crossover

sat (poly)

sat (exp)

unsat (exp)

“dynamical” transition(depends on the heuristic)

sat

unsat

G

Satisfiable, hard instances 3.003< < 4.3

Vardi et al. ’00Cocco, R.M. ‘01

Achlioptas, Beame, Molloy ‘02

Fluctuations of complexity for finite instance size

Histograms of solving times

Exponentialregime

Complexity= 2 0.035 N

Linear regime Very rare! frequence = 2-0.011 N

Application to Stop & Restart resolution

Contradictoryregion Easy resolution

trajectoriesmanage to survive in

the contradictory region!

Halt line for first branch = accumulation of unitary clauses

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 2 0.011 N

Cocco, R.M. ‘02

Analysis of the probability of survival (I)

Transition matrix

Probability of survival of the first branch with

clause populations C1, C2, C3

• 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).

*(1-C1/2/(N-T))C1-1

Analysis of the probability of survival (II)

(PDE)

tt+dt

= log(probability)/N = 0

• Safe regime: c1= 0

• Dangerous regime: c1=O(1) , < 0