It’s all about the support:
a new perspective on the satisfiability problem
Danny Vilenchik
SAT – Basic Notions
3CNF form:
F = (x1Çx2Ǭx5) Æ (x3Ǭx4Ǭx1) Æ (x1Çx2Çx6) Æ…
Ã
x1
x2
x3
x4
x5
x6
F
F
T
F
F
T
F = ( F ÇF Ç T ) Æ ( T Ç T Ç T ) Æ ( T Ç F Ç T )Æ…
x5 supports this clause w.r.t.
Ã
The supporter (if exists) is
always unique
Goal: algorithm that produces optimal result, efficient, and works for all inputs
SAT – Some Background

Finding a satisfying assignment is NP Hard [Cook’71]

No approximation for MAX-SAT with factor better than 7/8 [Hastad’01]

How to proceed?

Hardness results only show that there exist hard instances

The heuristical approach - relaxes the universality requirement
Heuristic is a polynomial time algorithm that produces optimal results
on typical instances

Typical instance?

One possibility: random models
Random 3SAT


Random 3SAT:

Fix m,n

Pick m clauses uniformly at random (over the n variables)
Threshold: there exists a constant d such that [Fri99]

m/n¸d: most 3CNFs are not satisfiable (4.506)

m/n<d : most 3CNFs are satisfiable (3.42)

Near-threshold 3CNFs are apparently “hard” for many SAT heuristics

Possible reason: complicated structure of solution space (clustering)
Motivation – part 1
RWalkSAT(F) [Papa91]
1. Set à à random assignment
2. While(Ã doesn't satisfy F)
a. Pick a random unsatisfied clause C
b. Flip a random variable in C

Experimental results show: takes super-polynomial time for m/n¸2.65
WalkSAT(F)
1. Set à à random assignment
2. While(Ã doesn't satisfy F)
a. Pick a random unsatisfied clause C
b. If C contains x s.t. supportÃ(x)=0, flip x
c. Else,
- w.p. p flip variable with least support
- w.p. (1-p) flip a random variable
Simulated annealing
Motivation cont.

WalkSAT was suggested by Seizt, Alava, Orponen, 2005.

For p=0.57, experimental results show polynomial time for m/n¸4.2

Already above the clustering threshold (~3.92)

What makes this possible?

We suggest: taking the support into account
Support Paradigm: a simulated-annealing heuristic H is part of
the Support Paradigm if it bases its greedy decisions (which variable
to flip) on the notion of support.

WalkSAT is part of the Support Paradigm
Our Result – Part 1

We present a simulated-annealing algorithm – SupportSAT

In some sense a variation on RWalkSAT that considers the support

The algorithm is part of the Support Paradigm
Theorem: the algorithm SupportSAT finds whp a satisfying assignment
for 3CNF formulas in the planted 3SAT distribution with m/n¸c0,
c0 some sufficiently large constant.

Proving rigorously that WalkSAT “works” is a very ambitious task

Improving lower bound on the sat threshold from 3.42 to 4.21
Our Result cont.


RWalkSAT disregards the support – fails on such instances [AB04]
Staying at distance ¸ n/3 form any satisfying assignment

This rigorously mirrors the near-threshold picture for Random 3SAT

Experimental results show (random 3SAT):


RWalkSAT takes super-polynomial time for m/n¸2.65

WalkSAT (Support Paradigm) stays efficient until m/n¸4.21
Rigorous results show (Planted 3SAT) :

RWalkSAT takes super-polynomial time for m/n¸c0

SupportSAT finds whp a satisfying assignment in polynomial time
The Algorithms
WalkSAT(F)
1. Set à à random assignment
2. While(Ã doesn't satisfy F)
a. Pick a random unsatisfied clause C
b. If C contains x s.t. supportÃ(x)=0, flip x
c. Else,
- w.p. p flip variable with least support
- w.p. (1-p) flip a random variable
RWalkSAT(F)
1. Set à à random assignment
2. While(Ã doesn't satisfy F)
a. Pick a random unsatisfied clause C
b. Flip a random variable in C
SupportSAT(F)
1. Start with a random assignment
2. Iteratively flip variables with low support (O(log n) steps)
3. Exhaustively search the subformula induced by “suspicious” variables
The Planted Distribution

Planted 3SAT distribution with parameters m,n:

Fix an assignment 

Pick u.a.r. m clauses out of all clauses that are satisfied by 

We consider the case m/n=O(1)

Planted models also “fashionable” for graph coloring, max clique, max
independent set, min bisection, SAT …

Planted 3SAT was analyzed in several papers


[Fla03] shows a spectral algorithm for solving sparse instances
We use the planted 3SAT as a case study for rigorous analysis to show:

“Simple” algorithms can be rigorously analyzed

Analysis of an algorithm in the Support Paradigm
Analysis Sampler

Every variable x is expected to appear in 3m/n clauses

x supports (w.r.t. planted) a clause in which it appears w.p. 1/7



E[Support(x)]=3m/(7n) (O(1) in our setting)
Take à s.t. Ã(x)(x), but equal otherwise

SupportÃ(x)=0 then x will be flipped

After flip, SupportÃ(x) is typically large: x will not be flipped again

If Support(x) is small, then also after flip in à – remains small
In “reality”: Ã is random and therefore Ã(x)(x)
forsuspicious
half of the x’s
x is
Analysis Sampler cont.
SupportSAT(F)
1. Iteratively flip variables with low support (O(log n) steps)
2. Exhaustively search the subformula induced by “suspicious” variables

Step 1 sets correctly the typical variables (large support w.r.t. )

Step 2 completes the assignment of suspicious variables

There are whp e-£(m/n)n suspicious variables

They tpyically induce a “simple” formula (efficiently searchable)

One may not expect any “sophisticated” procedure to set them
Motivation – part 2
Conjectured solution space of Random 3SAT just below the threshold:
(rigorously proved for k¸8, [AR06,MMZ05])
All assignments within a
cluster are “close”
 A linear number of
variables are “frozen”

Every two clusters are “far”
from each other
 Exponentially many clusters

Motivation – cont.
[A. Coja-Oghlan, M. Krivelevich, V. 2007]
Solution space of Planted 3SAT, m/n some constant above the threshold:
(and also uniformly random satisfiable 3CNFs with same ratio)



Single cluster of satisfying assignments
Size of the cluster is exponential in n
(1-e-(m/n))n variables are frozen
Our Result – part 2

Observation: if 9Ã SupportÃ(x)=0, x can not be frozen in that cluster

Going over the proof in [CKV07]:


Combinatorial characterization of the single frozen cluster is
completely based on the support
The analysis of SupportSAT reveals:

The “WalkSAT” part of SupportSAT sets the frozen variables in the
cluster correctly - (1-e-(m/n))n variables

The non-frozen variables induce a simple formula:

can be identified and searched efficiently
Further Research


Rigorous analysis of “simple” and “practical” heuristics

WalkSAT on near-threshold random 3SAT

For starters, analyze it on the planted distribution
See if the components used in SupportSAT can be useful in practice


For example, for near-threhold random 3SAT
Any other interesting phenomena can be explained by the support ?
Motivation – Part 3 (philosophical)

3CNF formulas can be seen as physical objects (spin glass system)

Every assignment corresponds to a an energy level of the system

EF(Ã)= the number of unsatisfied clauses by à (free energy)

Goal: reach 0 temperature (freeze)

Connection to support?

Flipping variable with 0 support: making a move that can only
decrease energy

Flipping variable with lowest support: a move which incurs the
least increment of free energy

Flipping a variable in an unsatisfied clause: if it reduces the
energy of the system then it increases the support of this variable