practicalsat solving · 2 16.07.2019 lectureoutline derandomizinglocalsearch sat upperbounds...

KIT – University of the State of Baden-Wuerttemberg and National Research Center of the Helmholtz Association

Institute for Theoretical Computer ScienceResearch Group “Verification meets Algorithm Engineering”

www.kit.edu

Practical SAT Solving

Lecture 13

Carsten Sinz, Tomas Balyo | July 16, 2019

2 16.07.2019

Lecture Outline

Derandomizing Local SearchSAT Upper BoundsCovering Codes

Carsten Sinz, Tomas Balyo – SAT Solving

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Repetition: Local Search


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Open questions

How often should the inner/outerloop be repeated?How good are the chances to find a satisfying assignment?Can the algorithm be modified toprove unsatisfiability?Is there a general upper bound fordeciding SAT that is better than2n?


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Complexity of k-SAT: Upper Bounds

Source: Iwama&Tamaki: Improved Upper Bounds for 3-SAT (2003)


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Algorithm of Dantsin et al.Maybe[Assignment] Local-Search(ClauseSet S){

for i = 1 to MAX_TRIES doα = random-assignmentfor j = 1 to MAX_FLIPS do

if α satisfies all clauses in S return αchoose variable x to flipflip x

return Nothing}

Variant of local search algorithmus (as above)Input: Formula in k-CNF with n variables.

Idea:1. Make inner loop deterministic: systematically check all

assignments in the „neighborhood“ of α.2. Choose starting assignment α in outer loop deterministically.


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Idea 1: Hamming Ball Algorithm

Hamming distance between two assignments α, β:Def.: d(α,β) := |{x | α(x) ≠ β(x)}|Ex.: α=(0,1,1,0,0), β=(1,0,1,0,1), d(α,β) = 3

Hamming ball of radius r around α:Def.: H(α,r) := { β | d(α,β)≤r }

Volume V(r) of the Hamming ball: (given n variables, i.e. assignments are vectors of length n):

Define normalized radius ρ := r/n (i.e. 0≤ρ≤1)

independent of α.𝑉 𝑟 ≔ 𝐻 𝛼, 𝑟 =()*+

,𝑛𝑖


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Illustration: Hamming Ball

Example: n=4

(0011) (1100)

(0000)

(0001) (0010) (0100) (1000)

(1111)Ball around α=(0001)with radius 1 und radius 2.


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Hamming Ball Algorithm

Known approximation of the Hamming ball volume (frominformation theory, Shannon) for 𝜌 ≤ 1/2:

where(Entropy function; log = logarithm for base 2).I.e.: V(ρn) ≈ 2h(ρ)n (within polynomial factor)

18𝑛𝜌(1 − 𝜌)

28 9 : ≤ 𝑉(𝑟) ≤ 28 9 :

ℎ 𝜌 ≔ − 𝜌 log 𝜌 − (1 − 𝜌 )log(1 − 𝜌)


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Search in Hamming BallMaybe[Assignment] HB-Search(α,r){

if(α ⊨ F) then return αif (r = 0) return Nothingchoose C=(L1∨...∨Lk) with α(C)=0for i = 1 to k do

αi = α with α(Li)=1β = HB-Search(αi, r-1)if(β ≠ Nothing) return β

return Nothing}

Run time HB-Search: kr

Note: kr can be considerably smaller than V(r).E.g.: k = 3, r = n/2:V(r) = 2n-1, kr = 3n/2 ≈ 1.7321n


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Search in Hamming Ball

HB-Search provides simple deterministic algorithm for 3-SAT:

Execute two HB searches around α0=(0,...,0) and α1=(1,...,1) withradius n/2.Runtime (as seen): 1.7321n

Can we improve that?Idea: Choose radius r = ρn with ρ<1/2.Of course, more than two starting points for α in the outer loop areneeded then.


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Probabilistic Variant of HB LS AlgorithmMaybe[Assignment] Local-Search-HB-Prob{

for i = 1 to w doαi = random-assignment

β = HB-Search(αi, ρ⋅n)if(β ≠ Nothing) return β

return Nothing}

Choose αi uniformly and independently at random from the 2n

possibilities, w iterations.Runtime: w⋅kρn.Goal: Choose w and ρ such that (a) error probability is neglectablysmall and (b) runtime is optimized.


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Probabilistic Algorithm: Error Probability (I)

Only possible error: F satisfiable, algorithm doesn’t find a satisfyingassignmentAssumption: F satisfiable, α ⊨ F.HB-Search(αi, ρn) finds α, if α∈H(αi, ρn).Probability for that: V(ρn)⋅2-n

With Shannon approximation V(ρn) ≈ 2h(ρ)n: probability approx. 2h(ρ)n⋅2-n = 2h(ρ)n-n = 2-(1-h(ρ))n

Probability that α is not found in w iterations:

(with approximation 1+x ≤ ex for all x)

1 − 2B CB8 9 : D≤ 𝑒BDFGH IHJ K L


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Error Probability and Runtime Optimization

Probability that α is not found in w iterations:

(with approximation 1+x ≤ ex for all x)If error shall be below e-c: iterations needed.Runtime thus: (within polynomial factor):

(*)(with def. entropy: )Minimize runtime by differentiating (*) w.r.t. ρ and solving for ρ.Result: ρ = 1/(k+1).Substituing that in (*) provides runtime:


1 − 2B CB8 9 : D≤ 𝑒BDFGH IHJ K L

𝑤 F 𝑘9: ≈ 2 CB8 9 F 𝑘9:= 2 F 𝜌9 F (1 − 𝜌)(CB9)F 𝑘9 :

ℎ 𝜌 = − 𝜌 log 𝜌 − (1 − 𝜌 )log(1 − 𝜌)

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Result for Probabilistic Algorithm

Runtime:

Base for small values of k:

Next step: Derandomize this algorithm (i.e. outer loop)

k 3 4 5 6 72k/(k+1) 1.5 1.6 1.66667 1.71429 1.75


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Covering Codes

Derandomization: Deterministic algorithm for selecting startingassignments αi.Definition: A set C of assignments is a covering code of radius r, if foreach assignment α: α∈H(γ,r) for at least one γ∈C.Example:

Covering code ofradius 1.


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Construction of Covering Codes

Problem is a special case of SET-COVER:

Given: Set S and family F ⊆ 2S of subsets of S

Wanted: Subset C ⊆ F, which covers S, i.e.,

Goal: Minimize |C|

Greedy algorithm provides log|S|-approximation for SET-COVER


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Construction of Covering Codes

Problem is a special case of SET-COVER:

Given: Set S and family F ⊆ 2S of subsets of S

Wanted: Subset C ⊆ F, which covers S, i.e.,

Goal: Minimize |C|

Greedy algorithm provides log|S|-approximation for SET-COVER

Set of all possibleassignments α

Hamming balls ofradius r

Hamming balls chosen bythe algorithm


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Greedy algorithm for SET-COVER1. Set C=∅ .2. As long as there are not-covered elements in S:

a) Choose c ∈ F, covering largest number of yet not-coveredelements from S.

b) Add c to C.3. If ∪C = S, then C is a solution; otherwise none exists.

Known:Result of greedy alg. is log|S|-approximation for SET-COVER, i.e.. |C| ≤ log|S|⋅|C*|, where C* is optimal solution.(in our case: |S|=2n, i.e. log|S| ∈ O(n))

Needed for approximation of |C|:Upper bound for C*


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Upper Bound for C*

Obviously, for all covering codes C (for a given radius r) the followingholds: 𝐶 ≥ TGL

U(,).Next lemma: optimal code is at most by a factor of n larger.


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Lemma: Size of Optimal Covering Code

For all n and r = ρn with 0≤ρ≤1/2, there is a covering code for radius r

of size .

Proof: probabilistic methodA randomly selected set of s assignments is with high probability a covering code. Thus, in particular, such a code must exist.


𝑠 ≔ 𝑛 F 8𝑛𝜌(1 − 𝜌) F 2 CB8(9) :

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Proof of Lemma

β fixed assignment, α selected at random.Then: β∈H(α,r) with probability V(r)/2n.C: set of s independently selected random assignments.Then: Probability that β is not covered by C:

In total: probabilty that C is covering code:≥ (approaches 1 for n → ∞)

(with approximation 1+x ≤ ex for all x,

and )


1 −𝑉(𝑟)2:

W

≤ 𝑒BWU(,)GL ≤ 𝑒

BW GJ K HI L

X:9(CB9) ≤ 𝑒B:

𝑉(𝑟) ≥28 9 :

8𝑛𝜌(1 − 𝜌)𝑠 = 𝑛 F 8𝑛𝜌(1 − 𝜌) F 2 CB8 9 :

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Analysis of the Greedy Algorithm

Optimal code: |C*| ≤ sGreedy algorithm:

|C| ≤ O(n)⋅s = O((5/2)n)⋅2(1-h(ρ))n

Runtime of greedy algorithm: 23n (without proof)Result: |C| ok, but runtime of SET-COVER algorithm too high!Idea:

Reduce runtime by allowing increased |C|.


24 16.07.2019

Block Codes

Idea:Let C be a covering code of radius r for n variablesObtain covering code Cd of radius d⋅r for d⋅n variables by repeatingcovering code C for d times

Example:

d=4, |C|=3: covering code for 4n variables with |C|4 elements.In general:

Block code Cd has size |C|d and radius r⋅d.

× ××


:

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Deterministic Hamming-Ball Algorithm

Choose d=6 and use block code C6.Generate covering code C for n/6 variables with radius ρn/6 usinggreedy SET-COVER algorithm

Runtime: 23n/6 ≤ 1.414n

Size of covering code C6:

Same as non-blockcode, ignoring polynomial factor

Complete run time of algorithm:1. Generating covering code: O(1.414n)

(once befoe start of Hamming-ball search)2. Hamming-ball search with radius r = n/(1+k): O(2k/(k+1))n

--> The latter dominates for k>=3, thus: O(2k/(k+1))n (i.e. same run-time asprobabilististic algorithm, e.g. 1.5n for 3-SAT)


𝑂𝑛6

[/GF 2 CB8 9 :

\

\

= 𝑂(𝑛C[) F 2 CB8 9 :

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Further Improvements

Hamming-ball search can also be done in time 2.848r (instead of 3r). Refined algorithm: Dantsin et al., 2002.

Therefore:Run-time of 1.481n for determinististic 3-SAT algorithm (best upper bound for determinististic 3-SAT so far)


practicalsat solving · 2 16.07.2019 lectureoutline derandomizinglocalsearch sat upperbounds...

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