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Algorithms for SAT Based on Search in Hamming Balls. Author : Evgeny Dantsin, Edward A. Hirsch, and Alexander Wolpert Speaker : 張經略, 吳冠賢, 羅正偉. Outline. Introduction Definitions and notation Randomized Algorithm Derandomization. Introduction.

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Algorithms for SAT Based on Search in Hamming Balls

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Algorithms for sat based on search in hamming balls

Algorithms for SAT Based on Search in Hamming Balls

Author : Evgeny Dantsin, Edward A. Hirsch,

and Alexander Wolpert

Speaker : 張經略, 吳冠賢, 羅正偉


Outline

Outline

  • Introduction

  • Definitions and notation

  • Randomized Algorithm

  • Derandomization


Introduction

Introduction

  • In this paper a randomized algorithm for SAT is given, and its derandomized version is the first non-trivial bound for a deterministic SAT algorithm with no restriction on clause length.

  • For k-SAT, Schuler’s algorithm is better than this one.


Notation

Notation


Fact about h

Fact about H

  • The graph of H is


A bound of v n r

A bound of V(n,R)


Ball checking algorithms

Ball-Checking algorithms


Observations

Observations

  • The recursion depth is at most R

  • Any literal is altered at most once during execution of Ball-Checking, and at any time, those remaining variables are assigned as in the original assignment


Lemma 1

Lemma 1

  • There is a satisfying assignment in B(A,R) iff Ball-Checking(F,A,R) returns a satisfying assignment in B(A,R)

  • Proof. The lemma is correct for R=0. For the induction step, assume that Ball-Checking(Fi,Ai,R-1) finds a satisfying assignment in B(Ai,R-1), if any. Furthermore, the assignment found can be different from Ai only at those variables that appear in Fi.


Lemma2

Lemma2

  • The running time of Ball-Checking(F,A,R) is at most , where k is the maximum length of clauses occurring at step 3 in all recursive calls.

  • Proof. The recursion depth is at most R and the maximum degree of branching is at most k.


Full ball checking

Full Ball Checking

  • Procedure Full-Ball-Checking(F,A,R)Input: formula F over variables

    assignment A, number ROutput: satisfying assignment or “no”

  • 1.Try each assignment A’ in B(A,R), if it satisfies F, return it.

  • 2.Return “no”


Observation

Observation

  • Full-Ball-Checking runs in time poly(n)mV(n,R)


Randomized algorithm

Randomized algorithm


Correctness

Correctness

  • Lemma 3. For any R,l,(a)If F is unsatisfiable, then Random-Balls returns “no”, (b)else Random-Balls finds a satisfying assignment with probabability 1/2


Proofs of lemma 3

Proofs of lemma 3


Amplifying prob of correctness

Amplifying prob. of correctness

  • Choosing N to be n time larger will reduce the probability of error to less than .Note this doesn’t ruin the time complexity we need.


Lemma 4

Lemma 4

  • Consider the execution of Random-Balls(F,R,l) that invokes Ball-Checking. For any input R,l, the maximum length of clauses chosen at step 3 of Procedure Ball-Checking is less than l.


Proof of lemma 4

Proof of lemma 4


Lemma 5

Lemma 5

  • For any R,l, let p be the probability (taken over random assignment A) that Random-Balls invokes Full-Ball-Checking. Then


Proof of lemma 5

Proof of lemma 5


Proof of lemma 5 conti

Proof of lemma 5(conti.)


Theorem 1

Theorem 1


Proof of theorem 1

Proof of theorem 1


Proof of theorem 1 conti

Proof of Theorem 1(conti.)


Proof of theorem 1 conti1

Proof of Theorem 1 (conti.)

  • Assign R=a ,l=b ,where a < b constants. We use the fact ln(1+x)=x+o(x).


Proof of theorem1 conti

Proof of theorem1(conti.)


Proof of theorem 1 conti2

Proof of Theorem 1 (conti.)

  • Taking a=0.339, b=1.87, we have Φ ,ψ>0.712 , proving the theorem.


Derandomization

Derandomization

  • From “A deterministic Algorithm for k-SAT based on local search”Lemma.

  • Let R < n/2,β=β(n,R)= ,存在nβ 個R ball cover


Proof of the lemma

Proof of the lemma


Approximation for ball covering

Approximation for Ball covering

  • We give a greedy algorithm for choosing a near optimal number of covering R-balls.At each step, choose the R-ball that covers the most yet-uncovered elements.


Time complexity

Time complexity


Approximation ratio

Approximation ratio

  • Let OPT be the optimal number of covering R-balls. At each iteration, the yet-uncovered elements can be covered by OPT R-balls. Thus some R-ball covers at least 1/OPT fraction of the yet-uncovered elements. So the # of yet-uncovered elements becomes less than (1-1/OPT) times the original value.


Approximation ratio conti

Approximation Ratio(conti.)


Lemma 6

Lemma 6


Proof of lemma 6

Proof of Lemma 6


Proof of lemma 6 conti

Proof of Lemma 6(conti.)


Derandomized algorithm

Derandomized Algorithm


Correctness1

Correctness

  • The correctness follows since C is a covering of .


Theorem 2

Theorem 2


Proof of theorem 2

Proof of Theorem 2


Proof of theorem 2 conti

Proof of Theorem 2(conti.)


Proof of theorem 2 conti1

Proof of theorem 2(conti.)


Proof of theorem 2 conti2

Proof of theorem 2(conti.)


Proof of theorem 2 conti3

Proof of theorem 2(conti.)

We now estimate S2 as follows:


Proof of theorem 2 conti4

Proof of theorem 2(conti.)


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