Private Approximation of Search Problems

Amos BeimelPaz Carmi

Kobbi NissimEnav Weinreb

Ben Gurion University

Research partially Supported by the Frankel Center for Computer Science.

Vertex Cover

Input: undirected graph G=<E,V>. A set is a vertex cover of G,

if: For every , or .

Vertex Cover size: Return the size of a smallest vertex

cover of G. Vertex Cover (Search):

Return a minimum vertex cover of G.

VC

Evu , CvCu

Vertex Cover - Example

1 2

3

4 5

Vertex Cover size: 2

Vertex Cover (search): {2,3} or {3,5}

6

G“Would you tell me the

Vertex Cover size

of your graph?”

“I would, but it is hard to

compute.”

“So, tell me an approximation!”

“Hmmm…”

Maximal Matching Approximation

Find maximal matching.

Its vertices form a cover.

2-approximation: solution size is at most 2 times the optimal solution.

1 2

3

4 5

6

G“So, tell me an

approximation!”“Hmmm…”

1 2

3

4 56

1 2

3

4 56

VC

Matching

2 2

24

Talk Overview

Definitions and Previous Work Impossibility Result for Vertex Cover Algorithms that Leak (Little)

Information Positive Result for MAX-3SAT

Conclusions and Open Problems

Previous Work

Private approximation of Vertex Cover size. [HKKN STOC01]

mVC(G1)=mVC(G2) A(G1)=A(G2) Results:

If NP BPP there is no polynomial private n1-ε-approximation algorithm for Vertex Cover size.

There is a 4-approximation algorithm for Vertex Cover size that leaks 1 bit of information.

Previous Work (cont.)

Private Multiparty Computation of Approximations [FIMNSW ICALP01] Private Approximation of Hamming

distance in communication ~ .(Improved to polylog by [IW STOC05])

Private approximation algorithm for the Permanent.

n

The Search Problem

What is the right definition of privacy?!

Many solutions for one input.

mVC(G1)=mVC(G2)A(G1)=A(G2)? NO!!!

1 2

3

4 56

1 2

3

4 56

Private Algorithms - Definition

R – Equivalence Relation over {0,1}*

A - Algorithm

A is private with respect to R if:

x y

A( ) A( )≈c

x y

Example – Vertex Cover (Search)

G1 ≈VC G2 if they have the same set of minimum vertex covers.

A is a private approximation algorithm for Vertex Cover if: A is an approximation algorithm for

vertex cover. G1 ≈VC G2 A(G1) A(G2)≈c

G“Would you give me a

Vertex Cover of

your graph?”

“I would, but it is hard to compute.”

“So, give me an approximation!

”

“Hmmm…, Private

Approximation”

(At least)

≈VC

1 2

3

4 5

6

1 2

3

4 5

6

Vertex Cover (search): {2,3} or {3,5}

Relation to previous work

A new framework where all previous results fit in.

Search problem – HUGE number of equivalence classes.

Previous works relied on having small number of equivalence classes.

Talk Overview

Definitions and Previous Work Impossibility Result for Vertex Cover Private Algorithms that Leak (Little)

Information Positive Result for MAX-3SAT

Conclusions and Open Problems

Vertex Cover (Search) - Reminder

G1 ≈VC G2 if they have the same set of minimum vertex covers.

A is a private approximation algorithm for Vertex Cover if: A is an approximation algorithm for

vertex cover. G1 ≈VC G2 A(G1) A(G2) If A is deterministic:

G1 ≈VC G2 A(G1) = A(G2)

≈c

Vertex Cover - Impossibility Result

Theorem 1If P ≠NP there is no deterministic

polynomial time n1-ε-approximation algorithm that is private with respect to ≈VC.

Proof Idea:Given a private n1-ε-approximation

algorithm A, we exactly solve Vertex Cover.

Definitions for the Proof

Let be a graph. A vertex is critical for if

every minimum vertex cover of contains .

A vertex is relevant for if there exists a minimum vertex cover of that contains .

Every vertex is non-critical or relevant (or both).

v

GVv EVG ,

G

v

GVv

G

Claim 1

Let be a graph and .If then both and are

non critical for .

Note: The claim is useless if .

1vG )(GAW

2vWvv 21,G

VW

Proof of Claim 1

Let be a graph and .If then both and are

non critical for .1v

G )(GAW 2vWvv 21,

GG

)(GAW 2v

1v

2v

1v

*G

*)(* GAW

*WW *)()( GVCGVC is non critical for .G1v

Privacy

Relevant / Non-Critical Algorithm

Input Graph G Vertex v.

Output - one of the following: v is Relevant for G. v is Non-Critical for G.

Enables a greedy algorithm for Vertex Cover.

'G

vG

I

vG

'' Wv

Relevant / Non-Critical Algorithm

vG

I

vG

)'(' GAW

'' Wv*G

*).(* GAW Let Is ?*' Wv

*)(* GAW

I is a big set of isolated vertices.

Case 1 *' Wv

'G

vG

I

vG

'' Wv

vG

I

vG

)'(' GAW

'' Wv*G *)(* GAW

'* WW )'(*)( GVCGVC

is non critical for .v G

Privacy

(If was critical for , both copies of would be critical for .)

Gv'Gv

Case 2 *' Wv

'G

vG

I

vG

'' Wv

vG

I

vG

)'(' GAW

'' Wv*G

*)(* GAW

By Claim 1 is not critical for .'v *G

Let be the size of the minimum vertex cover of . G

c

Thus, there exists a minimum cover of that does not contain .*G

*C'v

Case 2 (cont.)*' WvAssume is not relevant for .Gv

must contain both copies of as it does not contain .

*C'v

v

Thus, contains two non-optimal vertex covers of

*CG 22|*| cC

'G

vG

I

vG

'' Wv

vG

I

vG

)'(' GAW

'' Wv*G

*)(* GAW

Case 2 (cont. ii)*' Wv

However, taking two minimal covers of and adding results in a cover for of size .G

*G'v12 c

Contradiction to the minimality of .*CHence, is relevant for . Gv

'G

vG

I

vG

'' Wv

vG

I

vG

)'(' GAW

'' Wv*G

*)(* GAW

Relevant / Non-Critical (Summary)

On input ( ): Define the graph . Compute . Choose . Define the graph . If , output:

“ is Non-Critical for .” If , output:

“ is Relevant for .”

vG,'G

'' Wv*G

*)(' GAvG

)'(' GAW

v*)(' GAv

Gv

'G

vG

I

vG

)'(' GAW

'' Wv

vG

I

vG

'' Wv*G

*)(* GAW

Greedy Vertex Cover

Choose an arbitrary vertex . Execute the Relevant/Non-Critical

algorithm on . If is Relevant, take and delete

all edges adjacent to . If is Non-Critical, take and

delete all edges adjacent to . Continue recursively.

v

vG,

v

vvv

)(vN)(vN

Vertex Cover - Impossibility Result

Theorem 1

If P ≠NP there is no deterministic polynomial time n1-ε-approximation algorithm that is private with respect to ≈VC.

randomizedNP BPP

Definitions and Previous Work Impossibility Result for Vertex Cover Algorithms that Leak (Little)

Information Positive Result for MAX-3SAT

Conclusions and Open Problems

Talk Overview

Given a 3CNF formula find an assignment that satisfies a maximum fraction of its clauses.

Best approximation ratio: 7/8. if and have the same

set of maximum satisfying assignments.

Again, no private approximation!

MAX-3SAT

121 SAT2

Almost-Private Algorithms

x y

A( ) A( )≈c

wz

A( ) A( )?

x yz w

Almost-Private Algorithms

is k-private with respect to if there exists such that:

1. .2. Every equivalence class of is a

union of at most equivalence classes of .

3. is private with respect to .

RA

RR ''R

'R

Rk2

'RA

Almost Private Approximation for MAX-3SAT

SAT

and have the same set of maximum satisfying assignments.

1 2

1 2

1

2

3

nlog

…

Every equivalence class of is divided into subclasses.)(lognO

SAT

…

Almost Private Approximation for MAX-3SAT

Lemma 1There is a set of assignments

such that for every 3SAT formula on n variables there exists an that satisfies of the clauses in .

)(log1,..., nO

i

)(lognO

8/7SAT

1 2

1

2

3

nlog

……

Almost Private Approximation for MAX-3SAT

Theorem 2There exists a -private

-approximation algorithm for MAX-3SAT.

Proof:We use from Lemma 1.Given a formula return the first

that satisfies at least of the clauses in .

)log(log nO

)(log1,..., nO

i

)8/7(

)8/7(

Proof of Lemma 1

There is a set of assignments such that for every 3SAT formula on n variables there exists an that satisfies of the clauses in .

ProofConstruct almost 3-wise [AGHP]

independent variables . Number of assignments: .

)(log1,..., nO

i

)(lognO

nxx ,...,1

8/7

)( lognO

Proof of Lemma 1(cont.)

For every 3 random variables and every 3 Boolean values :

Conclusion 1: For each clause and assignment :

Conclusion 2: For every formula there is an assignment that satisfies of its clauses.

321 ,, xxx

8/1]Pr[8/1 332211 bxbxbx321 ,, bbb

8/7]by satisfied is Pr[C

8/7

C

Solution-List Paradigm

A short list of solutions. Every input has a good

approximation in the list. k solutions logk-private algorithm

Definitions and Previous Work Impossibility Result for Vertex Cover Algorithms that Leak (Little)

Information Positive Result for MAX-3SAT

Conclusions and Open Problems

Talk Overview

Further Results

Solution list algorithm for Vertex Cover: -private (exponential list size) -approximation ratio. Optimal with respect to solution-list

algorithms. Impossibility result for (any)

1-private -approximation algorithm for Vertex Cover.

)( nO)( nO

1n

Further Results (cont.)

Impossibility result for a private -approximation algorithm for MAX-3SAT.

Impossibility result for a solution-list algorithm for MAX-3SAT that is: -private -approximation ratio.

Problems in P. Positive and negative results.

)2/1( )log(log no

n

Can We Do Better?

Open Problem:Are there stronger k-private

algorithms than solution-list algorithms?

SAT

1 2

1

2

3

4

Can We Do Better?

Open Problem:Are there stronger k-private

algorithms than solution-list algorithms?

Positive: design non-solution-list private approximation algorithms.

Negative: improve impossibility result for any almost-private algorithm.