private approximation of search problems amos beimel paz carmi kobbi nissim enav weinreb ben gurion...
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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.