Private Approximation of Search Problems Amos Beimel Paz Carmi Kobbi Nissim Enav Weinreb Ben Gurion University Research partially Supported by the Frankel Center for Computer Science.

Vertex Cover  Input: undirected graph G=.  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.

Vertex Cover - Example 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

G “ So, tell me an approximation! ” “ Hmmm …” VC Matching 22 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(G 1 )=mVC(G 2 )  A(G 1 )=A(G 2 )  Results: If NP BPP there is no polynomial private n 1-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.

The Search Problem  What is the right definition of privacy?!  Many solutions for one input.  mVC(G 1 )=mVC(G 2 )  A(G 1 )=A(G 2 ) ? NO!!!

Private Algorithms - Definition R – Equivalence Relation over {0,1}* A - Algorithm A is private with respect to R if: x y A( ) ≈ c x y

Example – Vertex Cover (Search)  G 1 ≈ VC G 2 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. G 1 ≈ VC G 2  A(G 1 ) A(G 2 ) ≈ 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 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  G 1 ≈ VC G 2 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. G 1 ≈ VC G 2  A(G 1 ) A(G 2 ) If A is deterministic: G 1 ≈ VC G 2  A(G 1 ) = A(G 2 ) ≈ c

Vertex Cover - Impossibility Result Theorem 1 If P ≠NP there is no deterministic polynomial time n 1-ε -approximation algorithm that is private with respect to ≈ VC. Proof Idea: Given a private n 1-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).

Claim 1 Let be a graph and. If then both and are non critical for. Note: The claim is useless if.

Proof of Claim 1 Let be a graph and. If then both and are non critical for. is non critical for. 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.

Relevant / Non-Critical Algorithm Let Is ? I is a big set of isolated vertices.

Case 1 is non critical for. Privacy (If was critical for, both copies of would be critical for.)

Case 2 By Claim 1 is not critical for.Claim 1 Let be the size of the minimum vertex cover of. Thus, there exists a minimum cover of that does not contain.

Case 2 (cont.) Assume is not relevant for. must contain both copies of as it does not contain. Thus, contains two non-optimal vertex covers of

Case 2 (cont. ii) However, taking two minimal covers of and adding results in a cover for of size. Contradiction to the minimality of Hence, is relevant for.

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. ”

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.

Vertex Cover - Impossibility Result Theorem 1 If P ≠NP there is no deterministic polynomial time n 1-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

Almost-Private Algorithms x y A( ) ≈ c w z ? x y z w

Almost-Private Algorithms is k-private with respect to if there exists such that: Every equivalence class of is a union of at most equivalence classes of. 3. is private with respect to.

Almost Private Approximation for MAX-3SAT and have the same set of maximum satisfying assignments. … Every equivalence class of is divided into subclasses. …

Almost Private Approximation for MAX-3SAT 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. … …

Almost Private Approximation for MAX-3SAT Theorem 2 There 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.

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. Proof Construct almost 3-wise [AGHP] independent variables. Number of assignments:.

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.

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.

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.

Can We Do Better? Open Problem: Are there stronger k-private algorithms than solution-list algorithms?

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.