Loading in 5 sec....

Bypassing the Unique Games Conjecture for two geometric problemsPowerPoint Presentation

Bypassing the Unique Games Conjecture for two geometric problems

Download Presentation

Bypassing the Unique Games Conjecture for two geometric problems

Loading in 2 Seconds...

- 134 Views
- Uploaded on
- Presentation posted in: General

Bypassing the Unique Games Conjecture for two geometric problems

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Bypassing the Unique Games Conjecture for two geometric problems

Yi Wu

IBM Almaden Research

Based on joint work with

VenkatesanGuruswami Prasad Raghavendra Rishi Saket

CMUGeorgia Tech IBM

- Unique Games Conjecture [Khot 02]
- For every there is an integer such that it is NP-hard to decide whether a UG instance on labels has:
- (YES instance)
- (NO instance)

Implications of UGC

For a large class of optimization problems, Semidefinite Programming (SDP) gives the best polynomial time approximation.

MAX 2SAT

MAX 2LIN

MAX 2AND

Max Cut

MAX 3CSP

MultiCut

0-EXTENSION

Max 3 SAT

Multiway Cut

MAX 3SAT

Max 4 SAT

Max 2 SAT

- Lower bound: strong SDP integrality gap instance exists. [KV05, KS09,RS09, BGHMRS]
- Upper bound: [Arora-Barak-Steurer 11] can be solved in time .
- The reduction from SAT (of size to prove UGC needs to have size blowup if SAT does not have sub-exponential algorithm.

- What if UGC is false? The optimality of SDP may not hold.
- very few result on the optimality of SDP without UGC.

- It is not clear whether Unique Games Conjecture is a necessary assumption for all the hardness results.

- For two natural geometric problems, we prove that Semidefinite Programming gives the best polynomial time approximation withoutassuming UGC.
- same UG-hardness results known previously.

- Input: a set of points , a number Some constant
- Algorithmic task: finding the best dimensional subspace minimize the norm of its Euclidean distance to the points.

is the Euclidean distance between and

- Objective function:

- least square regression.
- : Minimum enclosing ball.
In this work, we study the problem for

where

Let be the -th norm of a Gaussian

- Previous result: [Deshpande-Tulsiani-Vishnoi11] :
- UG hardnessof approximation
- approximation by SDP.

- Our result: NP hardness of approximation.

- Input: a symmetric matrix
- Algorithmic goal:
Subject to for

- : calculating the largest eigenvalue.
- : the Grothendieck problem on complete graph.
In this work, we study the problem for

[Kindler-Naor-Schechtman 06] :

- UG hardness
- approximation by SDP.

- NP-hardnessof approximation.
- approximation by SDP.
- independently by [Naor-Schechtman]

- While both problems have nothing to do with Gaussian, involves Gaussian Distribution in a fundamental way.
- Gaussian Distribution also occurs fundamentally in UG hard ness proof, coincidence?

- Evidence that SDP can be the best algorithm for optimization problems without UGC.
- the approximation threshold is : unlikely to have a simple alternative combinatorial algorithm?

- Our hardness reduction have size blow up matching the Arora-Barak-Steurer algorithm’s requirement.

- A instance of subspace approximation over and .
Equivalent problem: finding

| is the distance from to subspace orthogonal to

- Completeness: for every depends only on 1 coordinate (, is less than
- Soundness: for every that depends only on a lot of coordinates, is above

If we have a dictator test instance, then it is UG-hard get better than-approximation.

- Let be all the points on
- (Completeness) When ,
- (Soundness, informal proof) When

by CLT

Label sets

For edge ,

satisfies if,

Theorem [Khot 02] : (soundness), s.t. given an instance with label sets it is NP-hard to decide,

OPT() (YES) or OPT() (NO)

where satisfies the following property,

(smoothness) the set of projections is a good hash family.

- Composing the Smooth Label Cover with the dictator test.

- Other geometric problem with only UG hardness are known.
- Kernel Clustering
- Learning halfspacesby degree polynomials
- Matrix Norm (SSE hardness).