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Bypassing the Unique Games Conjecture for two geometric problemsPowerPoint Presentation

Bypassing the Unique Games Conjecture for two geometric problems

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Bypassing the Unique Games Conjecture for two geometric problems

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Bypassing the Unique Games Conjecture for two geometric problems

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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).