Bypassing the Unique Games Conjecture for two geometric problems

Yi WuIBM Almaden Research

Based on joint work with

Venkatesan Guruswami Prasad Raghavendra Rishi Saket CMU Georgia 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)

Unique Games Conjecture

𝑢𝑣

𝑒

Max 3 SAT

Max 2 SAT

Max Cut MAX 3CSP

Max 4 SAT

MAX 2AND

0-EXTENSIONMultiway Cut

MAX 2SAT MAX 2LIN

MAX 3SAT

MultiCut

Implications of UGC

For a large class of optimization problems, Semidefinite Programming (SDP) gives the

best polynomial time approximation.

Status of the UGC

• 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.

Skepticism of UGC

• 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.

Overview of our work

• For two natural geometric problems, we prove that Semidefinite Programming gives the best polynomial time approximation without assuming UGC.– same UG-hardness results known previously.

Problem 1: Subspace approximation

• 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

Special case

• least square regression.• : Minimum enclosing ball.

In this work, we study the problem for

Objective function:

Our results

Let be the -th norm of a Gaussian• Previous result: [Deshpande-Tulsiani-Vishnoi

11] : – UG hardness of approximation – approximation by SDP.

• Our result: NP hardness of approximation.

where

Problem 2: Quadratic Maximization

• Input: a symmetric matrix

• Algorithmic goal:

Subject to for

Special case

• : calculating the largest eigenvalue.• : the Grothendieck problem on complete

graph.

In this work, we study the problem for

𝑚𝑎𝑥|𝑥|𝑝=1

𝑥𝑇 𝐴𝑥

Previous Result:

[Kindler-Naor-Schechtman 06] :• UG hardness• approximation by SDP.

𝑚𝑎𝑥|𝑥|𝑝=1

𝑥𝑇 𝐴𝑥

Our Result

• NP-hardness of approximation.

• approximation by SDP.– independently by [Naor-Schechtman]

Remarks on our results

• 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.

Proof overview forsubspace approximation

Main Gadget: Dictator Test

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

| is the distance from to subspace orthogonal to

A Dictator Test instance

• 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.

A -Dictator Test instance

• Let be all the points on – (Completeness) When ,

– (Soundness, informal proof) When by CLT

Reduction from Smooth Label Cover

Label sets

For edge ,

satisfies if,

[𝑀 ][𝑀 ]

[𝑁 ]

𝑣𝑢 𝑒

π : [𝑀 ]→[𝑁 ] π ′ : [𝑀 ]→[𝑁 ]

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

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

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

where satisfies the following property,𝑣

[𝑀 ]𝑎 𝑏

[𝑁 ]

Rest of the proof

• Composing the Smooth Label Cover with the dictator test.

Future Work

• Other geometric problem with only UG hardness are known.– Kernel Clustering– Learning halfspaces by degree polynomials– Matrix Norm (SSE hardness).