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![Page 1: Bypassing the Unique Games Conjecture for two geometric problems Yi Wu IBM Almaden Research Based on joint work with Venkatesan Guruswami Prasad Raghavendra.](https://reader035.fdocuments.net/reader035/viewer/2022062801/56649e715503460f94b70400/html5/thumbnails/1.jpg)
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
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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
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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.
![Page 4: Bypassing the Unique Games Conjecture for two geometric problems Yi Wu IBM Almaden Research Based on joint work with Venkatesan Guruswami Prasad Raghavendra.](https://reader035.fdocuments.net/reader035/viewer/2022062801/56649e715503460f94b70400/html5/thumbnails/4.jpg)
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.
![Page 5: Bypassing the Unique Games Conjecture for two geometric problems Yi Wu IBM Almaden Research Based on joint work with Venkatesan Guruswami Prasad Raghavendra.](https://reader035.fdocuments.net/reader035/viewer/2022062801/56649e715503460f94b70400/html5/thumbnails/5.jpg)
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.
![Page 6: Bypassing the Unique Games Conjecture for two geometric problems Yi Wu IBM Almaden Research Based on joint work with Venkatesan Guruswami Prasad Raghavendra.](https://reader035.fdocuments.net/reader035/viewer/2022062801/56649e715503460f94b70400/html5/thumbnails/6.jpg)
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.
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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
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Special case
• least square regression.• : Minimum enclosing ball.
In this work, we study the problem for
Objective function:
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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
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Problem 2: Quadratic Maximization
• Input: a symmetric matrix
• Algorithmic goal:
Subject to for
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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
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Previous Result:
[Kindler-Naor-Schechtman 06] :• UG hardness• approximation by SDP.
𝑚𝑎𝑥|𝑥|𝑝=1
𝑥𝑇 𝐴𝑥
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Our Result
• NP-hardness of approximation.
• approximation by SDP.– independently by [Naor-Schechtman]
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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.
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Proof overview forsubspace approximation
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Main Gadget: Dictator Test
• A instance of subspace approximation over and . Equivalent problem: finding
| is the distance from to subspace orthogonal to
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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.
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A -Dictator Test instance
• Let be all the points on – (Completeness) When ,
– (Soundness, informal proof) When by CLT
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Reduction from Smooth Label Cover
Label sets
For edge ,
satisfies if,
[𝑀 ][𝑀 ]
[𝑁 ]
𝑣𝑢 𝑒
π : [𝑀 ]→[𝑁 ] π ′ : [𝑀 ]→[𝑁 ]
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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 ]
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Rest of the proof
• Composing the Smooth Label Cover with the dictator test.
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Future Work
• Other geometric problem with only UG hardness are known.– Kernel Clustering– Learning halfspaces by degree polynomials– Matrix Norm (SSE hardness).