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Bypassing the Unique Games Conjecture for two geometric problems. Yi Wu IBM Almaden Research. Based on joint work with Venkatesan Guruswami Prasad Raghavendra Rishi Saket CMU Georgia Tech IBM . Unique Games Conjecture. Unique Games Conjecture [ Khot 02]
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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 • 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
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 withoutassuming 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 • Objective function: • least square regression. • : Minimum enclosing ball. In this work, we study the problem for
Our results 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.
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
Previous Result: [Kindler-Naor-Schechtman 06] : • UG hardness • approximation by SDP.
Our Result • NP-hardnessof 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.
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,
Smooth Label Cover 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.
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 halfspacesby degree polynomials • Matrix Norm (SSE hardness).