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<p class=MsoNormal><span style='font-size:9.0pt;font-family:"Arial","sans-serif";
color:blue'>Moritz Hardt will present his research seminar/general exam on
Monday May 11 at 2PM <o:p></o:p></span></p>
<p class=MsoNormal><span style='font-size:9.0pt;font-family:"Arial","sans-serif";
color:blue'>in Room 302 (note room). The members of his committee
are: Boaz Barak, advisor, <o:p></o:p></span></p>
<p class=MsoNormal><span style='font-size:9.0pt;font-family:"Arial","sans-serif";
color:blue'>Sanjeev Arora, and Bernard Chazelle. Everyone is invited to
attend his talk, and those <o:p></o:p></span></p>
<p class=MsoNormal><span style='font-size:9.0pt;font-family:"Arial","sans-serif";
color:blue'>faculty wishing to remain for the oral exam following are welcome
to do so. His abstract <o:p></o:p></span></p>
<p class=MsoNormal><span style='font-size:9.0pt;font-family:"Arial","sans-serif";
color:blue'>and reading list follow below.<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-size:9.0pt;font-family:"Arial","sans-serif";
color:blue'>--------------------------------------------------<o:p></o:p></span></p>
<p class=MsoNormal style='margin-bottom:12.0pt'><br>
I. Title + abstract<br>
<br>
What are plausible hard instances for Unique Games?<br>
<br>
Unique Games are a family of constraint satisfaction problems conjectured to<br>
be NP-hard. This conjecture due to Khot implies that simple semidefinite<br>
programs (SDP) achieve optimal approximation ratios for several important<br>
problems including Max-Cut and Vertex Cover. Unfortunately, little evidence in<br>
support of the conjecture exists. Indeed, random instances of Unique Games are<br>
known to be easy. In this work we thus address the question: What are<br>
plausible hard instances for Unique Games? <br>
<br>
Our approach is to consider the refutation problem induced by subsampling a<br>
given SDP gap instance to constant degree. While subsampling preserves global<br>
parameters such as the integrality gap, the lack of local structure may lead<br>
to computational hardness. In support of our methodology, we observe that a<br>
constant degree subsample of essentially any unsatisfiable k-CSP instance<br>
gives rise to strong lower bounds in the Lasserre SDP hierarchy so long as k<br>
is greater than 3. Based on the work of Charikar, Makarychev and Makarychev,<br>
we also show that for 2-CSPs and Unique Games a constant degree subsample of<br>
any unsatisfiable instance gives strong lower bounds in the Sherali-Adams<br>
linear programming hierarchy. Unfortunately, our approach can only go so far.<br>
Appealing to Grothendieck's inequality, we show that for any 2-CSP the induced<br>
refutation problem is solvable in polynomial time.<br>
<br>
We then consider the question whether known SDP gaps withstand additional<br>
constraints when subsampled to constant degree. Somewhat surprisingly, we can<br>
show that already the triangle inequalities annihilate the integrality gap of<br>
the well known Feige-Schechtman gap instance for Max-Cut. This remains true<br>
even in a constant degree subsample. As for a future research direction,<br>
similar ideas could apply to all known Unique Games gap instances.<br>
<br>
My talk is based on ongoing research with Boaz Barak, Thomas Holenstein and<br>
David Steurer.<br>
<br>
<br>
II. Reading list<br>
<br>
Books:<br>
<br>
* Arora, Barak. Computational Complexity.
Chapters 11,13,14,18--23<br>
<br>
Papers:<br>
<br>
[Ale03] Alekhnovich. More on average case vs approximation complexity<br>
<br>
[ABW09] Applebaum, Barak, Wigderson. Public Key Cryptography from<br>
Different Assumptions <br>
<br>
[AKK+08] Arora, Khot, Kolla, Steurer, Tulsiani and Vishnoi. Unique Games <br>
on Expanding Constraint Graphs
are Easy<br>
<br>
[CMM09] Charikar, Makarychev, Makarychev. Integrality Gaps for Sherali-<br>
Adams Relaxations<br>
<br>
[Fei02] Feige. Relations between Average Case Complexity and <br>
Approximation Complexity<br>
<br>
[FS02] Feige and Schechtman. On the optimality of the random
hyperplane<br>
rounding technique for MAX CUT<br>
<br>
[Hol07] Holenstein. Parallel Repetition: Simplifications and the <br>
no-signaling case<br>
<br>
[KKMO04] Khot, Kindler, Mossel, O'Donnell. Optimal inapproximability <br>
results for max-cut and other
2-variable CSPs?<br>
<br>
[KV05] Khot, Vishnoi. The unique games conjecture, integrality gap
for <br>
cut problems, and
embeddability of negative type metrics into l1.<br>
<br>
[Rag08] Raghavendra. Optimal Algorithms and Inapproximability Results for
<br>
Every CSP?<br>
<br>
[Sch08] Schoenebeck. Linear Level Lasserre Lower Bounds for Certain k-CSPs
<br>
<br>
[TTV09] Trevisan, Tulsiani, Vadhan. Regularity, Boosting, and Efficiently<br>
<br>
<o:p></o:p></p>
<p class=MsoNormal><o:p> </o:p></p>
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