Name
Abstract | ||
|---|---|---|
The long code is a central tool in hardness of approximation, especially in questions related to the unique games conjecture. We construct a new code that is exponentially more efficient, but can still be used in many of these applications. Using the new code we obtain exponential improvements over several known results, including the following: 1) For any ε >; 0, we show the existence of an n vertex graph G where every set of o(n) vertices has expansion 1 - ε, but G's adjacency matrix has more than exp(logδ n) eigenvalues larger than 1 - ε, where δ depends only on ε. This answers an open question of Arora, Barak and Steurer (FOCS 2010) who asked whether one can improve over the noise graph on the Boolean hypercube that has poly(log n) such eigenvalues. 2) A gadget that reduces unique games instances with linear constraints modulo K into instances with alphabet k with a blowup of Kpolylog(K), improving over the previously known gadget with blowup of 2Ω(K). 3) An n variable integrality gap for Unique Games that survives exp(poly(log log n)) rounds of the SDP + Sherali Adams hierarchy, improving on the previously known bound of poly(log log n). We show a connection between the local testability of linear codes and small set expansion in certain related Cayley graphs, and use this connection to derandomize the noise graph on the Boolean hypercube. |
Year | DOI | Venue |
|---|---|---|
2012 | 10.1109/FOCS.2012.83 | SIAM: SIAM Journal on Computing |
Keywords | Field | DocType |
noise graph,cayley graphs,boolean hypercube,new code,unique games conjecture,eigenvalues,long code local testability,long code,approximation theory,log n,boolean algebra,matrix algebra,adjacency matrix,variable integrality gap,certain related cayley graph,small set expansion,linear code,game theory,sdp-sherali adams hierarchy,vertex graph,graph theory,known result,locally testable codes,linear constraints modulo,eigenvalues and eigenfunctions,hypercube networks,approximation hardness,hardness of approximation,computer science,cayley graph | Adjacency matrix,Graph theory,Discrete mathematics,Combinatorics,Unique games conjecture,Vertex (geometry),Hardness of approximation,Vertex (graph theory),Cayley graph,Mathematics,Hypercube | Conference |
Volume | Issue | ISSN |
44 | 5 | 0272-5428 |
ISBN | Citations | PageRank |
978-1-4673-4383-1 | 20 | 0.72 |
References | Authors | |
11 | 6 | |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Boaz Barak | 1 | 2563 | 127.61 |
| Parikshit Gopalan | 2 | 1186 | 61.52 |
| Johan Håstad | 3 | 3586 | 557.23 |
| Raghu Meka | 4 | 485 | 37.05 |
| Prasad Raghavendra | 5 | 1013 | 50.58 |
| David Steurer | 6 | 934 | 44.91 |