Dana Moshkovitz MIT

 Take-home message: Prove hardness results assuming The Projection Games Conjecture (PGC)! Theorem: If SAT requires exponential time, and the PGC holds, then (1-  )lnN approximating Set-Cover instances of size N requires time 2 N  (  ) (tight; stronger than Feige’s).

AB AA BB  e :  A  B An edge e=(a,b)  E is satisfied by assignments f A :A  A, f B :B  B, if  e (f A (a))=f B (b). PG: Given a game G=(G=(A,B,E),  A,  B, {  e } e ), Find f A :A  A, f B :B  B maximizing P e (e satisfied).

 There exists c>0, such that for any  1/n c, a Boolean formula , |  |=n can be efficiently transformed to a projection game with graph size n 1+o(1) poly(1/  ) and alphabet size poly(1/  ): ◦ If  is satisfiable, then there exist assignments to the projection game that satisfy all edges. ◦ If  is not satisfiable, then any assignments satisfy at most  fraction of the edges.

 Parallel repetition [Raz94]: graph size n  (log(1/  )) and alphabet size poly(1/  ).  [M-Raz08]: graph size n 1+o(1) poly(1/  ) and alphabet size exp(1/  ).

1.Greedy: lnN-lnlnN+O(1)- approx in poly time [C79,S96] 2.Linear programming: lnN–approx; poly time [S99] 3.Sub-exponential: (1-  )lnN-approx in 2 N O(  ) -time [CKW09]

Who proved?Approx factorTime assuming ETH Comments Lund-Yannakakis, 93log N/42 2 logN  (1) Bellare-Goldwasser- Lund-Russell 93 log N/C2 N  (1/loglogN) Feige 96(1-  )lnN2 N  (  /loglogN) Raz-Safra/Arora- Sudan 97 log N/C2 N  (1) Alon-M-Safra 06~0.22lnN2 N  (1) Current work(1-  )lnN2N()2N() Assuming PGC

All the results use the scheme of Lund- Yannakakis that has two components: 1. Projection game with  =1/(logn) Combinatorial set-cover gadget. PCP  time lower bound. Combinatorics  approximation factor.

sub-universe per b  B subset per a  A,  A For every e=(a,b)  E, covers a subset of the sub-universe of b that is associated with  e (  ). Combinatorial gadget: the only ways to cover a sub-universe: 1.Pick  B and use subsets associated with . 2.Pick more than lnN subsets.

 In the non-sat case, for any f A :A  A, for (1-  ) fraction of the b  B, all b’s neighbors a  A have different projections  (a,b) (f A (a)) AB AA BB 1 4 2

 We show how to transform any projection game to a Feige/rainbow game (without parallel repetition!) … Standard projection game: At most  fraction of neighbors agree … Feige/rainbow game: All neighbors disagree

 A bipartite graph (N,C,E); for every coloring of the N vertices, where no color set is of fraction > , at least (1-  ) fraction of the C vertices have all their neighbors from different colors.  The construction: incidence graph of lines and points NC

 Prove hardness results assuming the projection games conjecture, e.g., for Clique, Chromatic Number, Shortest-Vector-Problem (SVP), Group-Steiner-Tree…