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The Projection Games Conjecture & The Hardness of lnN -Approximating Set-Cover

The Projection Games Conjecture & The Hardness of lnN -Approximating Set-Cover. Dana Moshkovitz MIT. This Talk. Take-home message: Prove hardness results assuming The Projection Games Conjecture (PGC)!

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The Projection Games Conjecture & The Hardness of lnN -Approximating Set-Cover

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  1. The Projection Games Conjecture &The Hardness of lnN-Approximating Set-Cover Dana Moshkovitz MIT

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

  3. Projection Games A B A B e:AB An edge e=(a,b)E is satisfied by assignments fA:AA, fB:BB, if e(fA(a))=fB(b). PG: Given a game G=(G=(A,B,E),A,B, {e}e), Find fA:AA, fB:BBmaximizingPe(e satisfied). . . . . . .

  4. The Projection Games Conjecture • There exists c>0, such that for any 1/nc, a Boolean formula , ||=n can be efficiently transformed to a projection game with graph size n1+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.

  5. What’s Known • Parallel repetition [Raz94]: graph size n(log(1/)) and alphabet size poly(1/). • [M-Raz08]: graph size n1+o(1)poly(1/) and alphabet size exp(1/).

  6. Set-Cover • Greedy: lnN-lnlnN+O(1)-approx in poly time[C79,S96] • Linear programming: lnN–approx; poly time [S99] • Sub-exponential: (1-)lnN-approx in 2NO()-time [CKW09]

  7. The Hardness of Set-Cover

  8. How To Prove Hardness? All the results use the scheme of Lund-Yannakakisthat has two components: • Projection game with =1/(logn)3. • Combinatorial set-cover gadget. PCP time lower bound. Combinatoricsapproximation factor.

  9. The Lund-Yannakakis Scheme 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: Pick Band use subsets associated with . Pick more thanlnNsubsets.

  10. The Feige Game • In the non-sat case, for any fA:AA, for (1-) fraction of the bB, all b’s neighbors aA have different projections (a,b)(fA(a)). A B A B 1 4 2 . . . . . .

  11. The Current Work • 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

  12. Our Tool • 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. N C . . .

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

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