3 research outputs found
Improved NP-Inapproximability for 2-Variable Linear Equations
An instance of the 2-Lin(2) problem is a system of equations of the form "x_i + x_j = b (mod 2)". Given such a system in which it\u27s possible to satisfy all but an epsilon fraction of the equations, we show it is NP-hard to satisfy all but a C*epsilon fraction of the equations, for any C < 11/8 = 1.375 (and any 0 < epsilon <= 1/8). The previous best result, standing for over 15 years, had 5/4 in place of 11/8. Our result provides the best known NP-hardness even for the Unique Games problem, and it also holds for the special case of Max-Cut. The precise factor 11/8 is unlikely to be best possible; we also give a conjecture concerning analysis of Boolean functions which, if true, would yield a larger hardness factor of 3/2.
Our proof is by a modified gadget reduction from a pairwise-independent predicate. We also show an inherent limitation to this type of gadget reduction. In particular, any such reduction can never establish a hardness factor C greater than 2.54. Previously, no such limitation on gadget reductions was known
Reduction from Non-Unique Games to Boolean Unique Games
We reduce the problem of proving a "Boolean Unique Games Conjecture" (with gap 1-? vs. 1-C?, for any C > 1, and sufficiently small ? > 0) to the problem of proving a PCP Theorem for a certain non-unique game. In a previous work, Khot and Moshkovitz suggested an inefficient candidate reduction (i.e., without a proof of soundness). The current work is the first to provide an efficient reduction along with a proof of soundness. The non-unique game we reduce from is similar to non-unique games for which PCP theorems are known.
Our proof relies on a new concentration theorem for functions in Gaussian space that are restricted to a random hyperplane. We bound the typical Euclidean distance between the low degree part of the restriction of the function to the hyperplane and the restriction to the hyperplane of the low degree part of the function
Strong Parallel Repetition for Unique Games on Small Set Expanders
Strong Parallel Repetition for Unique Games on Small Set Expanders
The strong parallel repetition problem for unique games is to efficiently
reduce the 1-delta vs. 1-C*delta gap problem of Boolean unique games (where C>1
is a sufficiently large constant) to the 1-epsilon vs. epsilon gap problem of
unique games over large alphabet. Due to its importance to the Unique Games
Conjecture, this problem garnered a great deal of interest from the research
community. There are positive results for certain easy unique games (e.g.,
unique games on expanders), and an impossibility result for hard unique games.
In this paper we show how to bypass the impossibility result by enlarging the
alphabet sufficiently before repetition. We consider the case of unique games
on small set expanders for two setups: (i) Strong small set expanders that
yield easy unique games. (ii) Weaker small set expanders underlying possibly
hard unique games as long as the game is mildly fortified. We show how to
fortify unique games in both cases, i.e., how to transform the game so
sufficiently large induced sub-games have bounded value. We then prove strong
parallel repetition for the fortified games. Prior to this work fortification
was known for projection games but seemed hopeless for unique games