### Improved NP-Inapproximability for 2-Variable Linear Equations

Johan Håstad, Sangxia Huang, Rajsekar Manokaran, Ryan O'Donnell, John Wright, Marc Herbstritt
2015 International Workshop on Approximation Algorithms for Combinatorial Optimization
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's possible to satisfy all but an fraction of the equations, we show it is NP-hard to satisfy all but a C fraction of equations, for any C < 11 8 = 1.375 (and any 0 < ≤ 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
more » ... l 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 it was possible for the hardness factor to grow to infinity as the arity of the predicate grows. The well known constraint satisfaction problem (CSP) 2-Lin(q) is defined as follows: Given n variables x 1 , . . . , x n , as well as a system of equations (constraints) of the form " for constants b ∈ Z q , the task is to assign values from Z q to the variables so that there are as few unsatisfied constraints as possible. It is known [KKMO07, MOO10] that, from an approximability standpoint, this problem is equivalent to the notorious Unique-Games problem [Kho02] . The special case of q = 2 is particularly interesting and can be equivalently stated as follows: Given a "supply graph" G and a "demand graph" H over the same set of vertices V , partition V into two parts so as to minimize the total number of cut supply edges and uncut demand edges. The further special case when the supply graph G is empty (i.e., every equation is of the form x i − x j = 1 (mod 2)) is equivalent to the Max-Cut problem. Let's say that an algorithm guarantees an ( , )-approximation if, given any instance in which the best solution falsifies at most an -fraction of the constraints, the algorithm finds a solution falsifying at most an -fraction of the constraints. If an algorithm guarantees ( , C )-approximation for every then we also say that it is a factor-C approximation. To illustrate the notation we recall two simple facts. On one hand, for each fixed q, there is a trivial greedy algorithm which (0, 0)approximates 2-Lin(q). On the other hand, ( , )-approximation is NP-hard for every 0 < < 1 q ; in particular, factor-1 approximation is NP-hard. We remark here that we are prioritizing the so-called "Min-Deletion" version of the 2-Lin(2) problem. We feel it is the more natural parameterization. For example, in the more traditional "Max-2-Lin(2)" formulation, the discrepancy between known algorithms and NP-hardness involves two quirky factors, 0.878 and 0.912. However, this disguises what we feel is the really interesting question -the same key open question that arises for the highly analogous Sparsest-Cut problem: Is there an efficient ( , O( ))-approximation, or even one that improves on the known ( , O( √ log n) )-and ( , O( √ ))-approximations? The relative importance of the "Min-Deletion" version is even more pronounced for the 2-Lin(q) problem. As we describe below, this version of the problem is essentially equivalent to the highly notorious Unique-Games problem. By way of contrast, the traditional maximization approximation factor measure for Unique-Games is not particularly interesting -it's known [FR04] that there is no constant-factor approximation for "Max-Unique-Games", but this appears to have no relevance for the Unique Games Conjecture.