Inapproximability of NP-complete Problems, Discrete Fourier Analysis, and Geometry

Abstract

Abstract. This article gives a survey of recent results that connect three areas in computer science and mathematics: (1) (Hardness of) computing approximate solutions to NP-complete problems. (2) Fourier analysis of boolean functions on boolean hypercube. (3) Certain problems in geometry, especially related to isoperimetry and embeddings between metric spaces. Mathematics Subject Classification (2000). Primary 68Q17.

On Khot’s unique games conjecture

“…In 2002, Khot [23] formulated a conjecture that considerably simplifies (2), essentially making it of difficulty comparable to (1).…”

“…Khot formulated the Unique Games Conjecture in a very influential 2002 paper [23]. In the subsequent eight years, the conjecture has motivated and enabled a large body of work on the computational complexity of approximating combinatorial optimization problems (the original context of the conjecture) and on the quality of approximation provided by "semidefinite programming" convex relaxations (a somewhat unexpected by-product).…”

Reoptimization of constraint satisfaction problems with approximation resistant predicates

“…(2) c c c In what follows, we will use information from [19]. Let P be an arbitrary optimization (for definiteness, maximization) problem.…”

On the Unique Games Conjecture

“…We present a complete reduction from GapUG to GapMin-2SAT-Deletion as an illustration. We refer to the author's recent article [62] for a more detailed survey of such reductions.…”

“…4 A random function is intuitively very far from being a dictatorship, but has all influences close to 1 2 (same holds for the PARITY function whose all influences are 1). This is a somewhat annoying issue and one gets around it by using a more refined notion of "low degree" influence (see [62]). In this article, however, we choose not to present this refined notion for the sake of simplicity.…”