# The Approximability of Learning and Constraint Satisfaction Problems

An α-approximation algorithm is an algorithm guaranteed to output a solution

that is within an α ratio of the optimal solution. We are interested in the

following question: Given an NP-hard optimization problem, what is the best

approximation guarantee that any polynomial time algorithm could achieve?

We mostly focus on studying the approximability of two classes of NP-hard

problems: Constraint Satisfaction Problems (CSPs) and Computational Learning Problems.

For CSPs, we mainly study the approximability of MAX CUT, MAX 3-CSP,

MAX 2-LIN_{R}, VERTEX-PRICING, as well as serval variants of the UNIQUEGAMES.

• The problem of MAX CUT is to find a partition of a graph so as to maximize

the number of edges between the two partitions. Assuming the

Unique Games Conjecture, we give a complete characterization of the approximation

curve of the MAX CUT problem: for every optimum value of

the instance, we show that certain SDP algorithm with RPR^{2} rounding

always achieve the optimal approximation curve.

• The input to a 3-CSP is a set of Boolean constraints such that each constraint

contains at most 3 Boolean variables. The goal is to find an assignment

to these variables to maximize the number of satisfied constraints.

We are interested in the case when a 3-CSP is satisfiable, i.e.,

there does exist an assignment that satisfies every constraint. Assuming

the d-to-1 conjecture (a variant of the Unique Games Conjecture), we

prove that it is NP-hard to give a better than 5/8-approximation for the

problem. Such a result matches a SDP algorithm by Zwick which gives

a 5/8-approximation problem for satisfiable 3-CSP. In addition, our result

also conditionally resolves a fundamental open problem in PCP theory on

the optimal soundness for a 3-query nonadaptive PCP system for NP with

perfect completeness.

• The problem of MAX 2-LINZ involves a linear systems of integer equations;

these equations are so simple such that each equation contains at

most 2 variables. The goal is to find an assignment to the variables so as

to maximize the total number of satisfied equations. It is a natural generalization

of the Unique Games Conjecture which address the hardness of

the same equation systems over finite fields. We show that assuming the

Unique Games Conjecture, for a MAX 2-LINZ instance, even that there

exists a solution that satisfies 1−ε of the equations, it is NP-hard to find

one that satisfies ² of the equations for any ε > 0.