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 RPR² 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.