## Hardness of Constraint Satisfaction and Hypergraph Coloring

### Constructions of Probabilistically Checkable Proofs with Perfect Completeness

Sangxia Huang

[**PDF**] [**Errata**]

**Abstract**

A Probabilistically Checkable Proof (PCP) of a mathematical statement is a proof written in a special manner that allows for efficient probabilistic verification. The celebrated PCP Theorem states that for every family of statements in NP, there is a probabilistic verification procedure that checks the validity of a PCP proof by reading only 3 bits from it. This landmark theorem, and the works leading up to it, laid the foundation for many subsequent works in computational complexity theory, the most prominent among them being the study of inapproximability of combinatorial optimization problems.

This thesis focuses on a broad class of combinatorial optimization problems called Constraint Satisfaction Problems (CSPs). In an instance of a CSP problem of arity *k*, we are given a set of variables taking values from some finite domain, and a set of constraints each involving a subset of at most *k* variables. The goal is to find an assignment that simultaneously satisfies as many constraints as possible. An alternative formulation of the goal that is commonly used is Gap-CSP, where the goal is to decide whether a CSP instance is satisfiable or far from satisfiable, where the exact meaning of being far from satisfiable varies depending on the problems.We first study Boolean CSPs, where the domain of the variables is {0,1}. The main question we study is the hardness of distinguishing satisfiable Boolean CSP instances from those for which no assignment satisfies more than some epsilon fraction of the constraints. Intuitively, as the arity increases, the CSP gets more complex and thus the hardness parameter epsilon should decrease. We show that for Boolean CSPs of arity *k*, it is NP-hard to distinguish satisfiable instances from those that are at most *2 ^{Õ(k1/3)}/2^{k}*-satisfiable.

We also study coloring of graphs and hypergraphs. Given a graph or a hypergraph, a coloring is an assignment of colors to vertices, such that all edges or hyperedges are non-monochromatic. The gap problem is to distinguish instances that are colorable with a small number of colors, from those that require a large number of colors. For graphs, we prove that there exists a constant *K _{0}>0*, such that for any

*K ≥ K*, it is NP-hard to distinguish

_{0}*K*-colorable graphs from those that require

*2*colors. For hypergraphs, we prove that it is quasi-NP-hard to distinguish 2-colorable 8-uniform hypergraphs of size

^{Ω(K1/3)}*N*from those that require

*2*colors.

^{(log N)1/10-o(1)}In terms of techniques, all these results are based on constructions of PCPs with perfect completeness, that is, PCPs where the probabilistic proof verification procedure always accepts a correct proof. Not only is this a very natural property for proofs, but it can also be an essential requirement in many applications. It has always been particularly challenging to construct PCPs with perfect completeness for NP statements due to limitations in techniques. Our improved hardness results build on and extend many of the current approaches. Our Boolean CSP result and GraphColoring result were proved by adapting the Direct Sum of PCPs idea by Siu On Chan to the perfect completeness setting. Our proof for hypergraph coloring hardness improves and simplifies the recent work by Khot and Saket, in which they proposed the notion of superposition complexity of CSPs.

*Abstract in Swedish available on KTH publication database DiVA.*