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Given a fixed arity $k \geq 2$, Min-$k$-CSP on complete instances involves a set of $n$ variables $V$ and one nontrivial constraint for every $k$-subset of variables (so there are $\binom{n}{k}$ constraints). The goal is to find an…

In this paper, we consider lower bounds on the query complexity for testing CSPs in the bounded-degree model. First, for any ``symmetric'' predicate $P:{0,1}^{k} \to {0,1}$ except \equ where $k\geq 3$, we show that every (randomized)…

In this paper, we present a polynomial-time algorithm that approximates sufficiently high-value Max 2-CSPs on sufficiently dense graphs to within $O(N^{\varepsilon})$ approximation ratio for any constant $\varepsilon > 0$. Using this…

In this paper, we prove an almost-optimal hardness for Max $k$-CSP$_R$ based on Khot's Unique Games Conjecture (UGC). In Max $k$-CSP$_R$, we are given a set of predicates each of which depends on exactly $k$ variables. Each variable can…

Raghavendra (STOC 2008) gave an elegant and surprising result: if Khot's Unique Games Conjecture (STOC 2002) is true, then for every constraint satisfaction problem (CSP), the best approximation ratio is attained by a certain simple…

Motivated by the pervasiveness of strong inapproximability results for Max-CSPs, we introduce a relaxed notion of an approximate solution of a Max-CSP. In this relaxed version, loosely speaking, the algorithm is allowed to replace the…

We consider the question of approximating Max 2-CSP where each variable appears in at most $d$ constraints (but with possibly arbitrarily large alphabet). There is a simple $(\frac{d+1}{2})$-approximation algorithm for the problem. We prove…

This thesis investigates the extent to which the optimal value of a constraint satisfaction problem (CSP) can be approximated by some sentence of fixed point logic with counting (FPC). It is known that, assuming $\mathsf{P} \neq…

Let $\mathcal{G}$ be a $k$-player game with value $<1$, whose query distribution is such that no marginal on $k-1$ players admits a non-trivial Abelian embedding. We show that for every $n\geq N$, the value of the $n$-fold parallel…

We consider the problem of approximately solving constraint satisfaction problems with arity $k > 2$ ($k$-CSPs) on instances satisfying certain expansion properties, when viewed as hypergraphs. Random instances of $k$-CSPs, which are also…

We develop a polynomial time $\Omega\left ( \frac 1R \log R \right)$ approximate algorithm for Max 2CSP-$R$, the problem where we are given a collection of constraints, each involving two variables, where each variable ranges over a set of…

Random constraint satisfaction problems (CSPs) such as random $3$-SAT are conjectured to be computationally intractable. The average case hardness of random $3$-SAT and other CSPs has broad and far-reaching implications on problems in…

We introduce and study a new model of interactive proofs: AM(k), or Arthur-Merlin with k non-communicating Merlins. Unlike with the better-known MIP, here the assumption is that each Merlin receives an independent random challenge from…

We establish the satisfiability threshold for random $k$-SAT for all $k\ge k_0$, with $k_0$ an absolute constant. That is, there exists a limiting density $\alpha_*(k)$ such that a random $k$-SAT formula of clause density $\alpha$ is with…

We study a birthday inequality in random geometric graphs: the probability of the empty graph is upper bounded by the product of the probabilities that each edge is absent. We show the birthday inequality holds at low densities, but does…

We show that for every $k\in\mathbb{N}$ and $\varepsilon>0$, for large enough alphabet $R$, given a $k$-CSP with alphabet size $R$, it is NP-hard to distinguish between the case that there is an assignment satisfying at least…

The subspace approximation problem Subspace($k$,$p$) asks for a $k$-dimensional linear subspace that fits a given set of points optimally, where the error for fitting is a generalization of the least squares fit and uses the $\ell_{p}$ norm…

Given $k$ collections of 2SAT clauses on the same set of variables $V$, can we find one assignment that satisfies a large fraction of clauses from each collection? We consider such simultaneous constraint satisfaction problems, and design…

Constraint satisfaction problems (CSPs) are ubiquitous in theoretical computer science. We study the problem of StrongCSPs, i.e. instances where a large induced sub-instance has a satisfying assignment. More formally, given a CSP instance…

Given a predicate $P: \{-1, 1\}^k \to \{-1, 1\}$, let $CSP(P)$ be the set of constraint satisfaction problems whose constraints are of the form $P$. We say that $P$ is approximable if given a nearly satisfiable instance of $CSP(P)$, there…