d-To-1 Hardness of Coloring 3-Colorable Graphs with O(1) Colors

The d-to-1 conjecture of Khot asserts that it is NP-hard to satisfy an ? fraction of constraints of a satisfiable d-to-1 Label Cover instance, for arbitrarily small ? > 0. We prove that the d-to-1 conjecture for any fixed d implies the hardness of coloring a 3-colorable graph with C colors for arbitrarily large integers C. Earlier, the hardness of O(1)-coloring a 4-colorable graphs is known under the 2-to-1 conjecture, which is the strongest in the family of d-to-1 conjectures, and the hardness for 3-colorable graphs is known under a certain "fish-shaped" variant of the 2-to-1 conjecture

On the Usefulness of Predicates

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 constraints of an instance by some other (possibly real-valued) constraints, and then only needs to satisfy as many of the new constraints as possible. To be more precise, we introduce the following notion of a predicate $P$ being \emph{useful} for a (real-valued) objective $Q$: given an almost satisfiable Max-$P$ instance, there is an algorithm that beats a random assignment on the corresponding Max-$Q$ instance applied to the same sets of literals. The standard notion of a nontrivial approximation algorithm for a Max-CSP with predicate $P$ is exactly the same as saying that $P$ is useful for $P$ itself. We say that $P$ is useless if it is not useful for any $Q$. This turns out to be equivalent to the following pseudo-randomness property: given an almost satisfiable instance of Max-$P$ it is hard to find an assignment such that the induced distribution on $k$-bit strings defined by the instance is not essentially uniform. Under the Unique Games Conjecture, we give a complete and simple characterization of useful Max-CSPs defined by a predicate: such a Max-CSP is useless if and only if there is a pairwise independent distribution supported on the satisfying assignments of the predicate. It is natural to also consider the case when no negations are allowed in the CSP instance, and we derive a similar complete characterization (under the UGC) there as well. Finally, we also include some results and examples shedding additional light on the approximability of certain Max-CSPs

Near-Optimal UGC-hardness of Approximating Max k-CSP_R

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 take any value from $1, 2, \dots, R$. The goal is to find an assignment to variables that maximizes the number of satisfied predicates. Assuming the Unique Games Conjecture, we show that it is NP-hard to approximate Max $k$-CSP$_R$ to within factor $2^{O(k \log k)}(\log R)^{k/2}/R^{k - 1}$ for any $k, R$. To the best of our knowledge, this result improves on all the known hardness of approximation results when $3 \leq k = o(\log R/\log \log R)$. In this case, the previous best hardness result was NP-hardness of approximating within a factor $O(k/R^{k-2})$ by Chan. When $k = 2$, our result matches the best known UGC-hardness result of Khot, Kindler, Mossel and O'Donnell. In addition, by extending an algorithm for Max 2-CSP$_R$ by Kindler, Kolla and Trevisan, we provide an $\Omega(\log R/R^{k - 1})$-approximation algorithm for Max $k$-CSP$_R$. This algorithm implies that our inapproximability result is tight up to a factor of $2^{O(k \log k)}(\log R)^{k/2 - 1}$. In comparison, when $3 \leq k$ is a constant, the previously known gap was $O(R)$, which is significantly larger than our gap of $O(\text{polylog } R)$. Finally, we show that we can replace the Unique Games Conjecture assumption with Khot's $d$-to-1 Conjecture and still get asymptotically the same hardness of approximation

An Improved Dictatorship Test with Perfect Completeness

A Boolean function f:{0,1}^n->{0,1} is called a dictator if it depends on exactly one variable i.e f(x_1, x_2, ..., x_n) = x_i for some i in [n]. In this work, we study a k-query dictatorship test. Dictatorship tests are central in proving many hardness results for constraint satisfaction problems. The dictatorship test is said to have perfect completeness if it accepts any dictator function. The soundness of a test is the maximum probability with which it accepts any function far from a dictator. Our main result is a k-query dictatorship test with perfect completeness and soundness (2k + 1)/(2^k), where k is of the form 2^t -1 for any integer t > 2. This improves upon the result of [Tamaki-Yoshida, Random Structures & Algorithms, 2015] which gave a dictatorship test with soundness (2k + 3)/(2^k)

The Quest for Strong Inapproximability Results with Perfect Completeness

The Unique Games Conjecture (UGC) has pinned down the approximability of all constraint satisfaction problems (CSPs), showing that a natural semidefinite programming relaxation offers the optimal worst-case approximation ratio for any CSP. This elegant picture, however, does not apply for CSP instances that are perfectly satisfiable, due to the imperfect completeness inherent in the UGC. For the important case when the input CSP instance admits a satisfying assignment, it therefore remains wide open to understand how well it can be approximated. This work is motivated by the pursuit of a better understanding of the inapproximability of perfectly satisfiable instances of CSPs. Our main conceptual contribution is the formulation of a (hypergraph) version of Label Cover which we call "V label cover." Assuming a conjecture concerning the inapproximability of V label cover on perfectly satisfiable instances, we prove the following implications: * There is an absolute constant c0 such that for k >= 3, given a satisfiable instance of Boolean k-CSP, it is hard to find an assignment satisfying more than c0 k^2/2^k fraction of the constraints. * Given a k-uniform hypergraph, k >= 2, for all epsilon > 0, it is hard to tell if it is q-strongly colorable or has no independent set with an epsilon fraction of vertices, where q = ceiling[k + sqrt(k) - 0.5]. * Given a k-uniform hypergraph, k >= 3, for all epsilon > 0, it is hard to tell if it is (k-1)-rainbow colorable or has no independent set with an epsilon fraction of vertices. We further supplement the above results with a proof that an ``almost Unique\u27\u27 version of Label Cover can be approximated within a constant factor on satisfiable instances

Conditional Hardness for Satisfiable 3-CSPs

In this paper we study a fundamental open problem in the area of probabilistic checkable proofs: What is the smallest s such that NP ⊆ naPCP1,s[O(log n),3]? In the language of hardness of approximation, this problem is equivalent to determining the smallest s such that getting an s-approximation for satisfiable 3-bit constraint satisfaction problems ("3-CSPs") is NP-hard. The previous best upper bound and lower bound for s are 20/27+µ by Khot and Saket [KS06], and 5/8 (assuming NP subseteq BPP) by Zwick [Zwi98]. In this paper we close the gap assuming Khot's d-to-1 Conjecture. Formally, we prove that if Khot's d-to-1 Conjecture holds for any finite constant integer d, then NP naPCP1,5/8+ µ[O(log n),3] for any constant µ > 0. Our conditional result also solves Hastad's open question [Has01] on determining the inapproximability of satisfiable Max-NTW ("Not Two") instances and confirms Zwick's conjecture [Zwi98] that the 5/8-approximation algorithm for satisfiable 3-CSPs is optimal