A Hypergraph Dictatorship Test with Perfect Completeness

A hypergraph dictatorship test is first introduced by Samorodnitsky and Trevisan and serves as a key component in their unique games based \PCP construction. Such a test has oracle access to a collection of functions and determines whether all the functions are the same dictatorship, or all their low degree influences are $o(1).$ Their test makes $q\geq3$ queries and has amortized query complexity $1+O(\frac{\log q}{q})$ but has an inherent loss of perfect completeness. In this paper we give an adaptive hypergraph dictatorship test that achieves both perfect completeness and amortized query complexity $1+O(\frac{\log q}{q})$.Comment: Some minor correction

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

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)

Ignorance is Almost Bliss: Near-Optimal Stochastic Matching With Few Queries

The stochastic matching problem deals with finding a maximum matching in a graph whose edges are unknown but can be accessed via queries. This is a special case of stochastic $k$-set packing, where the problem is to find a maximum packing of sets, each of which exists with some probability. In this paper, we provide edge and set query algorithms for these two problems, respectively, that provably achieve some fraction of the omniscient optimal solution. Our main theoretical result for the stochastic matching (i.e., $2$-set packing) problem is the design of an \emph{adaptive} algorithm that queries only a constant number of edges per vertex and achieves a $(1-\epsilon)$ fraction of the omniscient optimal solution, for an arbitrarily small $\epsilon>0$. Moreover, this adaptive algorithm performs the queries in only a constant number of rounds. We complement this result with a \emph{non-adaptive} (i.e., one round of queries) algorithm that achieves a $(0.5 - \epsilon)$ fraction of the omniscient optimum. We also extend both our results to stochastic $k$-set packing by designing an adaptive algorithm that achieves a $(\frac{2}{k} - \epsilon)$ fraction of the omniscient optimal solution, again with only $O(1)$ queries per element. This guarantee is close to the best known polynomial-time approximation ratio of $\frac{3}{k+1} -\epsilon$ for the \emph{deterministic} $k$-set packing problem [Furer and Yu, 2013] We empirically explore the application of (adaptations of) these algorithms to the kidney exchange problem, where patients with end-stage renal failure swap willing but incompatible donors. We show on both generated data and on real data from the first 169 match runs of the UNOS nationwide kidney exchange that even a very small number of non-adaptive edge queries per vertex results in large gains in expected successful matches

Improved Inapproximability Results for Maximum k-Colorable Subgraph

We study the maximization version of the fundamental graph coloring problem. Here the goal is to color the vertices of a k-colorable graph with k colors so that a maximum fraction of edges are properly colored (i.e. their endpoints receive different colors). A random k-coloring properly colors an expected fraction 1-1/k of edges. We prove that given a graph promised to be k-colorable, it is NP-hard to find a k-coloring that properly colors more than a fraction ~1-O(1/k} of edges. Previously, only a hardness factor of 1-O(1/k^2) was known. Our result pins down the correct asymptotic dependence of the approximation factor on k. Along the way, we prove that approximating the Maximum 3-colorable subgraph problem within a factor greater than 32/33 is NP-hard. Using semidefinite programming, it is known that one can do better than a random coloring and properly color a fraction 1-1/k +2 ln k/k^2 of edges in polynomial time. We show that, assuming the 2-to-1 conjecture, it is hard to properly color (using k colors) more than a fraction 1-1/k + O(ln k/ k^2) of edges of a k-colorable graph.Comment: 16 pages, 2 figure

Average-Case Complexity

We survey the average-case complexity of problems in NP. We discuss various notions of good-on-average algorithms, and present completeness results due to Impagliazzo and Levin. Such completeness results establish the fact that if a certain specific (but somewhat artificial) NP problem is easy-on-average with respect to the uniform distribution, then all problems in NP are easy-on-average with respect to all samplable distributions. Applying the theory to natural distributional problems remain an outstanding open question. We review some natural distributional problems whose average-case complexity is of particular interest and that do not yet fit into this theory. A major open question whether the existence of hard-on-average problems in NP can be based on the P$\neq$NP assumption or on related worst-case assumptions. We review negative results showing that certain proof techniques cannot prove such a result. While the relation between worst-case and average-case complexity for general NP problems remains open, there has been progress in understanding the relation between different ``degrees'' of average-case complexity. We discuss some of these ``hardness amplification'' results

Perfect zero knowledge for quantum multiprover interactive proofs

In this work we consider the interplay between multiprover interactive proofs, quantum entanglement, and zero knowledge proofs - notions that are central pillars of complexity theory, quantum information and cryptography. In particular, we study the relationship between the complexity class MIP$^*$, the set of languages decidable by multiprover interactive proofs with quantumly entangled provers, and the class PZKMIP$^*$, which is the set of languages decidable by MIP$^*$ protocols that furthermore possess the perfect zero knowledge property. Our main result is that the two classes are equal, i.e., MIP$^* =$ PZKMIP$^*$. This result provides a quantum analogue of the celebrated result of Ben-Or, Goldwasser, Kilian, and Wigderson (STOC 1988) who show that MIP $=$ PZKMIP (in other words, all classical multiprover interactive protocols can be made zero knowledge). We prove our result by showing that every MIP$^*$ protocol can be efficiently transformed into an equivalent zero knowledge MIP$^*$ protocol in a manner that preserves the completeness-soundness gap. Combining our transformation with previous results by Slofstra (Forum of Mathematics, Pi 2019) and Fitzsimons, Ji, Vidick and Yuen (STOC 2019), we obtain the corollary that all co-recursively enumerable languages (which include undecidable problems as well as all decidable problems) have zero knowledge MIP$^*$ protocols with vanishing promise gap