A PCP Characterization of AM

We introduce a 2-round stochastic constraint-satisfaction problem, and show that its approximation version is complete for (the promise version of) the complexity class AM. This gives a `PCP characterization' of AM analogous to the PCP Theorem for NP. Similar characterizations have been given for higher levels of the Polynomial Hierarchy, and for PSPACE; however, we suggest that the result for AM might be of particular significance for attempts to derandomize this class. To test this notion, we pose some `Randomized Optimization Hypotheses' related to our stochastic CSPs that (in light of our result) would imply collapse results for AM. Unfortunately, the hypotheses appear over-strong, and we present evidence against them. In the process we show that, if some language in NP is hard-on-average against circuits of size 2^{Omega(n)}, then there exist hard-on-average optimization problems of a particularly elegant form. All our proofs use a powerful form of PCPs known as Probabilistically Checkable Proofs of Proximity, and demonstrate their versatility. We also use known results on randomness-efficient soundness- and hardness-amplification. In particular, we make essential use of the Impagliazzo-Wigderson generator; our analysis relies on a recent Chernoff-type theorem for expander walks.Comment: 18 page

Spatial Mixing of Coloring Random Graphs

We study the strong spatial mixing (decay of correlation) property of proper $q$-colorings of random graph $G(n, d/n)$ with a fixed $d$. The strong spatial mixing of coloring and related models have been extensively studied on graphs with bounded maximum degree. However, for typical classes of graphs with bounded average degree, such as $G(n, d/n)$, an easy counterexample shows that colorings do not exhibit strong spatial mixing with high probability. Nevertheless, we show that for $q\ge\alpha d+\beta$ with $\alpha>2$ and sufficiently large $\beta=O(1)$, with high probability proper $q$-colorings of random graph $G(n, d/n)$ exhibit strong spatial mixing with respect to an arbitrarily fixed vertex. This is the first strong spatial mixing result for colorings of graphs with unbounded maximum degree. Our analysis of strong spatial mixing establishes a block-wise correlation decay instead of the standard point-wise decay, which may be of interest by itself, especially for graphs with unbounded degree

Cake Cutting Algorithms for Piecewise Constant and Piecewise Uniform Valuations

Cake cutting is one of the most fundamental settings in fair division and mechanism design without money. In this paper, we consider different levels of three fundamental goals in cake cutting: fairness, Pareto optimality, and strategyproofness. In particular, we present robust versions of envy-freeness and proportionality that are not only stronger than their standard counter-parts but also have less information requirements. We then focus on cake cutting with piecewise constant valuations and present three desirable algorithms: CCEA (Controlled Cake Eating Algorithm), MEA (Market Equilibrium Algorithm) and CSD (Constrained Serial Dictatorship). CCEA is polynomial-time, robust envy-free, and non-wasteful. It relies on parametric network flows and recent generalizations of the probabilistic serial algorithm. For the subdomain of piecewise uniform valuations, we show that it is also group-strategyproof. Then, we show that there exists an algorithm (MEA) that is polynomial-time, envy-free, proportional, and Pareto optimal. MEA is based on computing a market-based equilibrium via a convex program and relies on the results of Reijnierse and Potters [24] and Devanur et al. [15]. Moreover, we show that MEA and CCEA are equivalent to mechanism 1 of Chen et. al. [12] for piecewise uniform valuations. We then present an algorithm CSD and a way to implement it via randomization that satisfies strategyproofness in expectation, robust proportionality, and unanimity for piecewise constant valuations. For the case of two agents, it is robust envy-free, robust proportional, strategyproof, and polynomial-time. Many of our results extend to more general settings in cake cutting that allow for variable claims and initial endowments. We also show a few impossibility results to complement our algorithms.Comment: 39 page

Quantum Algorithms for Learning and Testing Juntas

In this article we develop quantum algorithms for learning and testing juntas, i.e. Boolean functions which depend only on an unknown set of k out of n input variables. Our aim is to develop efficient algorithms: - whose sample complexity has no dependence on n, the dimension of the domain the Boolean functions are defined over; - with no access to any classical or quantum membership ("black-box") queries. Instead, our algorithms use only classical examples generated uniformly at random and fixed quantum superpositions of such classical examples; - which require only a few quantum examples but possibly many classical random examples (which are considered quite "cheap" relative to quantum examples). Our quantum algorithms are based on a subroutine FS which enables sampling according to the Fourier spectrum of f; the FS subroutine was used in earlier work of Bshouty and Jackson on quantum learning. Our results are as follows: - We give an algorithm for testing k-juntas to accuracy $\epsilon$ that uses $O(k/\epsilon)$ quantum examples. This improves on the number of examples used by the best known classical algorithm. - We establish the following lower bound: any FS-based k-junta testing algorithm requires $\Omega(\sqrt{k})$ queries. - We give an algorithm for learning $k$-juntas to accuracy $\epsilon$ that uses $O(\epsilon^{-1} k\log k)$ quantum examples and $O(2^k \log(1/\epsilon))$ random examples. We show that this learning algorithms is close to optimal by giving a related lower bound.Comment: 15 pages, 1 figure. Uses synttree package. To appear in Quantum Information Processin

Constrained Non-Monotone Submodular Maximization: Offline and Secretary Algorithms

Constrained submodular maximization problems have long been studied, with near-optimal results known under a variety of constraints when the submodular function is monotone. The case of non-monotone submodular maximization is less understood: the first approximation algorithms even for the unconstrainted setting were given by Feige et al. (FOCS '07). More recently, Lee et al. (STOC '09, APPROX '09) show how to approximately maximize non-monotone submodular functions when the constraints are given by the intersection of p matroid constraints; their algorithm is based on local-search procedures that consider p-swaps, and hence the running time may be n^Omega(p), implying their algorithm is polynomial-time only for constantly many matroids. In this paper, we give algorithms that work for p-independence systems (which generalize constraints given by the intersection of p matroids), where the running time is poly(n,p). Our algorithm essentially reduces the non-monotone maximization problem to multiple runs of the greedy algorithm previously used in the monotone case. Our idea of using existing algorithms for monotone functions to solve the non-monotone case also works for maximizing a submodular function with respect to a knapsack constraint: we get a simple greedy-based constant-factor approximation for this problem. With these simpler algorithms, we are able to adapt our approach to constrained non-monotone submodular maximization to the (online) secretary setting, where elements arrive one at a time in random order, and the algorithm must make irrevocable decisions about whether or not to select each element as it arrives. We give constant approximations in this secretary setting when the algorithm is constrained subject to a uniform matroid or a partition matroid, and give an O(log k) approximation when it is constrained by a general matroid of rank k.Comment: In the Proceedings of WINE 201

10-year follow-up of patients with rheumatoid arthritis and secondary Sjogren's syndrome or sicca symptoms in daily clinical practice

Objective. To evaluate the presence of sicca symptoms and secondary Sjogren's syndrome (SS) and the association with clinical characteristics, functional tests and patient-reported outcomes in patients with rheumatoid arthritis (RA) at baseline and after 10 years of follow-up. Methods. A cohort of RA patients was evaluated in 2008 and re-evaluated in 2018 with respect to sicca symptoms, presence of secondary SS according to AECG classification criteria, disease activity of RA and patient-reported outcomes. Patient characteristics were compared between the RA-non-sicca, RA-sicca and RA-SS groups. Results. Of the original 2008 cohort of 96 RA patients, 32 (33%) had sicca symptoms and 6 (6.3%) secondary SS. Of the 36 patients who agreed to be reevaluated in 2018, 6 (17%) had sicca symptoms and 2 (6%) developed secondary SS. In the majority of patients, sicca symptoms were reversible while the functional tests of salivary and lacrimal glands significantly decreased. 67% of RA-sicca patients had no sicca complaints at the second screening, while only two RA-sicca patients developed secondary SS. RA-SS patients and, to a slightly lesser extent, RA-sicca patients had significantly higher RA disease activity (DAS-28), lower lacrimal (Schirmer's test) and salivary gland function, more limitations in daily activities (HAQ), worse health-related quality of life (RAND-36), more fatigue (MFI) and more patient symptoms (ESSPRI) compared to RA-non-sicca patients. Conclusion. Secondary SS was found in a minor subset of the RA patients. Sicca symptoms of the eyes or mouth were more frequent, but their presence varied over time. Higher RA disease activity was associated with SS and sicca symptoms. These patients had lower gland function and worse patient-reported outcomes

10-year follow-up of patients with rheumatoid arthritis and secondary Sjogren's syndrome or sicca symptoms in daily clinical practice

Objective. To evaluate the presence of sicca symptoms and secondary Sjogren's syndrome (SS) and the association with clinical characteristics, functional tests and patient-reported outcomes in patients with rheumatoid arthritis (RA) at baseline and after 10 years of follow-up. Methods. A cohort of RA patients was evaluated in 2008 and re-evaluated in 2018 with respect to sicca symptoms, presence of secondary SS according to AECG classification criteria, disease activity of RA and patient-reported outcomes. Patient characteristics were compared between the RA-non-sicca, RA-sicca and RA-SS groups. Results. Of the original 2008 cohort of 96 RA patients, 32 (33%) had sicca symptoms and 6 (6.3%) secondary SS. Of the 36 patients who agreed to be reevaluated in 2018, 6 (17%) had sicca symptoms and 2 (6%) developed secondary SS. In the majority of patients, sicca symptoms were reversible while the functional tests of salivary and lacrimal glands significantly decreased. 67% of RA-sicca patients had no sicca complaints at the second screening, while only two RA-sicca patients developed secondary SS. RA-SS patients and, to a slightly lesser extent, RA-sicca patients had significantly higher RA disease activity (DAS-28), lower lacrimal (Schirmer's test) and salivary gland function, more limitations in daily activities (HAQ), worse health-related quality of life (RAND-36), more fatigue (MFI) and more patient symptoms (ESSPRI) compared to RA-non-sicca patients. Conclusion. Secondary SS was found in a minor subset of the RA patients. Sicca symptoms of the eyes or mouth were more frequent, but their presence varied over time. Higher RA disease activity was associated with SS and sicca symptoms. These patients had lower gland function and worse patient-reported outcomes

The theoretical capacity of the Parity Source Coder

The Parity Source Coder is a protocol for data compression which is based on a set of parity checks organized in a sparse random network. We consider here the case of memoryless unbiased binary sources. We show that the theoretical capacity saturate the Shannon limit at large K. We also find that the first corrections to the leading behavior are exponentially small, so that the behavior at finite K is very close to the optimal one.Comment: Added references, minor change