How to Play Unique Games on Expanders

In this note we improve a recent result by Arora, Khot, Kolla, Steurer, Tulsiani, and Vishnoi on solving the Unique Games problem on expanders. Given a $(1-\varepsilon)$-satisfiable instance of Unique Games with the constraint graph $G$, our algorithm finds an assignment satisfying at least a $1- C \varepsilon/h_G$ fraction of all constraints if $\varepsilon < c \lambda_G$ where $h_G$ is the edge expansion of $G$, $\lambda_G$ is the second smallest eigenvalue of the Laplacian of $G$, and $C$ and $c$ are some absolute constants

On the NP-Hardness of Approximating Ordering Constraint Satisfaction Problems

We show improved NP-hardness of approximating Ordering Constraint Satisfaction Problems (OCSPs). For the two most well-studied OCSPs, Maximum Acyclic Subgraph and Maximum Betweenness, we prove inapproximability of $14/15+\epsilon$ and $1/2+\epsilon$. An OCSP is said to be approximation resistant if it is hard to approximate better than taking a uniformly random ordering. We prove that the Maximum Non-Betweenness Problem is approximation resistant and that there are width-$m$ approximation-resistant OCSPs accepting only a fraction $1 / (m/2)!$ of assignments. These results provide the first examples of approximation-resistant OCSPs subject only to P $\neq$ \NP

Lower Bounds for the Graph Homomorphism Problem

The graph homomorphism problem (HOM) asks whether the vertices of a given $n$-vertex graph $G$ can be mapped to the vertices of a given $h$-vertex graph $H$ such that each edge of $G$ is mapped to an edge of $H$. The problem generalizes the graph coloring problem and at the same time can be viewed as a special case of the $2$-CSP problem. In this paper, we prove several lower bound for HOM under the Exponential Time Hypothesis (ETH) assumption. The main result is a lower bound $2^{\Omega\left( \frac{n \log h}{\log \log h}\right)}$. This rules out the existence of a single-exponential algorithm and shows that the trivial upper bound $2^{{\mathcal O}(n\log{h})}$ is almost asymptotically tight. We also investigate what properties of graphs $G$ and $H$ make it difficult to solve HOM$(G,H)$. An easy observation is that an ${\mathcal O}(h^n)$ upper bound can be improved to ${\mathcal O}(h^{\operatorname{vc}(G)})$ where $\operatorname{vc}(G)$ is the minimum size of a vertex cover of $G$. The second lower bound $h^{\Omega(\operatorname{vc}(G))}$ shows that the upper bound is asymptotically tight. As to the properties of the "right-hand side" graph $H$, it is known that HOM$(G,H)$ can be solved in time $(f(\Delta(H)))^n$ and $(f(\operatorname{tw}(H)))^n$ where $\Delta(H)$ is the maximum degree of $H$ and $\operatorname{tw}(H)$ is the treewidth of $H$. This gives single-exponential algorithms for graphs of bounded maximum degree or bounded treewidth. Since the chromatic number $\chi(H)$ does not exceed $\operatorname{tw}(H)$ and $\Delta(H)+1$, it is natural to ask whether similar upper bounds with respect to $\chi(H)$ can be obtained. We provide a negative answer to this question by establishing a lower bound $(f(\chi(H)))^n$ for any function $f$. We also observe that similar lower bounds can be obtained for locally injective homomorphisms.Comment: 19 page

Algebraic Theory of Promise Constraint Satisfaction Problems, First Steps

What makes a computational problem easy (e.g., in P, that is, solvable in polynomial time) or hard (e.g., NP-hard)? This fundamental question now has a satisfactory answer for a quite broad class of computational problems, so called fixed-template constraint satisfaction problems (CSPs) -- it has turned out that their complexity is captured by a certain specific form of symmetry. This paper explains an extension of this theory to a much broader class of computational problems, the promise CSPs, which includes relaxed versions of CSPs such as the problem of finding a 137-coloring of a 3-colorable graph

Inapproximability Results for Approximate Nash Equilibria.

We study the problem of finding approximate Nash equilibria that satisfy certain conditions, such as providing good social welfare. In particular, we study the problem $\epsilon$-NE $\delta$-SW: find an $\epsilon$-approximate Nash equilibrium ($\epsilon$-NE) that is within $\delta$ of the best social welfare achievable by an $\epsilon$-NE. Our main result is that, if the exponential-time hypothesis (ETH) is true, then solving $\left(\frac{1}{8} - \mathrm{O}(\delta)\right)$-NE $\mathrm{O}(\delta)$-SW for an $n\times n$ bimatrix game requires $n^{\mathrm{\widetilde \Omega}(\log n)}$ time. Building on this result, we show similar conditional running time lower bounds on a number of decision problems for approximate Nash equilibria that do not involve social welfare, including maximizing or minimizing a certain player's payoff, or finding approximate equilibria contained in a given pair of supports. We show quasi-polynomial lower bounds for these problems assuming that ETH holds, where these lower bounds apply to $\epsilon$-Nash equilibria for all $\epsilon < \frac{1}{8}$. The hardness of these other decision problems has so far only been studied in the context of exact equilibria.Comment: A short (14-page) version of this paper appeared at WINE 2016. Compared to that conference version, this new version improves the conditional lower bounds, which now rely on ETH rather than RETH (Randomized ETH

Correlated-Source Extractors and Cryptography with Correlated-Random Tapes

In this paper, we consider the setting where a party uses correlated random tapes across multiple executions of a cryptographic algorithm. We ask if the security properties could still be preserved in such a setting. As examples, we introduce the notion of correlated-tape zero knowledge, and, correlated-tape multi-party computation, where, the zero-knowledge property, and, the ideal/real model security must still be preserved even if a party uses correlated random tapes in multiple executions. Our constructions are based on a new type of randomness extractor which we call correlated-source extractors. Correlated-source extractors can be seen as a dual of non-malleable extractors, and, allow an adversary to choose several tampering functions which are applied to the randomness source. Correlated-source extractors guarantee that even given the output of the extractor on the tampered sources, the output on the original source is still uniformly random. Given (seeded) correlated-source extractors, and, resettably-secure computation protocols, we show how to directly get a positive result for both correlated-tape zero-knowledge and correlated-tape multi-party computation in the CRS model. This is tight considering the known impossibility results on cryptography with imperfect randomness. Our main technical contribution is an explicit construction of a correlated-source extractor where the length of the seed is independent of the number of tamperings. Additionally, we also provide a (non-explicit) existential result for correlated source extractors with almost optimal parameters

Computing Constrained Approximate Equilibria in Polymatrix Games

This paper is about computing constrained approximate Nash equilibria in polymatrix games, which are succinctly represented many-player games defined by an interaction graph between the players. In a recent breakthrough, Rubinstein showed that there exists a small constant $\epsilon$, such that it is PPAD-complete to find an (unconstrained) $\epsilon$-Nash equilibrium of a polymatrix game. In the first part of the paper, we show that is NP-hard to decide if a polymatrix game has a constrained approximate equilibrium for 9 natural constraints and any non-trivial approximation guarantee. These results hold even for planar bipartite polymatrix games with degree 3 and at most 7 strategies per player, and all non-trivial approximation guarantees. These results stand in contrast to similar results for bimatrix games, which obviously need a non-constant number of actions, and which rely on stronger complexity-theoretic conjectures such as the exponential time hypothesis. In the second part, we provide a deterministic QPTAS for interaction graphs with bounded treewidth and with logarithmically many actions per player that can compute constrained approximate equilibria for a wide family of constraints that cover many of the constraints dealt with in the first part

Blockwise pp-Tampering Attacks on Cryptographic Primitives, Extractors, and Learners

Austrin, Chung, Mahmoody, Pass and Seth (Crypto\u2714) studied the notion of bitwise $p$-tampering attacks over randomized algorithms in which an efficient `virus\u27 gets to control each bit of the randomness with independent probability $p$ in an online way. The work of Austrin et al. showed how to break certain `privacy primitives\u27 (e.g., encryption, commitments, etc.) through bitwise $p$-tampering, by giving a bitwise $p$-tampering biasing attack for increasing the average $E[f(U_n)]$ of any efficient function $f \colon \{0,1\}^n \to [-1,+1]$ by $\Omega(p \cdot Var[f(U_n)])$ where $Var[f(U_n)]$ is the variance of $f(U_n)$. In this work, we revisit and extend the bitwise tampering model of Austrin et al. to blockwise setting, where blocks of randomness becomes tamperable with independent probability $p$. Our main result is an efficient blockwise $p$-tampering attack to bias the average $E[f(X)]$ of any efficient function $f$ mapping arbitrary $X$ to $[-1,+1]$ by $\Omega(p \cdot Var[f(X)])$ regardless of how $X$ is partitioned into individually tamperable blocks $X=(X_1,\dots,X_n)$. Relying on previous works, our main biasing attack immediately implies efficient attacks against the privacy primitives as well as seedless multi-source extractors, in a model where the attacker gets to tamper with each block (or source) of the randomness with independent probability $p$. Further, we show how to increase the classification error of deterministic learners in the so called `targeted poisoning\u27 attack model under Valiant\u27s adversarial noise. In this model, an attacker has a `target\u27 test data $d$ in mind and wishes to increase the error of classifying $d$ while she gets to tamper with each training example with independent probability $p$ an in an online way

Complete tree subset difference broadcast encryption scheme and its analysis

The subset difference (SD) method proposed by Naor, Naor and Lotspiech is the most popular broadcast encryption (BE) scheme. It is suitable for real-time applications like Pay-TV and has been suggested for use by the AACS standard for digital rights management in Blu-Ray and HD-DVD discs. The SD method assumes the number of users to be a power of two. We propose the complete tree subset difference (CTSD) method that allows the system to support an arbitrary number of users. In particular, it subsumes the SD method and all results proved for the CTSD method also hold for the SD method. Recurrences are obtained for the CTSD scheme to count the number, N(n, r, h), of possible ways r users in the system of n users can be revoked to result in a transmission overhead or header length of h. The recurrences lead to a polynomial time dynamic programming algorithm for computing N(n, r, h). Further, they provide bounds on the maximum possible header length. A probabilistic analysis is performed to obtain an O(r log n) time algorithm to compute the expected header length in the CTSD scheme. Further, for the SD scheme we obtain an explicit limiting upper bound on the expected header length

Oksidacijski stres u lakirera izloženih niskim razinama olova

Lead toxicity is a public health problem particularly to the children and to occupationally exposed adults. Evidence is mounting successively regarding the adverse health effects of lead at low levels. This study was undertaken to assess the antioxidant status of lead-exposed residential and commercial painters of Lucknow city in Uttar Pradesh, India. Thirty-five painters aged 20 to 50 years who had blood lead levels ≤400 µg L-1 were selected for the study from a population of 56 male painters initially screened for blood lead. The control group included an equal number of subjects of the same age group without any occupational exposure to lead. We studied the association between low lead level exposure and antioxidant status and found that blood lead levels in painters were approximately seven times as high as in controls [(219.2 ± 61.9) µg L-1 vs. (30.6±10.1) µg L-1, respectively]. Among the biomarkers of lead toxicity a significant decrease in the level of delta-aminolevulinic acid dehydratase [(9.13±4.62) UL-1 vs. (39.38±5.05) UL-1] and an increase in the level of zinc protoporphyrin [(187.9±49.8) µg L-1 vs. (26.4±5.5) µg L-1] were observed in painters compared to controls. Among antioxidant enzymes, painters showed a significant decrease in catalase [(56.77±11.11) UL-1 vs. (230.30±42.55) UL-1] and superoxide dismutase [(0.64±0.19) UL-1 vs. (2.68±0.62) UL-1] compared to controls. Lipid peroxidation was monitored by measuring thiobarbituric acid reactive substances (TBARS) that were expressed in terms of malondialdehyde (MDA) equivalents. Concentration of MDA in plasma was higher in painters than in controls [(7.48±1.31) nmol mL-1 vs. (3.08±0.56) nmol mL-1]. Significant changes were also observed in reduced and oxidised glutathione levels. The strong association between blood lead levels and oxidative stress markers in this population suggests that oxidative stress should be considered in the pathogenesis of lead-related diseases among people with low level environmental exposure to lead.Toksičnost olova javnozdravstveni je problem, napose u djece i odraslih osoba koje su im izložene profesionalno. Sve je više dokaza o štetnom djelovanju olova pri niskim razinama. Svrha je ovog ispitivanja bila procijeniti antioksidacijski status u lakirera iz grada Lucknowa u indijskoj pokrajini Uttar Pradesh. Iz skupine od 56 muškaraca lakirera u dobi od 20 do 50 godina s pozitivnim početnim nalazima olova u krvi, za ispitivanje su izabrana 35-orica čije su razine iznosile ≤400 µg L-1. Izabran je i jednaki broj kontrolnih ispitanika iz iste dobne skupine, koji nisu bili profesionalno izloženi olovu. Ispitana je povezanost izme|u izloženosti niskim razinama olova i antioksidacijskoga stanja te je utvrđeno da su razine olova u krvi lakirera [(219,2±61,9) µg L-1] bile oko sedam puta više negoli u kontrolnih ispitanika [(30,6±10,1) µg L-1]. Od biopokazatelja toksičnosti olova u lakirera je zamijećen značajan pad razina delta- ALAD [(9,13±4,62) UL-1 prema (39,38±5,05) UL-1] te rast razina cinkova protoporfirina [(187,9±49,8) µg L-1 prema (26,4±5,5) µg L-1] u odnosu na kontrolne ispitanike. Od antioksidacijskih enzima u lakirera je značajno pala aktivnost katalaze [(56,77±11,11) UL-1 prema (230,30±42,55) UL-1] i superoksid dismutaze [(0,64±0.19) UL-1 prema (2,68±0,62) UL-1] u odnosu na kontrolu, dok je produkt lipidne peroksidacije u plazmi (izv. thiobarbituric acid reactive substances, TBARS) izražen kao koncentracija malondialdehida (MDA) porastao [(7,48±1,31) nmol mL-1 prema (3,08±0,56) nmol mL-1]. Značajne su promjene također zamijećene u smanjenim razinama glutationa i njihovoj oksidaciji. Snažna povezanost razina olova u krvi s pokazateljima oksidacijskoga stresa upućuje na to da u osoba s niskom razinom izloženosti olovu iz okoliša kod razmatranja patogeneze bolesti povezane s olovom u obzir valja uzeti oksidacijski stres