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

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

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

Optimal Inapproximability with Universal Factor Graphs

The factor graph of an instance of a constraint satisfaction problem (CSP) is the bipartite graph indicating which variables appear in each constraint. An instance of the CSP is given by the factor graph together with a list of which predicate is applied for each constraint. We establish that many Max-CSPs remains as hard to approximate as in the general case even when the factor graph is fixed (depending only on the size of the instance) and known in advance.Examples of results obtained for this restricted setting are:1. Optimal inapproximability for Max-3-Lin.2. Approximation resistance for predicates supporting pairwise independent subgroups.3. Hardness of the "(2+epsilon)-Sat" problem and other Promise CSPs.The main technical tool used to establish these results is a new way of folding the long code which we call "functional folding"

Dense subset sum may be the hardest

The SUBSET SUM problem asks whether a given set of n positive integers contains a subset of elements that sum up to a given target t. It is an outstanding open question whether the O^*(2^{n/2})-time algorithm for SUBSET SUM by Horowitz and Sahni [J. ACM 1974] can be beaten in the worst-case setting by a "truly faster", O^*(2^{(0.5-delta)*n})-time algorithm, with some constant delta > 0. Continuing an earlier work [STACS 2015], we study SUBSET SUM parameterized by the maximum bin size beta, defined as the largest number of subsets of the n input integers that yield the same sum. For every epsilon > 0 we give a truly faster algorithm for instances with beta = 2^{0.661n}. Consequently, we also obtain a characterization in terms of the popular density parameter n/log_2(t): if all instances of density at least 1.003 admit a truly faster algorithm, then so does every instance. This goes against the current intuition that instances of density 1 are the hardest, and therefore is a step toward answering the open question in the affirmative. Our results stem from a novel combinatorial analysis of mixings of earlier algorithms for SUBSET SUM and a study of an extremal question in additive combinatorics connected to the problem of Uniquely Decodable Code Pairs in information theory

Locally Stable Marriage with Strict Preferences

Abstract. We study two-sided matching markets with locality of in-formation and control. Each male (female) agent has an arbitrary strict preference list over all female (male) agents. In addition, each agent is a node in a fixed network. Agents learn about possible partners dynam-ically based on their current network neighborhood. We consider con-vergence of dynamics to locally stable matchings that are stable with respect to their imposed information structure in the network. While ex-istence of such states is guaranteed, we show that reachability becomes NP-hard to decide. This holds even when the network exists only among one side. In contrast, if only one side has no network and agents re-member a previous match every round, reachability is guaranteed and random dynamics converge with probability 1. We characterize this pos-itive result in various ways. For instance, it holds for random memory and for memory with the most recent partner, but not for memory with the best partner. Also, it is crucial which partition of the agents has memory. Finally, we conclude with results on approximating maximum locally stable matchings.

Privacy with Imperfect Randomness

Abstract. We revisit the impossibility of a variety of cryptographic tasks including privacy and differ-ential privacy with imperfect randomness. For traditional notions of privacy, such as security of encryp-tion, commitment or secret sharing schemes, dramatic impossibility results are known [MP90,DOPS04] for several concrete sources R, including a (seemingly) very “nice and friendly ” Santha-Vazirani (SV) source. Moreover, Bosley and Dodis [BD07] gave strong evidence that many traditional privacy tasks (e.g., encryption) inherently require an “extractable ” source of randomness. The common interpretation of these negative results is that privacy is impossible even with “very structured ” imperfect sources. Somewhat surprisingly, Dodis et al. [DLMV12] put a slight dent in this belief, by showing that non-trivial differential privacy is possible with the SV sources. This suggested a qualitative gap between traditional and differential privacy, and left open the question if differential privacy is possible with more realistic (i.e., less structured) sources than the SV sources. Motivated by solving this question, we introduce a new, modular framework for showing strong im-possibility results for (either traditional or differential) privacy under a general imperfect source R. In particular, we introduce natural and easy-to-understand concepts of expressiveness and separability of a given imperfect source R, and show the following results

Conference — TCC 2014. This is the full version. Continuous Non-malleable Codes

Non-malleable codes are a natural relaxation of error correcting/detecting codes that have useful applications in the context of tamper resilient cryptography. Informally, a code is non-malleable if an adversary trying to tamper with an encoding of a given message can only leave it unchanged or modify it to the encoding of a completely unrelated value. This paper introduces an extension of the standard non-malleability security notion – so-called continuous non-malleability – where we allow the adversary to tamper continuously with an encoding. This is in contrast to the standard notion of non-malleable codes where the adversary only is allowed to tamper a single time with an encoding. We show how to construct continuous non-malleable codes in the common split-state model where an encoding consist of two parts and the tampering can be arbitrary but has to be independent with both parts. Our main contributions are outlined below: 1. We propose a new uniqueness requirement of split-state codes which states that it is computationally hard to find two codewords C = (X0, X1) and C ′ = (X0, X ′ 1) such that both codwords are valid, but X0 is the same in both C and C ′. A simple attack shows that uniqueness is necessary to achieve continuous non-malleability in the split-state model. Moreover