The (low soundness) linearity testing problem for the middle slice of the Boolean cube is as follows. Let$\varepsilon > 0$and$f$be a function on the middle slice on the Boolean cube, such that when choosing a uniformly random quadruple$(x,y,\ z,x\oplus y\oplus z)$of vectors of$2n$bits with exactly$n$ones, the probability that$f(x\oplus y\oplus z)=f(x)\oplus f(y)\oplus f(z)$is at least$1/2+\epsilon$. The linearity testing problem, posed by [6], asks whether there must be an actual linear function that agrees with$f$on$1/2+\epsilon^{\prime}$fraction of the inputs, where$\varepsilon^{\prime}=\in^{\prime}(\in) > 0$. We solve this problem, showing that$f$must indeed be correlated with a linear function. To do so, we prove a dense model theorem for the middle slice of the Boolean hypercube for Gowers uniformity norms. Specifically, we show that for every$k\in \mathbb{N}$, the normalized indicator function of the middle slice of the Boolean hypercube$\{0,1\}^{2n}$is close in Gowers norm to the normalized indicator function of the union of all slices with weight$t=n(\text{mod}\ 2^{k-1})$. Using our techniques we also give a more general ‘low degree test’ and a biased rank theorem for the slice.

Abstract. Lionel Levine’s hat challenge has [Formula: see text] players, each with a (very large or infinite) stack of hats on their head, each hat independently colored at random black or white. The players are allowed to coordinate before the random colors are chosen, but not after. Each player sees all hats except for those on her own head. They then proceed to simultaneously try and each pick a black hat from their respective stacks. They are proclaimed successful only if they are all correct. Levine’s conjecture is that the success probability tends to zero when the number of players grows. We prove that this success probability is strictly decreasing in the number of players, and present some connections to problems in graph theory: relating the size of the largest independent set in a graph and in a random induced subgraph of it, and bounding the size of a set of vertices intersecting every maximum-size independent set in a graph.

Bidding games are extensive form games, where in each turn players bid in order to determine who will play next. Zero-sum bidding games (also known as Richman games) have been extensively studied, focusing on the fraction of the initial budget that can guaranty the victory of each player [Lazarus et al.'99, Develin & Payne '10].

We introduce a sealed bid auction of a single item in which the winner is chosen at random among the highest k bidders according to a fixed probability distribution, and the price for the chosen winner is the Vickrey-Clarke-Groves price. We call such an auction a cascade auction. Our analysis suggests that this type of auction may give higher revenues compared to second price auction in cases of collusion.

The Gibbard–Satterthwaite theorem states that every nondictatorial election rule among at least three alternatives can be strategically manipulated. We prove a quantitative version of the Gibbard–Satterthwaite theorem: a random manipulation by a single random voter will succeed with a nonnegligible probability for any election rule among three alternatives that is far from being a dictatorship and from having only two alternatives in its range.

The Gibbard-Satterthwaite theorem states that every non-trivial voting method among at least 3 alternatives can be strategically manipulated. We prove a quantitative version of the Gibbard-Satterthwaite theorem: a random manipulation by a single random voter will succeed with non-negligible probability for every neutral voting method among 3 alternatives that is far from being a dictatorship.

The distinct distribution of a binary code C is the sequence (B/sub i/)/sub i=0//sup n/ defined as follows: let B/sub i/(w) be the number of codewords at distance i from the codeword w, and let B/sub i/ be the average of B/sub i/(w) over all w in C. In this correspondence we study the distance distribution for codes of length n and minimal distance /spl delta/n, with /spl delta/>0 fixed and n/spl rarr//spl infin/. Our main aim is to relate the size of the code with the distribution of distances near the minimal distance.>

The authors prove a lower bound of Omega (log n/log log n) on the competitive ratio of any (deterministic or randomised) distributed algorithm for solving the mobile user problem on certain networks of n processors. The lower bound holds for various networks, including the hypercube, any network with sufficiently large girth, and any highly expanding graph. A similar Omega (log n/log log n) lower bound is proved for the competitive ratio of the maximum job delay of any distributed algorithm for solving a distributed scheduling problem on any of these networks. The proofs combine combinatorial techniques with tools from linear algebra and harmonic analysis and apply, in particular, a generalization of the vertex isoperimetric problem on the hypercube, which may be of independent interest.>

Article A subexponential randomized simplex algorithm (extended abstract) Share on Author: Gil Kalai Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, IsraelView Profile Authors Info & Claims STOC '92: Proceedings of the twenty-fourth annual ACM symposium on Theory of ComputingJuly 1992 Pages 475–482https://doi.org/10.1145/129712.129759Online:01 July 1992Publication History 80citation890DownloadsMetricsTotal Citations80Total Downloads890Last 12 Months29Last 6 weeks4 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

Methods from harmonic analysis are used to prove some general theorems on Boolean functions. These connections with harmonic analysis viewed by the authors are very promising; besides the results on Boolean functions they enable them to prove theorems on the rapid mixing of the random walk on the cube and in the extremal theory of finite sets.>