arXiv subjects
Publications and source records attributed to Guy Kindler.
In this paper, we investigate the validity of the Unique Games Conjecture when the constraint graph is the boolean hypercube. We construct an almost optimal integrality gap instance on the Hypercube for the Goemans-Williamson semidefinite program (SDP) for Max-2-LIN$(\mathbb{Z}_2)$. We conjecture that adding triangle inequalities to the SDP provides a polynomial time algorithm to solve Unique Games on the hypercube.
We prove a quantitative version of the Gibbard-Satterthwaite theorem. We show that a uniformly chosen voter profile for a neutral social choice function f of $q \ge 4$ alternatives and n voters will be manipulable with probability at least $10^{-4} \eps^2 n^{-3} q^{-30}$, where $\eps$ is the minimal statistical distance between f and the family of dictator functions. Our results extend those of FrKaNi:08, which were obtained for the case of 3 alternatives, and imply that the approach of masking manipulations behind computational hardness (as considered in BarthOrline:91, ConitzerS03b, ElkindL05, ProcacciaR06 and ConitzerS06) cannot hide manipulations completely. Our proof is geometric. More specifically it extends the method of canonical paths to show that the measure of the profiles that lie on the interface of 3 or more outcomes is large. To the best of our knowledge our result is the first isoperimetric result to establish interface of more than two bodies.
A Boolean function $f:\{0,1\}^n \to \{0,1\}$ is said to be noise sensitive if inserting a small random error in its argument makes the value of the function almost unpredictable. Benjamini, Kalai and Schramm showed that if the sum of squares of influences in $f$ is close to zero then $f$ must be noise sensitive. We show a quantitative version of this result which does not depend on $n$, and prove that it is tight for certain parameters. Our results hold also for a general product measure $μ_p$ on the discrete cube, as long as $\log 1/p \ll \log n$. We note that in [BKS], a quantitative relation between the sum of squares of the influences and the noise sensitivity was also shown, but only when the sum of squares is bounded by $n^{-c}$ for a constant $c$. Our results require a generalization of a lemma of Talagrand on the Fourier coefficients of monotone Boolean functions. In order to achieve it, we present a considerably shorter proof of Talagrand's lemma, which easily generalizes in various directions, including non-monotone functions.