Extremal sets with forbidden configurations and the independence ratio of geometric hypergraphs

No abstrac

Random restrictions of high-rank tensors and polynomial maps

Motivated by a problem in computational complexity, we consider the behavior of rank functions for tensors and polynomial maps under random coordinate restrictions. We show that, for a broad class of rank functions called natural rank functions, random coordinate restriction to a dense set will typically reduce the rank by at most a constant factor.Comment: 28 page

Geometrical sets with forbidden configurations

Given finite configurations $P_1, \dots , P_n \subset \mathbb {R}^d$ , let us denote by $\mathbf {m}_{\mathbb {R}^d}(P_1, \dots , P_n)$ the maximum density a set $A \subseteq \mathbb {R}^d$ can have without containing congruent copies of any $P_i$ . We will initiate the study of this geometrical parameter, called the independence density of the considered configurations, and give several results we believe are interesting. For instance we show that, under suitable size and nondegeneracy conditions, $\mathbf {m}_{\mathbb {R}^d}(t_1 P_1, t_2 P_2, \dots , t_n P_n)$ progressively ‘untangles’ and tends to $\prod _{i=1}^n \mathbf {m}_{\mathbb {R}^d}(P_i)$ as the ratios $t_{i+1}/t_i$ between consecutive dilation parameters grow large; this shows an exponential decay on the density when forbidding multiple dilates of a given configuration, and gives a common generalization of theorems by Bourgain and by Bukh in geometric Ramsey theory. We also consider the analogous parameter $\mathbf {m}_{\mathbb {S}^d}(P_1, \dots , P_n)$ in the more complicated framework of sets on the unit sphere $\mathbb {S}^d$ , obtaining the corresponding results in this setting

Optimization models for allocation of air strike assets with persistence

This thesis addresses the critical process of assigning strike aircraft to targets once the targets have been identified: How do we optimally employ available aircraft and weapons on the current set of targets, and how can we modify a previously optimized assignment list to face changes in the tactical situation? Our contribution to the strike-planning problem includes (1) a static allocation model in which each aircraft makes at most one sortie during the planning time horizon, (2) a dynamic model in which each aircraft may make more than one sortie during that horizon, and (3) extensions of these models with "persistence incentives," which discourage major plan changes in the results when partial but important changes in the tactical situation necessitate reoptimization. These optimization models are mixed-integer programs that solve in seconds on a personal computer for realistic scenarios with three weapons types, 156 aircraft at seven bases, and 100 potential targets. In a scenario in which two new high-priority targets arise and must be added to an air tasking order with eight original targets, persistence incentives reduce the number of major plan changes from five to two.http://archive.org/details/optimizationmode109454159Major, Brazilian Air ForceApproved for public release; distribution is unlimited

Noisy decoding by shallow circuits with parities: classical and quantum

We consider the problem of decoding corrupted error correcting codes with NC$^0[\oplus]$ circuits in the classical and quantum settings. We show that any such classical circuit can correctly recover only a vanishingly small fraction of messages, if the codewords are sent over a noisy channel with positive error rate. Previously this was known only for linear codes with large dual distance, whereas our result applies to any code. By contrast, we give a simple quantum circuit that correctly decodes the Hadamard code with probability $\Omega(\varepsilon^2)$ even if a $(1/2 - \varepsilon)$-fraction of a codeword is adversarially corrupted. Our classical hardness result is based on an equidistribution phenomenon for multivariate polynomials over a finite field under biased input-distributions. This is proved using a structure-versus-randomness strategy based on a new notion of rank for high-dimensional polynomial maps that may be of independent interest. Our quantum circuit is inspired by a non-local version of the Bernstein-Vazirani problem, a technique to generate ``poor man's cat states'' by Watts et al., and a constant-depth quantum circuit for the OR function by Takahashi and Tani.Comment: 39 pages; This is the full version of an extended abstract that will appear in the proceedings of ITCS'2

A recursive Lov\'asz theta number for simplex-avoiding sets

We recursively extend the Lov\'asz theta number to geometric hypergraphs on the unit sphere and on Euclidean space, obtaining an upper bound for the independence ratio of these hypergraphs. As an application we reprove a result in Euclidean Ramsey theory in the measurable setting, namely that every $k$-simplex is exponentially Ramsey, and we improve existing bounds for the base of the exponential.Comment: 13 pages, 3 figure

A recursive theta body for hypergraphs

The theta body of a graph, introduced by Gr\"otschel, Lov\'asz, and Schrijver in 1986, is a tractable relaxation of the independent-set polytope derived from the Lov\'asz theta number. In this paper, we recursively extend the theta body, and hence the theta number, to hypergraphs. We obtain fundamental properties of this extension and relate it to the high-dimensional Hoffman bound of Filmus, Golubev, and Lifshitz. We discuss two applications: triangle-free graphs and Mantel's theorem, and bounds on the density of triangle-avoiding sets in the Hamming cube.Comment: 23 pages, 2 figure

Raising the roof on the threshold for Szemerédi‘s theorem with random differences

Using recent developments on the theory of locally decodable codes, we prove that the critical size for Szemerédi’s theorem with random differences is bounded from above by N 1 − 2 k + o (1) for length- k progressions. This improves the previous best bounds of N 1 − 1 d k/ 2 e + o (1) for all odd k

Random restrictions of high-rank tensors and polynomial maps

Motivated by a problem in computational complexity, we consider the behavior of rank functions for tensors and polynomial maps under random coordinate restrictions. We show that, for a broad class of rank functions called natural rank functions, random coordinate restriction to a dense set will typically reduce the rank by at most a constant factor