Small Extended Formulation for Knapsack Cover Inequalities from Monotone Circuits

Initially developed for the min-knapsack problem, the knapsack cover inequalities are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield linear programming (LP) relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. In this paper we address this issue and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities. For the min-knapsack cover problem, our main result can be stated formally as follows: for any $\varepsilon >0$, there is a $(1/\varepsilon)^{O(1)}n^{O(\log n)}$-size LP relaxation with an integrality gap of at most $2+\varepsilon$, where $n$ is the number of items. Prior to this work, there was no known relaxation of subexponential size with a constant upper bound on the integrality gap. Our construction is inspired by a connection between extended formulations and monotone circuit complexity via Karchmer-Wigderson games. In particular, our LP is based on $O(\log^2 n)$-depth monotone circuits with fan-in~$2$ for evaluating weighted threshold functions with $n$ inputs, as constructed by Beimel and Weinreb. We believe that a further understanding of this connection may lead to more positive results complementing the numerous lower bounds recently proved for extended formulations.Comment: 21 page

A first step toward higher order chain rules in abelian functor calculus

One of the fundamental tools of undergraduate calculus is the chain rule. The notion of higher order directional derivatives was developed by Huang, Marcantognini, and Young, along with a corresponding higher order chain rule. When Johnson and McCarthy established abelian functor calculus, they proved a chain rule for functors that is analogous to the directional derivative chain rule when $n = 1$. In joint work with Bauer, Johnson, and Riehl, we defined an analogue of the iterated directional derivative and provided an inductive proof of the analogue to the chain rule of Huang et al. This paper consists of the initial investigation of the chain rule found in Bauer et al., which involves a concrete computation of the case when $n=2$. We describe how to obtain the second higher order directional derivative chain rule for abelian functors. This proof is fundamentally different in spirit from the proof given in Bauer et al. as it relies only on properties of cross effects and the linearization of functors

Session 4-2-C: Does Non-problem Gaming Have Any Negative Impact on Gamblers?

Outline Background Literature Review Data and Methodology Analysis and Discussio

Frictioned Micromotion Input for Touch Sensitive Devices

Techniques are provided for securing devices using wiggles, which can include small finger motions made on a touch screen associated with a device. The wiggles can be made by one or multiple fingers and can be made without lifting a finger from the touch screen or sliding the finger along the touch screen, which makes wiggles difficult to observe and therefore difficult to reproduce by another person. Wiggles can include directional motions, rotational motions, and multi-touch motions. For applications in device unlocking, a user can register a “wiggle path” and then later execute an input wiggle path discreetly. The device can compare the input wiggle path against the registered wiggle path. If a match is found, the device can grant access to the user

Anisotropic, multi-carrier transport at the (111) LaAlO3_3/SrTiO3_3 interface

The conducting gas that forms at the interface between LaAlO$_3$ and SrTiO$_3$ has proven to be a fertile playground for a wide variety of physical phenomena. The bulk of previous research has focused on the (001) and (110) crystal orientations. Here we report detailed measurements of the low-temperature electrical properties of (111) LAO/STO interface samples. We find that the low-temperature electrical transport properties are highly anisotropic, in that they differ significantly along two mutually orthogonal crystal orientations at the interface. While anisotropy in the resistivity has been reported in some (001) samples and in (110) samples, the anisotropy in the (111) samples reported here is much stronger, and also manifests itself in the Hall coefficient as well as the capacitance. In addition, the anisotropy is not present at room temperature and at liquid nitrogen temperatures, but only at liquid helium temperatures and below. The anisotropy is accentuated by exposure to ultraviolet light, which disproportionately affects transport along one surface crystal direction. Furthermore, analysis of the low-temperature Hall coefficient and the capacitance as a function of back gate voltage indicates that in addition to electrons, holes contribute to the electrical transport.Comment: 11 pages, 9 figure