We derive a new upper bound on the diameter of a polyhedron P = {x ∈ R n : Ax b}, where A ∈ Z m×n . The bound is polynomial in n and the largest absolute value of a sub-determinant of A, denoted by ∆. More precisely, we show that the diameter of P is bounded by O ∆ 2 n 4 log n∆ . If P is bounded, then we show that the diameter of P is at most O ∆ 2 n 3.5 log n∆ .For the special case in which A is a totally unimodular matrix, the bounds are O n 4 log n and O n 3.5 log n respectively. This improves over the previous best bound of O(m 16 n 3 (log mn) 3 ) due to Dyer and Frieze [DF94]. * An extended abstract of this paper was presented at the 28-th annual ACM symposium on Computational Geometry (SOCG 12) † LIX, École Polytechnique, Palaiseau and IBM,

Improving on the Voronoi cell based techniques of [28, 24], we give a Las Vegas O(2 n ) expected time and space algorithm for CVPP (the preprocessing version of the Closest Vector Problem, CVP). This improves on the O(4 n ) deterministic runtime of the Micciancio Voulgaris algorithm [24] (henceforth MV) for CVPP 1 at the cost of a polynomial amount of randomness (which only affects runtime, not correctness).As in MV, our algorithm proceeds by computing a short path on the Voronoi graph of the lattice, where lattice points are adjacent if their Voronoi cells share a common facet, from the origin to a closest lattice vector. Our main technical contribution is a randomized procedure that, given the Voronoi relevant vectors of a lattice -the lattice vectors inducing facets of the Voronoi cell -as preprocessing, and any "close enough" lattice point to the target, computes a path to a closest lattice vector of expected polynomial size. This improves on the O(2 n ) path length given by the MV algorithm. Furthermore, as in MV, each edge of the path can be computed using a single iteration over the Voronoi relevant vectors.As a byproduct of our work, we also give an optimal relationship between geometric and path distance on the Voronoi graph, which we believe to be of independent interest.

Background Older individuals receiving home assistance are at high risk for emergency visits and unplanned hospitalization. Anticipating their health difficulties could prevent these events. This study investigated the effectiveness of an at-home monitoring method using social workers’ observations to predict risk for 7- and 14-day emergency department (ED) visits. Methods This was a prospective cohort study of persons ≥75 years, living at home and receiving assistance from home care aides (HCA) at 6 French facilities. After each home visit, HCAs reported on participants’ functional status using a smartphone application that recorded 27 functional items about each participant (e.g., ability to stand, move, eat, mood, loneliness). We recorded ED visits. Finally, we used machine learning techniques (i.e., leveraging random forest predictors) to develop a 7- and 14-day predictive algorithm for the risk of ED visit. Results The study included 301 participants, and the HCA made 9,987 observations. Over the mean 10-month follow-up, 97 participants (32%) had at least one ED visit. Modeling techniques identified 9 contributory factors from the longitudinal records of the HCA and developed a predictive algorithm for the risk of ED visit. The predictive performance (i.e., the area under the ROC curve) was 0.70 at 7 days and 0.67 at 14 days. Interpretation For frail elders receiving in-home care, information on functional status collected by HCA helps predict the risk of ED visits 7 to 14 days in advance. A survey system for real-time identification of risks could be developed using this exploratory work.