Simplifying Activity-On-Edge Graphs

We formalize the simplification of activity-on-edge graphs used for visualizing project schedules, where the vertices of the graphs represent project milestones, and the edges represent either tasks of the project or timing constraints between milestones. In this framework, a timeline of the project can be constructed as a leveled drawing of the graph, where the levels of the vertices represent the time at which each milestone is scheduled to happen. We focus on the following problem: given an activity-on-edge graph representing a project, find an equivalent activity-on-edge graph (one with the same critical paths) that has the minimum possible number of milestone vertices among all equivalent activity-on-edge graphs. We provide a polynomial-time algorithm for solving this graph minimization problem

Improved Distributed Algorithms for Random Colorings

Markov Chain Monte Carlo (MCMC) algorithms are a widely-used algorithmic tool for sampling from high-dimensional distributions, a notable example is the equilibirum distribution of graphical models. The Glauber dynamics, also known as the Gibbs sampler, is the simplest example of an MCMC algorithm; the transitions of the chain update the configuration at a randomly chosen coordinate at each step. Several works have studied distributed versions of the Glauber dynamics and we extend these efforts to a more general family of Markov chains. An important combinatorial problem in the study of MCMC algorithms is random colorings. Given a graph $G$ of maximum degree $\Delta$ and an integer $k\geq\Delta+1$, the goal is to generate a random proper vertex $k$-coloring of $G$. Jerrum (1995) proved that the Glauber dynamics has $O(n\log{n})$ mixing time when $k>2\Delta$. Fischer and Ghaffari (2018), and independently Feng, Hayes, and Yin (2018), presented a parallel and distributed version of the Glauber dynamics which converges in $O(\log{n})$ rounds for $k>(2+\varepsilon)\Delta$ for any $\varepsilon>0$. We improve this result to $k>(11/6-\delta)\Delta$ for a fixed $\delta>0$. This matches the state of the art for randomly sampling colorings of general graphs in the sequential setting. Whereas previous works focused on distributed variants of the Glauber dynamics, our work presents a parallel and distributed version of the more general flip dynamics presented by Vigoda (2000) (and refined by Chen, Delcourt, Moitra, Perarnau, and Postle (2019)), which recolors local maximal two-colored components in each step.Comment: 25 pages, 2 figure

On the Treewidth of Hanoi Graphs

The objective of the well-known Towers of Hanoi puzzle is to move a set of disks one at a time from one of a set of pegs to another, while keeping the disks sorted on each peg. We propose an adversarial variation in which the first player forbids a set of states in the puzzle, and the second player must then convert one randomly-selected state to another without passing through forbidden states. Analyzing this version raises the question of the treewidth of Hanoi graphs. We find this number exactly for three-peg puzzles and provide nearly-tight asymptotic bounds for larger numbers of pegs

Angles of Arc-Polygons and Lombardi Drawings of Cacti

We characterize the triples of interior angles that are possible in non-self-crossing triangles with circular-arc sides, and we prove that a given cyclic sequence of angles can be realized by a non-self-crossing polygon with circular-arc sides whenever all angles are at most pi. As a consequence of these results, we prove that every cactus has a planar Lombardi drawing (a drawing with edges depicted as circular arcs, meeting at equal angles at each vertex) for its natural embedding in which every cycle of the cactus is a face of the drawing. However, there exist planar embeddings of cacti that do not have planar Lombardi drawings.Comment: 12 pages, 8 figures. To be published in Proc. 33rd Canadian Conference on Computational Geometry, 202

New Applications of Nearest-Neighbor Chains: Euclidean TSP and Motorcycle Graphs

We show new applications of the nearest-neighbor chain algorithm, a technique that originated in agglomerative hierarchical clustering. We use it to construct the greedy multi-fragment tour for Euclidean TSP in O(n log n) time in any fixed dimension and for Steiner TSP in planar graphs in O(n sqrt(n)log n) time; we compute motorcycle graphs, a central step in straight skeleton algorithms, in O(n^(4/3+epsilon)) time for any epsilon>0

Flow-Based Decomposition for Geometric and Combinatorial Markov Chain Mixing

We prove that the well-studied triangulation flip walk on a convex point set mixes in time O(n^3 log^3 n), the first progress since McShine and Tetali’s O(n^5 log n) bound in 1997. In the process we give lower and upper bounds of respectively Omega(1/(sqrt(n) log n)) and O(1/sqrt(n))---asymptotically tight up to an O(log n) factor---for the expansion of the associahedron graph K_n---the first o(1) expansion result for this graph. We show quasipolynomial mixing for the k-angulation flip walk on a convex point set, for fixed k >= 4, and a treewidth result for the flip graph on n x n lattice triangulations.We show that the hardcore Glauber dynamics---a random walk on the independent sets of an input graph---mixes rapidly in graphs of bounded treewdith for all fixed values of the standard parameter lambda > 0, giving a simple alternative to existing sampling algorithms for these structures. We also show rapid mixing for analogous Markov chains on dominating sets and b-edge covers (for fixed b >= 1 and lambda > 0) in bounded-treewidth graphs, and for Markov chains on the b-matchings (for fixed b >= 1 and lambda > 0), the maximal independent sets, and the maximal b-matchings of a graph (for fixed b >= 1), in graphs of bounded carving width.To obtain these results, we introduce a decomposition framework for showing rapid Markov chain mixing. This framework is a purely combinatorial analogue that in some settings gives better results than the projection-restriction technique of Jerrum, Son, Tetali, and Vigoda

Cell Therapy for Chronic TBI Interim Analysis of the Randomized Controlled STEMTRA Trial

Objective To determine whether chronic motor deficits secondary to traumatic brain injury (TBI) can be improved by implantation of allogeneic modified bone marrow-derived mesenchymal stromal/stem cells (SB623). Methods This 6-month interim analysis of the 1-year double-blind, randomized, surgical sham-controlled, phase 2 Stem Cell Therapy for Traumatic Brain Injury (STEMTRA) trial (NCT02416492) evaluated safety and efficacy of the stereotactic intracranial implantation of SB623 in patients with stable chronic motor deficits secondary to TBI. Patients in this multicenter trial (n = 63) underwent randomization in a 1:1:1: 1 ratio to 2.5 x 10(6), 5.0 x 10(6), or 10 x 10(6) SB623 cells or control. Safety was assessed in patients who underwent surgery (n = 61), and efficacy was assessed in the modified intent-to-treat population of randomized patients who underwent surgery (n = 61; SB623 = 46, control = 15). Results The primary efficacy endpoint of significant improvement from baseline of Fugl-Meyer Motor Scale score at 6 months for SB623-treated patients was achieved. SB623-treated patients improved by (least square [LS] mean) 8.3 (standard error 1.4) vs 2.3 (standard error 2.5) for control at 6months, the LS mean difference was 6.0 (95% confidence interval 0.3-11.8, p = 0.040). Secondary efficacy endpoints improved from baseline but were not statistically significant vs control at 6 months. There were no dose-limiting toxicities or deaths, and 100% of SB623-treated patients experienced treatment-emergent adverse events vs 93.3% of control patients (p = 0.25). Conclusions SB623 cell implantation appeared to be safe and well tolerated, and patients implanted with SB623 experienced significant improvement from baseline motor status at 6 months compared to controls. Classification of Evidence This study provides Class I evidence that implantation of SB623 was well tolerated and associated with improvement in motor status