IA Scholar Query: On the Graph of the Pedigree Polytope.
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Internet Archive Scholar query results feedeninfo@archive.orgTue, 12 Apr 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440On the meaning of the Critical Cost Efficiency Index
https://scholar.archive.org/work/opcvgqnudzadzd56sixoac4tki
This note provides a critical discussion of the Critical Cost-Efficiency Index (CCEI) as used to assess deviations from utility-maximizing behavior. I argue that the CCEI is hard to interpret, and that it can disagree with other plausible measures of "irrational" behavior. The common interpretation of CCEI as wasted income is questionable. Moreover, I show that one agent may have more unstable preferences than another, but seem more rational according to the CCEI. This calls into question the (now common) use of CCEI as an ordinal and cardinal measure of degrees of rationality.Federico Echeniquework_opcvgqnudzadzd56sixoac4tkiTue, 12 Apr 2022 00:00:00 GMTOn 1-skeleton of the polytope of pyramidal tours with step-backs
https://scholar.archive.org/work/kpphkxe5cfgmblhw7crjxqldy4
Pyramidal tours with step-backs are Hamiltonian tours of a special kind: the salesperson starts in city 1, then visits some cities in ascending order, reaches city $n$, and returns to city 1 visiting the remaining cities in descending order. However, in the ascending and descending direction, the order of neighboring cities can be inverted (a step-back). It is known that on pyramidal tours with step-backs the traveling salesperson problem can be solved by dynamic programming in polynomial time. We define the polytope of pyramidal tours with step-backs $\operatorname{PSB}(n)$ as the convex hull of the characteristic vectors of all possible pyramidal tours with step-backs in a complete directed graph. The 1-skeleton of $\operatorname{PSB}(n)$ is the graph whose vertex set is the vertex set of the polytope, and the edge set is the set of geometric edges or one-dimensional faces of the polytope. We present a linear-time algorithm to verify vertex adjacencies in 1-skeleton of the polytope $\operatorname{PSB}(n)$ and estimate the diameter and the clique number of 1-skeleton: the diameter is bounded above by 4 and the clique number grows quadratically in the parameter $n$.Andrei Nikolaevwork_kpphkxe5cfgmblhw7crjxqldy4Tue, 01 Feb 2022 00:00:00 GMTFinding a second Hamiltonian decomposition of a 4-regular multigraph by integer linear programming
https://scholar.archive.org/work/wn6syzebgze75j3q25qa3ffu5y
A Hamiltonian decomposition of a regular graph is a partition of its edge set into Hamiltonian cycles. We consider the second Hamiltonian decomposition problem: for a 4-regular multigraph find 2 edge-disjoint Hamiltonian cycles different from the given ones. This problem arises in polyhedral combinatorics as a sufficient condition for non-adjacency in the 1-skeleton of the travelling salesperson polytope. We introduce two integer linear programming models for the problem based on the classical Dantzig-Fulkerson-Johnson and Miller-Tucker-Zemlin formulations for the travelling salesperson problem. To enhance the performance on feasible problems, we supplement the algorithm with a variable neighbourhood descent heuristic w.r.t. two neighbourhood structures, and a chain edge fixing procedure. Based on the computational experiments, the Dantzig-Fulkerson-Johnson formulation showed the best results on directed multigraphs, while on undirected multigraphs, the variable neighbourhood descent heuristic was especially effective.Andrei V. Nikolaev, Egor V. Klimovwork_wn6syzebgze75j3q25qa3ffu5yTue, 11 Jan 2022 00:00:00 GMTThe threshold for stacked triangulations
https://scholar.archive.org/work/x7hwca36kbhb5m76zc4tgze4he
A stacked triangulation of a d-simplex 𝐨={1,...,d+1} (d≥ 2) is a triangulation obtained by repeatedly subdividing a d-simplex into d+1 new ones via a new vertex (the case d=2 is known as an Appolonian network). We study the occurrence of such a triangulation in the Linial–Meshulam model, i.e., for which p does the random simplicial complex Y∼𝒴_d(n,p) contain the faces of a stacked triangulation of the d-simplex 𝐨, with its internal vertices labeled in [n]. In the language of bootstrap percolation in hypergraphs, it pertains to the threshold for K_d+2^d+1, the (d+1)-uniform clique on d+2 vertices. Our main result identifies this threshold for every d≥ 2, showing it is asymptotically (α_d n)^-1/d, where α_d is the growth rate of the Fuss–Catalan numbers of order d. The proof hinges on a second moment argument in the supercritical regime, and on Kalai's algebraic shifting in the subcritical regime.Eyal Lubetzky, Yuval Peledwork_x7hwca36kbhb5m76zc4tgze4heMon, 10 Jan 2022 00:00:00 GMTBlade Envelopes Part I: Concept and Methodology
https://scholar.archive.org/work/zxksqdqnvrdezcjqz26nvkpoxy
Blades manufactured through flank and point milling will likely exhibit geometric variability. Gauging the aerodynamic repercussions of such variability, prior to manufacturing a component, is challenging enough, let alone trying to predict what the amplified impact of any in-service degradation will be. While rules of thumb that govern the tolerance band can be devised based on expected boundary layer characteristics at known regions and levels of degradation, it remains a challenge to translate these insights into quantitative bounds for manufacturing. In this work, we tackle this challenge by leveraging ideas from dimension reduction to construct low-dimensional representations of aerodynamic performance metrics. These low-dimensional models can identify a subspace which contains designs that are invariant in performance -- the inactive subspace. By sampling within this subspace, we design techniques for drafting manufacturing tolerances and for quantifying whether a scanned component should be used or scrapped. We introduce the blade envelope as a computational manufacturing guide for a blade that is also amenable to qualitative visualizations. In this paper, the first of two parts, we discuss its underlying concept and detail its computational methodology, assuming one is interested only in the single objective of ensuring that the loss of all manufactured blades remains constant. To demonstrate the utility of our ideas we devise a series of computational experiments with the Von Karman Institute's LS89 turbine blade.Chun Yui Wong, Pranay Seshadri, Ashley Scillitoe, Andrew B. Duncan, Geoffrey Parkswork_zxksqdqnvrdezcjqz26nvkpoxyFri, 31 Dec 2021 00:00:00 GMTBCD Nets: Scalable Variational Approaches for Bayesian Causal Discovery
https://scholar.archive.org/work/jd66o5oh3rezlnqc3qjco5ndia
A structural equation model (SEM) is an effective framework to reason over causal relationships represented via a directed acyclic graph (DAG). Recent advances have enabled effective maximum-likelihood point estimation of DAGs from observational data. However, a point estimate may not accurately capture the uncertainty in inferring the underlying graph in practical scenarios, wherein the true DAG is non-identifiable and/or the observed dataset is limited. We propose Bayesian Causal Discovery Nets (BCD Nets), a variational inference framework for estimating a distribution over DAGs characterizing a linear-Gaussian SEM. Developing a full Bayesian posterior over DAGs is challenging due to the the discrete and combinatorial nature of graphs. We analyse key design choices for scalable VI over DAGs, such as 1) the parametrization of DAGs via an expressive variational family, 2) a continuous relaxation that enables low-variance stochastic optimization, and 3) suitable priors over the latent variables. We provide a series of experiments on real and synthetic data showing that BCD Nets outperform maximum-likelihood methods on standard causal discovery metrics such as structural Hamming distance in low data regimes.Chris Cundy and Aditya Grover and Stefano Ermonwork_jd66o5oh3rezlnqc3qjco5ndiaMon, 06 Dec 2021 00:00:00 GMTSpace: The Re-Visioning Frontier of Biological Image Analysis with Graph Theory, Computational Geometry, and Spatial Statistics
https://scholar.archive.org/work/4stjek4ryzfxrb7sgq2ip5g2z4
Every biological image contains quantitative data that can be used to test hypotheses about how patterns were formed, what entities are associated with one another, and whether standard mathematical methods inform our understanding of biological phenomena. In particular, spatial point distributions and polygonal tessellations are particularly amendable to analysis with a variety of graph theoretic, computational geometric, and spatial statistical tools such as: Voronoi polygons; Delaunay triangulations; perpendicular bisectors; circumcenters; convex hulls; minimal spanning trees; Ulam trees; Pitteway violations; circularity; Clark-Evans spatial statistics; variance to mean ratios; Gabriel graphs; and, minimal spanning trees. Furthermore, biologists have developed a number of empirically related correlations for polygonal tessellations such as: Lewis's law (the number of edges of convex polygons are positively correlated with the areas of these polygons): Desch's Law (the number of edges of convex polygons are positively correlated with the perimeters of these polygons); and Errara's Law (daughter cell areas should be roughly half that of their parent cells' areas). We introduce a new Pitteway Law that the number of sides of the convex polygons in a Voronoi tessellation of biological epithelia is proportional to the minimal interior angle of the convex polygons as angles less than 90 degrees result in Pitteway violations of the Delaunay dual of the Voronoi tessellation.John R. Jungck, Michael J. Pelsmajer, Camron Chappel, Dylan Taylorwork_4stjek4ryzfxrb7sgq2ip5g2z4Wed, 27 Oct 2021 00:00:00 GMT