This erratum corrects the statements of Theorems 3.2, 3.3, 3.6, and 3.7 from Constantine, Dow, and Wang [SIAM J. Sci. Comput., 36 (2014), pp. A1500–A1524], all of which contain a similar minor error in the application of the triangle inequality. It also corrects a missing minus sign in (5.3). These errors do not change the main conclusions of the paper.

We discuss two special cases of the three-dimensional bottleneck assignment problemwhere a certain underlying cost function satisfies the triangle inequality. We present polynomial time 2-approximation algorithms for the broadest class of these special cases, and we prove that (unless P = NP) this factor 2 is best possible even in the highly restricted setting of the Euclidean plane.

These notes presents a string similarity measure which is a metric in the mathematical sense. In particular, the triangle inequality holds for this metric. The metric is based on the longest common subsequence (LCS) measure, and the complexity of any sensible implementation will be no worse than O(n).

Given a complete graph with edge weights that satisfy the triangle inequality, we bound the minimum cost of a subgraph which is the union of two spanning trees in terms of the minimum cost of a k-edge-connected subgraph, for k ≤ 4.

Ramanujan graphs have extremal spectral properties, which imply a remarkable combinatorial behavior. In this paper we compute the high-dimensional Laplace spectrum of Ramanujan triangle complexes, and show that it implies a combinatorial expansion property, and a pseudo-randomness result. For this purpose we prove a Cheeger-type inequality and a mixing lemma of independent interest.

We show that the norm of the commutator defines “almost a metric” on the quotient space of commuting matrices, in the sense that it is a semi-metric satisfying the triangle inequality asymptotically for large matrices drawn from a “good” distribution. We provide theoretical analysis of this results for several distributions of matrices, and show numerical experiments confirming this observation.

The well-known O ( n1−1/d ) behaviour of the optimal tour length for TSP in d-dimensional Cartesian space causes breaches of the triangle inequality. Other practical inadequacies of this model are discussed, including its use as basis for approximation of the TSP optimal tour length or bounds derivations, which I attempt to remedy.

We consider a Generalized, Multiple Depot Hamiltonian Path Problem (GMDHPP) and show that it has an algorithm with an approximation ratio of 32 if the costs are symmetric and satisfy the triangle inequality. This improves on the 2-approximation algorithm already available for the same.

We study transitivity properties of edge weights in complex networks. We show that enforcing transitivity leads to a transitivity inequality which is equivalent to ultra-metric inequality. This can be used to define transitive closure on weighted undirected graphs, which can be computed using a modified Floyd-Warshall algorithm. These new concepts are extended to dissimilarity graphs and triang...

This paper describes our recent research progress on generalizing triangle inequality (TI) to optimize Machine Learning algorithms that involve either vector dot products (e.g., Neural Networks) or distance calculations (e.g., KNN, KMeans). The progress includes a new form of TI named Angular Triangular Inequality, abstractions to enable unified treatment to various ML algorithms, and TOP, a co...