In this paper, we study the relationship between the Asymmetric Traveling Salesman Problem (ATSP) and the Cycle Cover Problem in terms of the strength of the triangle inequality on the edge costs in the given complete directed graph instance, G = (V, E). The strength of the triangle inequality is captured by parametrizing the triangle inequality as follows. A complete directed graph G = (V, E) ...

For a weighted directed multigraph, let fij be the total weight of spanning converging forests that have vertex i in a tree converging to j. We prove that fij fjk = fik fjj if and only if every directed path from i to k contains j (a graph bottleneck equality). Otherwise, fij fjk < fik fjj (a graph bottleneck inequality). In a companion paper [1] (P. Chebotarev, A new family of graph distances,...

The adequacy of substitution matrices to model evolutionary relationships between amino acid sequences can be numerically evaluated by checking the mathematical property of triangle inequality for all triplets of residues. By converting substitution scores into distances, one can verify that a direct path between two amino acids is shorter than a path passing through a third amino acid in the a...

We consider the problem of finding a cheapest Hamiltonian path of a complete graph satisfying a relaxed triangle inequality, i.e., such that for some parameter β > 1, the edge costs satisfy the inequality c({x, y}) ≤ β `

Semi-supervised clustering of images has been an interesting problem for machine learning and computer vision researchers for decades. Pairwise constrained clustering is a popular paradigm for semi supervision that uses knowledge about whether two images belong to the same category (must-link constraint) or not (can’t-link constraint). Performance of constrained clustering algorithms can be imp...

• Pre-Hilbert spaces: definition • Cauchy-Schwarz-Bunyakowski inequality • Example: spaces ` • Triangle inequality, associated metric, continuity issues • Hilbert spaces, completions, infinite sums • Minimum principle • Orthogonal projections to closed subspaces • Orthogonal complements W⊥ • Riesz-Fischer theorem on linear functionals • Orthonormal sets, separability • Parseval equality, Bessel...

Very recently, Lin et al. [1] introduced the notion of generalized quasi-metric inspired from the notion of generalized metric, defined by Branciari [2]. It is a very well-known fact that the concept of generalizedmetric can be derived from the definition ofmetric by replacing the triangle inequality with a weaker condition, namely, quadrilateral inequality. In spite of the analogy between the ...

Let X, d be a metric space and x, y ∈ X with l d x, y . A geodesic path from x to y is an isometry c : 0, l → X such that c 0 x and c l y. The image of a geodesic path is called a geodesic segment. A metric spaceX is a (uniquely) geodesic space if every two points ofX are joined by only one geodesic segment. A geodesic triangle x1, x2, x3 in a geodesic space X consists of three points x1, x2, x...