We mainly study Max TSP with two objective functions. We propose an algorithm which returns a single Hamiltonian cycle with performance guarantee on both objectives. The algorithm is analysed in three cases. When both (resp. at least one) objective function(s) fulfill(s) the triangle inequality, the approximation ratio is 5 12 − ε ≈ 0.41 (resp. 3 8 − ε). When the triangle inequality is not assu...

Several researchers have developed properties that ensure compatibility of a concept similarity or dissimilarity measure with the formal semantics of Description Logics. While these authors have highlighted the relevance of the triangle inequality, none of their proposed dissimilarity measures satisfy it. In this work we present a theoretical framework for dissimilarity measures with this prope...

X iv :m at h/ 04 06 10 8v 1 [ m at h. FA ] 6 J un 2 00 4 QUADRATIC REVERSES OF THE CONTINUOUS TRIANGLE INEQUALITY FOR BOCHNER INTEGRAL OF VECTOR-VALUED FUNCTIONS IN HILBERT SPACES SEVER S. DRAGOMIR Abstract. Some quadratic reverses of the continuous triangle inequality for Bochner integral of vector-valued functions in Hilbert spaces are given. Applications for complex-valued functions are prov...

If the distances of TSP satisfy the triangle inequality, the minimum-cost-spanning tree (MST) heuristics produces a tour whose length is guaranteed to be less than 2 times the optimum tour length and Christofides’ heuristics generates the 3/2 times the optimum tour length. Otherwise, the quality of the approximation is hard to evaluate. Here a four vertices and three lines inequality is used to...

We describe hashing of data bases as a problem of information and coding theory. It is shown that the triangle inequality for the Hamming distances between binary vectors may essentially decrease the computational eeorts of a search for a pattern in the data base. Introduction of the Lee distance in the space, which consists of the Hamming distances, leads to a new metric space where the triang...

While Bregman divergences have been used for clustering and embedding problems in recent years, the facts that they are asymmetric and do not satisfy triangle inequality have been a major concern. In this paper, we investigate the relationship between two families of symmetrized Bregman divergences and metrics, which satisfy the triangle inequality. The first family can be derived from any well...

as one can check using induction on l. The usual absolute value function |x| satisfies these conditions with the ordinary triangle inequality (4). If N(x) = 0 when x = 0 and N(x) = 1 when x 6= 0, then N(x) satisfies these conditions with the ultrametric version of the triangle inequality. For each prime number p, the p-adic absolute value of a rational number x is denoted |x|p and defined by |x...

In this paper we obtain some inequalities related to the generalized triangle and quadratic triangle inequalities for vectors in inner product spaces. Some results that employ the Ostrowski discrete inequality for vectors in normed linear spaces are also obtained.