We study approximation complexity of the Vertex Cover problem restricted to dense and subdense balanced k-partite k-uniform hypergraphs. The best known approximation algorithm for the general k-partite case achieves an approximation ratio of k2 which is the best possible assuming the Unique Game Conjecture. In this paper, we present approximation algorithms for the dense and the subdense nearly...

A paired comparison digraph D is a weighted digraph in which the sum of the weights of arcs, if any, joining two vertices exactly one. A one-to-one mapping from V (D) onto {1, 2, . . . , |V (D)|} is called a ranking ofD, and a ranking α ofD is optimal if the backward length of α is minimum. We say that D is r-partite if V (D) can be partitioned into V1 ∪ · · · ∪ Vr so that every arc of D joinin...

It is a popular idea that a multi-partite quantum system represented by a density matrix having no product eigenbasis possesses non-classical correlation. Supporting this idea, we define an entropic measure of non-classical correlation on the basis of the minimum uncertainty about a joint-system after we collect reports of particular local measurements. Numerical computation of this measure for...

Quantum entanglement is one of the most remarkable aspects of quantum physics. If two particles are in a entangled state, then, even if the particles are physically separated by a great distance, they behave in some repects as a single entity. Entanglement is a key resource for quantum information processing and spatially separated entangled pairs of particles have been used for numerous purpos...

We introduce a family of QKD protocols for distributing shared random keys within a network of n users. The advantage of these protocols is that any possible key structure needed within the network, including broadcast keys shared among subsets of users, can be implemented by using a particular multi-partite high-dimensional quantum state. This approach is more efficient in the number of quantu...

The notion of a graph theoretic Ramsey number is generalised by assuming that both the original graph whose edges are arbitrarily bi–coloured and the sought after monochromatic subgraphs are complete, balanced, multipartite graphs, instead of complete graphs as in the classical definition. We previously confined our attention to diagonal multipartite Ramsey numbers. In this paper the definition...

We determine the asymptotics of the largest family of qualitatively 2{independent k{ partitions of an n{set, for every k > 2. We generalize a Sperner-type theorem for 2{partite sets of KK orner and Simonyi to the k{partite case. Both results have the feature that the corresponding trivial information-theoretic upper bound is tight. The results follow from a more general Sperner capacity theorem...

Let $C_k$ be a cycle of order $k$, where $k\ge 3$. ex$(n, n, \{C_{3}, C_{4}\})$ the maximum number edges in balanced $3$-partite graph whose vertex set consists three parts, each has $n$ vertices that no subgraph isomorphic to $C_3$ or $C_4$. We construct dense 3-partite graphs without 3-cycles 4-cycles and show C_{4}\})\ge (\frac{6\sqrt{2}-8}{(\sqrt{2}-1)^{3/2}}+o(1))n^{3/2}$.