Congruences modulo 8 for $$(2,\, k)$$ ( 2 , k ) -regular overpartitions for odd $$k > 1$$ k > 1

Algorithm Sum of Densities Average Cases K = 2 K = 3 K = 4 K = 8 K = 1 0 K = 1 5 K = 2 0 Improvement

Top-down partitioning has focused on minimum cut or ratio cut objectives, while bottom-up clustering has focused on density-based objectives. In seeking a more uni ed perspective, we propose a new sum of densities measure for multi-way circuit decomposition, where the density of a subhypergraph is the ratio of the number of edges to the number of nodes in the subhypergraph. Finding a k-way part...

Catalan Numbers Modulo 2 k

In this paper, we develop a systematic tool to calculate the congruences of some combinatorial numbers involving n!. Using this tool, we re-prove Kummer’s and Lucas’ theorems in a unique concept, and classify the congruences of the Catalan numbers cn (mod 64). To achieve the second goal, cn (mod 8) and cn (mod 16) are also classified. Through the approach of these three congruence problems, we ...

IMAGINARY QUADRATIC FIELDS k WITH Cl2(k) ≃ (2, 2 m) AND RANK Cl2(k 1) = 2

Planar k-cycle resonant graphs with k=1, 2

A connected graph is said to be k-cycle resonant if, for 16 t6 k, any t disjoint cycles in G are mutually resonant, that is, there is a perfect matching M of G such that each of the t cycles is an M -alternating cycle. The concept of k-cycle resonant graphs was introduced by the present authors in 1994. Some necessary and su6cient conditions for a graph to be k-cycle resonant were also given. I...

Approximating k-Spanner Problems for k>2

Given a graph G = (V ,E), a subgraph G′ = (V ,H), H ⊆ E is a k-spanner of G if for any pair of vertices u,w ∈ V it satisfies dH (u,w) kdG(u,w). The basic k-spanner problem is to find a k-spanner of a given graph G with the smallest possible number of edges. This paper considers approximation algorithms for this and some related problems for k > 2, known to be (2log 1− n)inapproximable. The basi...