In the k-supplier problem, we are given a set of clients C and set of facilities F located in a metric (C∪F, d), along with a bound k. The goal is to open a subset of k facilities so as to minimize the maximum distance of a client to an open facility, i.e., minS⊆F :|S|=k maxv∈C d(v, S), where d(v, S) = minu∈S d(v, u) is the minimum distance of client v to any facility in S. We present a 1 + √ 3...

forest fragmentation results in a loss of forest interior and an increase in edge habitat. we studied how understorey bird community composition and habitat variables changed along an edge-to-interior gradient in a 1248-ha lowland rainforest patch in peninsular malaysia. birds and environmental variables such as vegetation structure and litter depth were detected within a 25-m radius of each of...

Given a simple graph G = (V,E), a subset of E is called a triangle cover if it intersects each triangle of G. Let νt(G) and τt(G) denote the maximum number of pairwise edge-disjoint triangles in G and the minimum cardinality of a triangle cover of G, respectively. Tuza conjectured in 1981 that τt(G)/νt(G) ≤ 2 holds for every graph G. In this paper, using a hypergraph approach, we design polynom...

Let G = (V,E) be a k-edge-connected graph with edge costs {c(e) : e ∈ E} and let 1 ≤ l ≤ k − 1. We show by a simple and short proof, that G contains an l-edge cover I such that: c(I) ≤ l k c(E) if G is bipartite, or if l|V | is even, or if |E| ≥ k|V | 2 + k 2l ; otherwise, c(I) ≤ (

Let G(V,E) be a graph, and let f be an integer function on V with 1 ≤ f(v) ≤ d(v) to each vertex v ∈ V . An f -edge cover coloring is an edge coloring C such that each color appears at each vertex v at least f(v) times. The f -edge cover chromatic index of G, denoted by χ ′ fc(G), is the maximum number of colors needed to f -edge cover color G. It is well known that min v∈V {bd(v)− μ(v) f(v) c ...

Let M = (V, E, A) be a mixed graph with vertex set V , edge set E and arc set A. A cycle cover of M is a family C = {C1, . . . ,Ck} of cycles of M such that each edge/arc of M belongs to at least one cycle in C. The weight of C is ∑k i=1 |Ci |. The minimum cycle cover problem is the following: given a strongly connected mixed graph M without bridges, find a cycle cover of M with weight as small...

We consider a generalized Path Traveling Salesman Problem where the distances are defined by a 2-edge-connected graph metric and a constant number of salesmen have to cover all the destinations by traveling along paths of minimum total length. We show that for this problem there is a polynomial algorithm with asymptotic approximation ratio of 2 .

We develop a polynomial time 3-2-approximation algorithm to solve the vertex cover problem on a class of graphs satisfying a property called “active edge hypothesis”. The algorithm also guarantees an optimal solution on specially structured graphs. Further, we give an extended algorithm which guarantees a vertex cover S1 on an arbitrary graph such that |S1| ≤ 32 |S*| + ξ where S* is an optimal ...

The tree and tour cover problems on an edge-weighted graph are to compute a minimum weight tree and closed walk, respectively, whose vertices form a vertex cover. Both problems are NP-hard. In this note we give strongly polynomial time, constant factor approximation algorithms for both problems. An interesting feature of our algorithms is how they combine approximations of other problems, namel...

We consider a generalized Path Traveling Salesman Problem where the distances are defined by a 2-edge-connected graph metric and a constant number of salesmen have to cover all the destinations by traveling along paths of minimum total length. We show that for this problem there is a polynomial algorithm with asymptotic approximation ratio of 2 .