2453836
On connectivity of consecutive-d digraphs
The concept of a consecutive-d digraph was proposed by Du, Hsu and Hwang as a generalization of de Bruijn digraphs, Kautz digraphs, and their generalizations given by Imase and Itoh and Reddy, Pradhan and Kuhl. In this paper, we determine the connectivity of consecutive-d digraphs and study how to modify consecutive-d digraphs to reach maximum connectivity. Our results will generalize and improve several existing results on the connectivity of de Bruijn digraphs, Kautz digraphs and Imase-Itoh digraphs.
44914304
Steiner minimal trees on sets of four points
LetS = {A, B, C, D} consist of the four corner points of a convex quadrilateral where diagonals [A, C] and [B, D] intersect at the pointO. There are two possible full Steiner trees forS, theAB-CD tree hasA andB adjacent to one Steiner point, andC andD to another; theAD-BC tree hasA andD adjacent to one Steiner point, andB andC to another. Pollak proved that if both full Steiner trees exist, then theAB-CD (AD-BC) tree is the Steiner minimal tree ifAOD>3 (<) 90°, and both are Steiner minimal trees ifAOD=90°. While the theorem has been crucially used in obtaining results on Steiner minimal trees in general, its applicability is sometimes restricted because of the condition that both full Steiner trees must exist. In this paper we remove this obstacle by showing: (i) Necessary and sufficient conditions for the existence of either full Steiner tree forS. (ii) IfAOD≥90°, then theAB-CD tree is the SMT even if theAD-BC tree does not exist. (iii) IfAOD<90° but theAD-BC tree does not exist, then theAB-CD tree cannot be ruled out as a Steiner minimal tree, though under certain broad conditions it can.
206027538
A coloring problem on the n-cube
Abstract In this paper, we consider a coloring problem on the n-cube that arises in the study of scalability of optical networks. Let χ k (n) be the minimum number of colors needed to color the vertices of the n-cube so that every two vertices with Hamming distance ⩽k have different colors. We show that for k=3, 2n⩽χ 3 (n)⩽2 ⌈ log 2 n⌉+1 . We also provide upper and lower bounds on χ k (n) for general k.
42394808
on Component-size Bounded Steiner Trees
A Steiner tree is a tree interconnecting a given set of points in a metric space such that all leaves are given points. A (full) component of a Steiner tree is a subtree which results from splitting the Steiner tree at some given points. A k-size Steiner tree is a Steiner tree in which every component has at most k given points. The k-Steiner ratio is the largest lower bound for the ratio between lengths of a minimum Steiner tree and a minimum k-size Steiner tree for the same set of points. In this paper, we determine the 3-Steiner ratio in weighted graphs.
30354625
A class of full Steiner minimal trees
A Steiner minimal tree for a given set of points in the plane is a tree which interconnects these points using lines of shortest possible total length. We construct an infinite class of trees which are the unique full Steiner minimal trees for their sets of endpoints (vertices of degree one).
24107558
Lower Bounds on the Minus Domination and k-Subdomination Numbers
A three-valued function f defined on the vertex set of a graph G = (V,E), f : V → {-1, 0, 1} is a minus dominating function if the sum of its function values over any closed neighborhood is at least one. That is, for every v ∈ V, f(N[v]) ≥ 1, where N[v] consists of v and all vertices adjacent to v. The weight of a minus function is f(V) = Σv∈V f(v). The minus domination number of a graph G, denoted by γ-(G), equals the minimum weight of a minus dominating function of G. In this paper, sharp lower bounds on minus domination of a bipartite graph are given. Thus, we prove a conjecture proposed by J. Dunbar etc. (Discrete Math. 199(1999) 35-47), and we give a lower bound on γks(G) of a graph G.
18860643
The multiroute maximum flow problem revisited
We are given a directed network G = (V, A, u) with vertex setV , arc setA, a source vertexs ∈ V , a destination vertex t ∈ V , a finite capacity vector u = {uij}ij∈A, and a positive integerm ∈ Z+. The multiroute maximum flow problem ( m-MFP) generalizes the ordinary maximum flow problem by seeking a maximum flow from s to t subject to not only the regular flow conservation constraints at the vertices (except s andt) and the flow capacity constraints at the arcs, but also the extra constraints that any flow must be routed alon g m arc-disjoints-t paths. In this paper, we devise two new combinatorial algorithms for m-MFP. One is based on Newton’s method and another is based on augmenting-path technique. We also show how the Newton-based algorithm unifies two existing algorithms, and how the augmen ting-path algorithm is strongly polynomial for case m = 2.
8600977
Optimal preemptive semi-online scheduling on two uniform processors
We investigate a preemptive semi-online scheduling problem. Jobs with sizes within a certain range [1, r] (r≥1 ) arrive one by one to be scheduled on two uniform parallel processors with speed 1 and s≥1, respectively. The objective is to minimize makespan. We characterize the optimal competitive ratio as a function of both s and r by devising a deterministic on-line scheduling algorithm along with a matching lower bound, which also holds for randomized algorithms.
16471624
Preemptive on-line scheduling for two uniform processors
This paper considers the problem of preemptive on-line scheduling for two uniform processors in which one of the processors has speed 1 and the other has speed s>=1. The objective is to minimize the makespan. A best possible algorithm with competitive ratio of (1+s)^2/(1+s+s^2) is proposed for this problem.
35793386
An improved algorithm for decomposing arc flows into multipath flows
We consider a multipath maximum flow problem introduced by Kishimoto (Networks 27(4)(1996)279-291). The focus is on efficient transformation from arc flows into multipath flows, where a multipath flow is a nonnegative combination of multipaths. A new algorithm that is more efficient than existing ones is proposed for the transformation.
16666020
On-line Network Synthesis
We consider on-line network synthesis problems. Let N = {1, , n} be a set of n sites. Trafﬁc ﬂow requirements between pairs of sites are revealed one by one. Whenever a new request rij = rji (i < j) between sites i and j is revealed, an on-line algorithm must install the additional necessary capacity without decreasing the existing network capacity such that all the trafﬁc requirements are met. The objective is to minimize the total capacity installed by the algorithm. The performance of an on-line algorithm is measured by the competitive ratio, deﬁned to be the worst-case ratio between the total capacity by the on-line algorithm and the total optimal (off-line) capacity assuming we have priori information on all the requirements initially. We distinguish between two on-line versions of the problem depending on whether the entire set of sites is known a prior or not. For the ﬁrst version where the entire set of site is unknown, we present a best possible algorithm along with a matching lower bound. For the second version where the entire set of sites is known a priori, we present a best possible algorithm for n ≤ 6. 
 
Key words: On-line algorithm; Approximation algorithm; Network synthesis; Competitive analysis