Connecting colored point sets

We study the following Ramsey-type problem. Let S=B∪R be a two-colored set of n points in the plane. We show how to construct, in time, a crossing-free spanning tree T(B) for B, and a crossing-free spanning tree T(R) for R, such that both the number of crossings between T(B) and T(R) and the diameters of T(B) and T(R) are kept small. The algorithm is conceptually simple and is implementable without using any non-trivial data structure. This improves over a previous method in Tokunaga [Intersection number of two connected geometric graphs, Inform. Process. Lett. 59 (1996) 331-333] that is less efficient in implementation and does not guarantee a diameter bound. Implicit to our approach is a new proof for the result in the reference above on the minimum number of crossings between T(B) and T(R).     

Tight Lower Bounds for Halfspace Range Searching

We establish two new lower bounds for the halfspace range searching problem: Given a set of n points in ℝ ^d , where each point is associated with a weight from a commutative semigroup, compute the semigroup sum of the weights of the points lying within any query halfspace. Letting m denote the space requirements, we prove a lower bound for general semigroups of [\varOmega\tilde](n^1-1/(d+1)/m^1/(d+1))\widetilde{\varOmega}(n^{1-1/(d+1)}/m^{1/(d+1)}) and for integral semigroups of [\varOmega\tilde](n/m^1/d)\widetilde{\varOmega}(n/m^{1/d}).     

Tight Lower Bounds on the Size of a Maximum Matching in a Regular Graph

In this paper we study tight lower bounds on the size of a maximum matching in a regular graph. For k ≥3, let G be a connected k-regular graph of order n and let α′(G) be the size of a maximum matching in G. We show that if k is even, then , while if k is odd, then . We show that both bounds are tight. Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal.     

Network reinforcement

We give an algorithm for the following problem: given a graph G=(V,E) with edge-weights and a nonnegative integer k, find a minimum cost set of edges that contains k disjoint spanning trees. This also solves the following reinforcement problem: given a network, a number k and a set of candidate edges, each of them with an associated cost, find a minimum cost set of candidate edges to be added to the network so it contains k disjoint spanning trees. The number k is seen as a measure of the invulnerability of a network. Our algorithm has the same asymptotic complexity as |V| applications of the minimum cut algorithm of Goldberg & Tarjan. Received: April, 2004     

图的线独立数的一个公式

本文中我们用等秩变换证明了连通图G的所有生成树的邻接矩阵的秩中最大者就是图G线独立数的两倍。特别,我们给出了连通图G具有完美匹配的一个新的充要条件。     

A polynomial time approximation scheme for the two-source minimum routing cost spanning trees

Let G be an undirected graph with nonnegative edge lengths. Given two vertices as sources and all vertices as destinations, we investigated the problem how to construct a spanning tree of G such that the sum of distances from sources to destinations is minimum. In the paper, we show the NP-hardness of the problem and present a polynomial time approximation scheme. For any >0, the approximation scheme finds a (1+)-approximation solution in O(n^1/+1) time. We also generalize the approximation algorithm to the weighted case for distances that form a metric space.     

Worst-case behavior of the MVCA heuristic for the minimum labeling spanning tree problem

In this paper, we review recent work on the minimum labeling spanning tree problem and obtain a new worst-case ratio for the MVCA heuristic. We also present a family of graphs in which the worst-case ratio can be attained. This implies that the new ratio cannot be improved any further.     

Unexpected behaviour of crossing sequences

The nth crossing number of a graph G, denoted ncr(G), is the minimum number of crossings in a drawing of G on an orientable surface of genus n. We prove that for every a>b>0, there exists a graph G for which ₀cr(G)=a, ₁cr(G)=b, and ₂cr(G)=0. This provides support for a conjecture of Archdeacon et al. and resolves a problem of Salazar.     

Bounded-degree spanning tree problems: models and new algorithms

Given a connected graph G, a vertex v of G is said to be a branch vertex if its degree is greater than 2. We consider two problems arising in the context of optical networks:

Asymptotic Upper Bounds for Ramsey Functions

 We show that for any graph G with N vertices and average degree d, if the average degree of any neighborhood induced subgraph is at most a, then the independence number of G is at least Nf _{a +1}(d), where f _{a +1}(d)=∫₀ ¹(((1−t)^{1/( a +1)})/(a+1+(d−a−1)t))dt. Based on this result, we prove that for any fixed k and l, there holds r(K _k + _l ,K _n )≤ (l+o(1))n ^k /(logn) ^{k −1}. In particular, r(K _k , K _n )≤(1+o(1))n ^{k −1}/(log n) ^{k −2}. Received: May 11, 1998 Final version received: March 24, 1999     

A visual proof of a result of Knuth on spanning trees of Aztec diamonds in the case of odd order

The even Aztec diamond AD_n is known to have precisely four times more spanning trees than the odd Aztec diamond OD_n—this was conjectured by Stanley and first proved by Knuth. We present a short combinatorial proof of this fact in the case of odd n. Our proof works also for the more general case of odd-by-odd Aztec rectangles.     

Factorizations of complete graphs into caterpillars of diameter 5

We examine factorizations of complete graphs K_2n into caterpillars of diameter 5. First we present a construction generalizing some previously known methods. Then we use the new method along with some previous partial results to give a complete characterization of caterpillars of diameter 5, which factorize the complete graph K_2n.     

The Tree of Hubs Location Problem

This paper presents the Tree of Hubs Location Problem. It is a network hub location problem with single assignment where a fixed number of hubs have to be located, with the particularity that it is required that the hubs are connected by means of a tree. The problem combines several aspects of location, network design and routing problems. Potential applications appear in telecommunications and transportation systems, when set-up costs for links between hubs are so high that full interconnection between hub nodes is prohibitive. We propose an integer programming formulation for the problem. Furthermore, we present some families of valid inequalities that reinforce the formulation and we give an exact separation procedure for them. Finally, we present computational results using the well-known AP and CAB data sets.     

Lower Bounds for Catalan's Equation

This paper begins with a short historical survey on Catalan's equation, namely x^p-y^q=1, where p andq are prime numbers and x, y are non-zero rational integers. It is conjectured that the only solution is the trivial solution 3²-2³=1. We prove that there is no non-trivial solution with p orq smaller than 30000. The tools to reach such a result are presented. A crucial role is played by a recent estimate of linear forms in two logarithms obtained by Laurent, Mignotte and Nestrenko. The criteria used are also quite recent. We give information on the enormous amount of computation needed for the verification.     

Relations,models and a memetic approach for three degree-dependent spanning tree problems

In this paper we take into account three different spanning tree problems with degree-dependent objective functions. The main application of these problems is in the field of optical network design. In particular, we propose the classical Minimum Leaves Spanning Tree problem as a relevant problem in this field and show its relations with the Minimum Branch Vertices and the Minimum Degree Sum Problems. We present a unified memetic algorithm for the three problems and show its effectiveness on a wide range of test instances.     

Spanning trees without adjacent vertices of degree 2

Albertson, Berman, Hutchinson, and Thomassen showed in 1990 that there exist highly connected graphs in which every spanning tree contains vertices of degree 2. Using a result of Alon and Wormald, we show that there exists a natural number $d$ such that every graph of minimum degree at least $d$ contains a spanning tree without adjacent vertices of degree 2. Moreover, we prove that every graph with minimum degree at least 3 has a spanning tree without three consecutive vertices of degree 2.