关于图的若干介值问题

对连通图G,以C_i(G),■(G)分别表G的有i条边的连通支撑子图之集与连通子图之集,以C~i(G),(?)(G)分别表G的顶点数为i的子树集与连通子图之集.本文讨论了这四类子图簇对若干基本参数及端点数的介值性,从而对已有的一些结果作了若干有意义的拓广.     

一些图的生成树数

图 G 的生成树是它的连通子图(子树).本文精确地计算出了一些图的生成树的数目, 例如双心轮图、双柄扇图等等.     

区间图上可带负权的2-中位选址问题（英文）

Abstract本文研究了区间图上可带负权的2-中位选址问题.根据目标函数的不同,可带负权的p-中位选址问题（p≥2）可分为两类：即MWD和WMD模型;前者是所有顶点与服务该顶点的设施之间的最小权重距离之和,后者是所有顶点与相应设施之间的权重最小距离之和.在本篇论文中,我们讨论了区间图上可带负权2-中位选址问题的两类模型,并分别设计时间复杂度为O（n~2）的多项式时间算法.     

两个逆网络选址问题的计算复杂性

本文考虑两个我们称之为逆网络选址的改进问题,它们是修改网络上各个边的长度,分别使得网络上某个给定的顶点到网络上所有点的最大距离以及该点到其它顶点的距离之和不大于预先给定的上界,并且所做的修改总量最小.我们将证明这两个逆网络选址问题都是强NP困难的.     

赋权树状网络中r-控制集问题和k-中心问题

图G=(V, E;f,ω)是顶点和边都赋权的树,f:V→R+,ω:E→R+.本文给出了顶点u与v之间距离的一种新的定义.在顶点和边都赋权的树中,研究在新距离条件下的r-控制集问题与k-中心问题.对于r-控制集问题,设计出了复杂性为Ο(n)的多项式时间算法;对于k-中心问题,设计出了Ο(n2log n)的多项式时间算法.     

子矩阵约束下矩阵方程组的双对称最小二乘解

矩阵方程组l∑j=1在控制与系统领域中具有广泛应用.该文构造了一种算法求解这个矩阵方程组,其中X_j∈R~(n_j×n_j)(j=1,2,…,l)为带有特殊中心主子矩阵约束的双对称矩阵.在没有舍入误差的情况下,该算法经过有限步迭代得到[X_1,X_2,…,X_l],使得t∑i=1||l∑j=1A_(ij)X_jB_(ij)-C_i||=min.实例表明这种方法是有效的.     

鲁棒动态设施选址问题的近似算法

设施选址问题是组合优化中重要问题之一。动态设施选址问题是传统设施选址问题的推广,其中度量空间中设施的开设费用和顾客的需求均随着时间的变化而变化。更多地,经典设施选址问题假设所有的顾客都需要被服务。在这个模型假设下,所有的顾客都需要服务。但事实上,有时为服务距离较远的顾客,需要单独开设设施,导致了资源的浪费。因此,在模型设置中,可以允许一些固定数目的顾客不被服务 (带异常点的设施选址问题),此外也可以通过支付一些顾客的惩罚费用以达到不服务的目的 (带惩罚的设施选址问题)。本文将综合以上两种鲁棒设置考虑同时带有异常点和惩罚的动态设施选址问题,通过原始-对偶框架得到近似比为3的近似算法。     

正n边形中四边形计数问题

文[1]探究了正n边形中三角形计数问题,受其启发笔者探究了正n边形中四边形计数问题.引理1圆内接四边形为平行四边形(矩形),当且仅当该四边形的两条对角线为该外接圆的两条直径.引理2圆内接四边形为菱形(正方形),当且仅当该四边形的两条对角线为该外接圆的两条互相垂直的直径.引理1,引理2由简单的平面几何知识即可得证,在此从略.问题1以正八边形的八个顶点为顶点可作多少个四边形?其中含有多少个梯形?多少平行四边形(含矩形)?多少个菱形(含正方形)?分析1)此正八边形的八个顶点中任意四点即可构成一个四边形,故四边形个数为C4=70.2)若构成梯…     

一种基于多级查找表的高效Huffman编码算法

本文研究了适用于数字信号处理器的一种高效的Huffman编码算法的问题.利用多级查找表建立规则,并将Huffman树分割为若干子树且为分割后的所有子树建立一个统一的查找表的方法,获得了可用较小的查找表来存储Huffman树且能对比特流进行快速解码的结果.     

一类树图的最短路的算法

本文研究了边点赋权图、顶点关于图的运输量及质心,利用比较两个相邻顶点的运输量的方法,得到了一个连通树图的顶点是质心的充要条件及质心个数不大于2的结果.同时给出了求质心及最小运输量的算法,其算法的时间复杂度为O(n2),有利于可建立树图模型的优化问题的求解.     

图的最小控制树问题（英文）

A dominating tree T of a graph G is a subtree of G which contains at least one neighbor of each vertex of G.The minimum dominating tree problem is to find a dominating tree of G with minimum number of vertices,which is an NP-hard problem.This paper studies some polynomially solvable cases,including interval graphs,Halin graphs,special outer-planar graphs and others.     

Branches in scale-free trees

In the paper the scale-free (preferential attachment) model of a random recursive tree is considered. We deal with the size and the distribution of vertex degrees in the kth branch of such a tree (which is the subtree rooted at vertex labeled k). A comparison of these results with analogous results for the whole tree shows that the k-branch of a scale-free tree can be considered as a scale-free tree itself with the number of vertices being random variables.     

Approximations to clustering and subgraph problems on trees

Given a tree with vertex weights, we derive fully polynomial approximation schemes for the following two NP-hard problems: (i) Partition the graph into connected cluster sets such that the weight of each set does not exceed a given bound and a monotone objective function is minimized. (ii) Find a subtree with bounded vertex weight such that a monotone objective function is maximized.     

Approximating the minimum clique cover and other hard problems in subtree filament graphs

Subtree filament graphs are the intersection graphs of subtree filaments in a tree. This class of graphs contains subtree overlap graphs, interval filament graphs, chordal graphs, circle graphs, circular-arc graphs, cocomparability graphs, and polygon-circle graphs. In this paper we show that, for circle graphs, the clique cover problem is NP-complete and the h-clique cover problem for fixed h is solvable in polynomial time. We then present a general scheme for developing approximation algorithms for subtree filament graphs, and give approximation algorithms developed from the scheme for the following problems which are NP-complete on circle graphs and therefore on subtree filament graphs: clique cover, vertex colouring, maximum k-colourable subgraph, and maximum h-coverable subgraph.     

圈的多重联图的邻点可区别E-全染色的一些结果

记χ_(at)~e(C_n_i)为n_i阶的圈C_n_i的邻点可区别E-全色数.若n_i≡0(mod 2)(i=1,2,3…,t),则χ_(at)~e(C_n_1+C_n_2+…+C_n_t)=2t;若n_i≡0(mod 2)(i=1,2,3…,r,l相似文献   

An Algorithmic Framework for the Exact Solution of the Prize-Collecting Steiner Tree Problem

The Prize-Collecting Steiner Tree Problem (PCST) on a graph with edge costs and vertex profits asks for a subtree minimizing the sum of the total cost of all edges in the subtree plus the total profit of all vertices not contained in the subtree. PCST appears frequently in the design of utility networks where profit generating customers and the network connecting them have to be chosen in the most profitable way. Our main contribution is the formulation and implementation of a branch-and-cut algorithm based on a directed graph model where we combine several state-of-the-art methods previously used for the Steiner tree problem. Our method outperforms the previously published results on the standard benchmark set of problems. We can solve all benchmark instances from the literature to optimality, including some of them for which the optimum was not known. Compared to a recent algorithm by Lucena and Resende, our new method is faster by more than two orders of magnitude. We also introduce a new class of more challenging instances and present computational results for them. Finally, for a set of large-scale real-world instances arising in the design of fiber optic networks, we also obtain optimal solution values. Received: April, 2004 This work has been partly supported by the RTNADONET, 504438, by the Doctoral Scholarship Program of the Austrian Academy of Sciences (DOC) and by CNR and MIUR, Italy.A preliminary version of this paper appeared as [21].     

Reverse selective obnoxious center location problems on tree graphs

In this paper, we investigate a variant of the reverse obnoxious center location problem on a tree graph \(T=(V,E)\) in which a selective subset of the vertex set V is considered as locations of the existing customers. The aim is to augment or reduce the edge lengths within a given budget with respect to modification bounds until a predetermined undesirable facility location becomes as far as possible from the customer points under the new edge lengths. An \({\mathcal {O}}(|E|^2)\) time combinatorial algorithm is developed for the problem with arbitrary modification costs. For the uniform-cost case, one obtains the improved \({\mathcal {O}}(|E|)\) time complexity. Moreover, optimal solution algorithms with \({\mathcal {O}}(|E|^2)\) and \({\mathcal {O}}(|E|)\) time complexities are proposed for the integer version of the problem with arbitrary and uniform cost coefficients, respectively.     

Linear time optimal approaches for reverse obnoxious center location problems on networks

This paper is concerned with a reverse obnoxious (undesirable) center location problem on networks in which the aim is to modify the edge lengths within an associated budget such that a predetermined facility location on the underlying network becomes as far as possible from the existing customer points under the new edge lengths. Exact combinatorial algorithms with linear time complexities are developed for the problem under the weighted rectilinear norm and the weighted Hamming distance. Furthermore, it is shown that the problem with integer decision variables can also be solved in linear time.     

A subexponential algorithm for the coloured tree partition problem

Given a tree of n vertices and a list of feasible colours for each vertex, the coloured tree partition problem (CTPP) consists in partitioning the tree into p vertex-disjoint subtrees of minimum total cost, and assigning to each subtree a different colour, which must be feasible for all of its vertices. The problem is strongly NP-hard on general graphs, as well as on grid and bipartite graphs. This paper deals with the previously open case of tree graphs, showing that it is strongly NP-complete to determine whether a feasible solution exists. It presents reduction, decomposition and bounding procedures to simplify the problem and an exact algorithm of complexity (with ) for the special case in which a vertex of each subtree is given.     

Invariant subsets of scattered trees and the tree alternative property of Bonato and Tardif

A tree is scattered if it does not contain a subdivision of the complete binary tree as a subtree. We show that every scattered tree contains a vertex, an edge, or a set of at most two ends preserved by every embedding of T. This extends results of Halin, Polat and Sabidussi. Calling two trees equimorphic if each embeds in the other, we then prove that either every tree that is equimorphic to a scattered tree T is isomorphic to T, or there are infinitely many pairwise non-isomorphic trees which are equimorphic to T. This proves the tree alternative conjecture of Bonato and Tardif for scattered trees, and a conjecture of Tyomkyn for locally finite scattered trees.