覆盖数不超过3的图上渡河问题

1000多年前,英国著名学者Alcuin曾提出一个古老的渡河问题,即狼、羊和卷心菜的渡河问题。2006年,Prisner把该问题推广到任意的冲突图上,考虑了一类情况更一般的渡河运输问题。所谓冲突图是指一个图G=(V,E),这里V代表某些物品的集合,V中的两个点有边连结当且仅当这两个点是冲突的,即在无人监管的情况下不允许留在一起的点。图G=(V,E)的一个可行运输方案是指在保证不发生任何冲突的前提下,把V的点所代表的物品全部摆渡到河对岸的一个运输方案。图G的Alcuin数定义为它存在可行运输方案时所需船的最小容量。本文讨论了覆盖数不超过3的连通图的Alcuin数,给出了该类图Alcuin数的完全刻画。     

最大度为5的图的Alcuin数

1000多年前,英国著名学者Alcuin曾提出过一个古老的渡河问题,即狼、羊和卷心菜的渡河问题.最近,Prisner和Csorba等人把这一问题推广到任意的"冲突图"G=(V,E)上,考虑了一类情况更一般的运输计划问题.现在监管者欲运输V中的所有"物品/点"渡河,这里V的两个点邻接当且仅当这两个点为冲突点.冲突点是指不能在无人监管的情况下留在一起的点.特别地,Alcuin渡河问题可转化成"冲突路"P_3上是否存在可行运输方案问题.图G的Alcuin数是指图G具有可行运输方案(即把V的点代表的"物品"全部运到河对岸)时船的最小容量.最大度为5且覆盖数至少为5的图和最大度Δ(G)≤4且覆盖数不小于Δ(G)-1的图的Alcuin数已经被确定.本文给出最大度为4且覆盖数不超过2和最大度为5且覆盖数不超过4的图的Alcuin数.至此,最大度不超过5的图的Alcuin数被完全确定.     

超图的Alcuin数与其横贯数的关系

1000多年前, 英国著名学者Alcuin曾提出过一个古老的渡河问题, 即狼、羊和卷心菜的渡河问题. 最近, Prisner和Csorba等考虑了一般``冲突图"上的渡河问题. 将这一问题推广到超图$H=(V,\mathcal{E})$\,上, 考虑一类情况更一般的运输计划问题. 现在监管者 欲运输超图中的所有点\,(代表``items")\,渡河, 这里$V$的点子 形成超边 当且仅当这些点代表的``items"在无人监管的情况下不能留在一起. 超图$H$的Alcuin数是指超图$H$具有可行运输方案\,(即把$V$的点代表的``items" 全部运到河对岸)\,时船的最小容量. 给出了 $r$-一致完全二部超图和它的伴随超图, 以及$r$-一致超图的Alcuin数, 同时证明了判断$r$-一致超图是否为小船图是NP 困难的.     

Alcuin数列与整数三角形

<正>1问题背景阿尔昆(Alcuin 732-804)著有一本数学问题集,名为 Propositiones ad acuendos juvenes("磨利年轻人的问题"),其中包含与现实相结合的数学例题.Alcuin数学问题集中的第十二个问题便是烧瓶分配问题.著名的数列,Alcuin数列,就是受到这个问题的启发.烧瓶分配问题可以看作是三个变量在一定限制条件下的性质,从而转化为整数三角形的个数问题.给定周长下的整数三角形的数量可以直接连接到烧瓶的分配方式的数量.     

Kautz网络中的最小反馈点集

对简单有向图D=(V，E)，顶点子集F∈V，如果由V＼F导出的子图不含有向圈，则称F是D的反馈点集。点数最小的子集F称为最小反馈点集。最小的点数称为反馈数。本利用Kautz最小轨道的方法确定出了Kautz有向图K(d，k)反馈数的一个下界和上界。并且具体给出了当k≤3时的反馈数。     

一类带多重核元集在线装箱问题

本文讨论了一类在线变尺寸装箱问题,假定箱子的尺寸可以是不同的.箱子是在线到达的,仅当箱子到达后其尺寸才知道.给定一个带有核元的物品表及其上的核元关系图.我们的目标是要将表中元素装入到达的箱子中,保证任何箱子所装物品不互为核元,即所装物品对应的点所导出的子图是个空图,并使得所用的箱子总长最小.我们证明了该问题是NPHard的,并给出了基于图的点染色、图的团分解和基于背包问题的近似算法,给出了算法的时间复杂度和性能界.     

折叠立方体网络的最小反馈点集

对简单图G=(V,E),顶点子集F V,如果由V\F导出的子图不含圈,则称F是G的反馈点集。点数最小的反馈点集称图的最小反馈点集,最小的点数称为反馈数。一个k维折叠立方体是由一个k维超立方体加上所有的互补边构成的图。本文证明了k维折叠立方体网络的反馈数f(k)=c.2k-1(k 2),其中c∈k-1     

顶点覆盖约束下的同类机排序算法研究

给定m台同类机和n个工件,其中第j台机器的速度为s_j,第i个工件的加工时间为p_i并且在第j台机器上的负载为p_i/s_j.构造一个顶点赋权无向图G=(V,E;w),其中图G的n个顶点代表这n个工件,顶点权重代表相应工件的加工时间.本文研究顶点覆盖约束下的同类机排序问题.该问题是两个组合最优化问题的组合问题,其目标为首先确定图G的一个顶点覆盖,即图的一个顶点子集,使得图中每一条边都至少存在一个顶点属于该子集;然后把这个子集所代表的相应工件集放到m台同类机上加工,使得最大完工时间最小.该问题是NP-hard的.本文基于分层算法和LSPT算法设计一个■-近似算法,当所有机器的速度都相差不大时,该算法的近似效果较好.     

数学复习中的多学科沟通

随着高考改革的不断深入,命题中突出了数学的工具性,加强了知识间的联系,特别是注意了数学与其他学科的沟通,所以在高考复习中要重视数学与其他学科的联系.一、与物理的沟通物理考试不会刻意追求数学,但应该使学生懂得,如何利用数学知识来解决物理问题.其实物理中的很多问题都和数学有关,如物理中的最值问题、受力分析、运动问题、变力做功问题、气态变化问题等等.例1河宽H,船速为V船,水流速度为V水,船速V船与河岸的夹角为θ,如图1所示.①求渡河所用的时间,并讨论θ=?时渡河时间最短.②怎样渡河,船的合位移最小?分析①用船静水中的分运动…     

无爪图上团横贯数的界

设 G=(V,E) 为简单图,图 G 的每个至少有两个顶点的极大完全子图称为 G 的一个团. 一个顶点子集 S\subseteq V 称为图 G 的团横贯集, 如果 S 与 G 的所有团都相交,即对于 G 的任意的团 C 有 S\cap{V(C)}\neq\emptyset. 图 G 的团横贯数是图 G 的最小团横贯集所含顶点的数目,记为~${\large\tau}_{C}(G)$. 证明了棱柱图的补图(除5-圈外)、非奇圈的圆弧区间图和 Hex-连接图这三类无爪图的团横贯数不超过其阶数的一半.     

The Ferry Cover Problem on Regular Graphs and Small-Degree GraphsThe Ferry Cover Problem on Regular Graphs and Small-Degree Graphs

The ferry problem may be viewed as generalizations of the classical wolf-goatcabbage puzzle. The ferry cover problem is to determine the minimum required boat capacity to safely transport n items represented by a conflict graph. The Alcuin number of a conflict graph is the smallest capacity of a boat for which the graph possesses a feasible ferry schedule. In this paper the authors determine the Alcuin number of regular graphs and graphs with maximum degree at most five.     

The Ferry Cover Problem on Regular Graphs and Small-Degree Graphs

The ferry problem may be viewed as generalizations of the classical wolf-goat-cabbage puzzle. The ferry cover problem is to determine the minimum required boat capacity to safely transport n items represented by a conflict graph. The Alcuin number of a conflict graph is the smallest capacity of a boat for which the graph possesses a feasible ferry schedule. In this paper the authors determine the Alcuin number of regular graphs and graphs with maximum degree at most five.     

Two maps on one surface

Two embeddings of a graph in a surface S are said to be “equivalent” if they are identical under an homeomorphism of S that is orientation‐preserving for orientable S. Two graphs cellularly embedded simultaneously in S are said to be “jointly embedded” if the only points of intersection involve an edge of one graph transversally crossing an edge of the other. The problem is to find equivalent embeddings of the two graphs that minimize the number of these edge‐crossings; this minimum we call the “joint crossing number” of the two graphs. In this paper, we calculate the exact value for the joint crossing number for two graphs simultaneously embedded in the projective plane. Furthermore, we give upper and lower bounds when the surface is the torus, which in many cases give an exact answer. In particular, we give a construction for re‐embedding (equivalently) the graphs in the torus so that the number of crossings is best possible up to a constant factor. Finally, we show that if one of the embeddings is replaced by its “mirror image,” then the joint crossing number can decrease, but not by more than 6.066%. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 198–216, 2001     

Non-zero component union graph of a finite-dimensional vector space

In this paper, we introduce a graph structure, called non-zero component union graph on finite-dimensional vector spaces. We show that the graph is connected and find its domination number, clique number and chromatic number. It is shown that two non-zero component union graphs are isomorphic if and only if the base vector spaces are isomorphic. In case of finite fields, we study the edge-connectivity and condition under which the graph is Eulerian. Moreover, we provide a lower bound for the independence number of the graph. Finally, we come up with a structural characterization of non-zero component union graph.     

Bounds for rectilinear crossing numbers

A rectilinear drawing of a graph is one where each edge is drawn as a straight-line segment, and the rectilinear crossing number of a graph is the minimum number of crossings over all rectilinear drawings. We describe, for every integer k ≥ 4, a class of graphs of crossing number k, but unbounded rectilinear crossing number. This is best possible since the rectilinear crossing number is equal to the crossing number whenever the latter is at most 3. Further, if we consider drawings where each edge is drawn as a polygonal line segment with at most one break point, then the resulting crossing number is at most quadratic in the regular crossing number. © 1993 John Wiley & Sons, Inc.     

Paths, trees and matchings under disjunctive constraints

We study the minimum spanning tree problem, the maximum matching problem and the shortest path problem subject to binary disjunctive constraints: A negative disjunctive constraint states that a certain pair of edges cannot be contained simultaneously in a feasible solution. It is convenient to represent these negative disjunctive constraints in terms of a so-called conflict graph whose vertices correspond to the edges of the underlying graph, and whose edges encode the constraints.We prove that the minimum spanning tree problem is strongly NP-hard, even if every connected component of the conflict graph is a path of length two. On the positive side, this problem is polynomially solvable if every connected component is a single edge (that is, a path of length one). The maximum matching problem is NP-hard for conflict graphs where every connected component is a single edge.Furthermore we will also investigate these graph problems under positive disjunctive constraints: In this setting for certain pairs of edges, a feasible solution must contain at least one edge from every pair. We establish a number of complexity results for these variants including APX-hardness for the shortest path problem.     

On the complexity of crossings in permutations

We investigate crossing minimization problems for a set of permutations, where a crossing expresses a disarrangement between elements. The goal is a common permutation π^∗ which minimizes the number of crossings. In voting and social science theory this is known as the Kemeny optimal aggregation problem minimizing the Kendall-τ distance. This rank aggregation problem can be phrased as a one-sided two-layer crossing minimization problem for a series of bipartite graphs or for an edge coloured bipartite graph, where crossings are counted only for monochromatic edges. We contribute the max version of the crossing minimization problem, which attempts to minimize the discrimination against any permutation. As our results, we correct the construction from [C. Dwork, R. Kumar, M. Noar, D. Sivakumar, Rank aggregation methods for the Web, Proc. WWW10 (2001) 613-622] and prove the NP-hardness of the common crossing minimization problem for k=4 permutations. Then we establish a 2−2/k-approximation, improving the previous factor of 2. The max version is shown NP-hard for every k≥4, and there is a 2-approximation. Both approximations are optimal, if the common permutation is selected from the given ones. For two permutations crossing minimization is solved by inspecting the drawings, whereas it remains open for three permutations.