Improved bounds for minimal feedback vertex sets in tournaments

We study feedback vertex sets (FVS) in tournaments, which are orientations of complete graphs. As our main result, we show that any tournament on n nodes has at most 1.5949ⁿ minimal FVS. This significantly improves the previously best upper bound of 1.6667ⁿ by Fomin et al. [STOC 2016] and 1.6740ⁿ by Gaspers and Mnich [J. Graph Theory 72 (1):72–89, 2013]. Our new upper bound almost matches the best‐known lower bound of , due to Gaspers and Mnich. Our proof is algorithmic, and shows that all minimal FVS of tournaments can be enumerated in time .     

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。     

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     

A Linear Time Algorithm for the Minimum-weight Feedback Vertex Set Problem in Series-parallel Graphs

A feedback vertex set is a subset of vertices in a graph, whose deletion from the graph makes the resulting graph acyclic. In this paper, we study the minimum-weight feedback vertex set problem in seriesparallel graphs and present a linear-time exact algorithm to solve it.     

一个求外平面图最小顶点赋权反馈点集的线性时间算法

若从一个图中去掉某些顶点后得到的导出子图是无圈图,则所去的那些顶点组成的集合就是原图的反馈点集.本文主要考虑外平面图中的反馈点集并给出了一个求外平面图最小顶点赋权反馈点集的线性时间算法.     

Cycle Transversals in Tournaments with Few Vertex Disjoint Cycles

We prove that every tournament T with no three disjoint cycles contains a set X of at most four vertices such that is acyclic.     

Planar Digraphs without Large Acyclic Sets

Given a directed graph, an acyclic set is a set of vertices inducing a directed subgraph with no directed cycle. In this note, we show that for all integers , there exist oriented planar graphs of order n and digirth g for which the size of the maximum acyclic set is at most . When this result disproves a conjecture of Harutyunyan and shows that a question of Albertson is best possible.     

3-一致超图的反馈数研究

超图H=(V,E)顶点集为V,边集为E.S■V是H的顶点子集,如果H/S不含有圈,则称S是H的点反馈数,记τc(H)是H的最小点反馈数.本文证明了:(i)如果H是线性3-一致超图,边数为m,则τc(H)≤m/3;(ii)如果H是3-一致超图,边数为m,则τc(H)≤m/2并且等式成立当且仅当H任何一个连通分支是孤立顶点或者长度为2的圈.A■V是H的边子集,如果H\A不含有圈,则称A是H的边反馈数,记τc′(H)是H的最小边反馈数.本文证明了如果H是含有p个连通分支的3-一致超图,则τc’(H)≤2m-n+p.     

On the number of minimal separators in graphs

Weconsider the largest number of minimal separators a graph on n vertices can have.

• – We give a new proof that this number is in .
• – We prove that this number is in , improving on the previous best lower bound of .

This gives also an improved lower bound on the number of potential maximal cliques in a graph. We would like to emphasize that our proofs are short, simple, and elementary.     

Maximal Induced Matchings in Triangle‐Free Graphs

An induced matching in a graph is a set of edges whose endpoints induce a 1‐regular subgraph. It is known that every n‐vertex graph has at most  maximal induced matchings, and this bound is the best possible. We prove that every n‐vertex triangle‐free graph has at most  maximal induced matchings; this bound is attained by every disjoint union of copies of the complete bipartite graph K_{3, 3}. Our result implies that all maximal induced matchings in an n‐vertex triangle‐free graph can be listed in time , yielding the fastest known algorithm for finding a maximum induced matching in a triangle‐free graph.     

Arc‐Disjoint Cycles and Feedback Arc Sets

Isaak posed the following problem. Suppose T is a tournament having a minimum feedback arc set, which induces an acyclic digraph with a hamiltonian path. Is it true that the maximum number of arc‐disjoint cycles in T equals the cardinality of minimum feedback arc set of T? We prove that the answer to the problem is in the negative.     

偏序集最小顶点割算法与最小费用赶工问题

在网络计划问题中,一项大的工程由许多工序合成。工序与工序之间存在着一定的前后关系,每个工序有着自己的正常加工时间和通过赶工所能达到的最短加工时间以及每赶工一天的赶工费用。设一项工程的正常工期为T天,通过对所有可能的工序赶工,整个工程能达到的最短工期为S天。本文的问题是,对于任意给定的t(S t相似文献   

Small Blocking Sets in PG(2, p3)

In this paper we show that a small minimalblocking set in PG(2,p³), with p 7,is of Rédei type.     

Data structures for maintaining set partitions

Efficiently maintaining the partition induced by a set of features is an important problem in building decision‐tree classifiers. In order to identify a small set of discriminating features, we need the capability of efficiently adding and removing specific features and determining the effect of these changes on the induced classification or partition. In this paper we introduce a variety of randomized and deterministic data structures to support these operations on both general and geometrically induced set partitions. We give both Monte Carlo and Las Vegas data structures that realize near‐optimal time bounds and are practical to implement. We then provide a faster solution to this problem in the geometric setting. Finally, we present a data structure that efficiently estimates the number of partitions separating elements. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

Constraint Handling in Genetic Algorithms: The Set Partitioning Problem

In this paper we present a genetic algorithm-based heuristic for solving the set partitioning problem (SPP). The SPP is an important combinatorial optimisation problem used by many airlines as a mathematical model for flight crew scheduling.A key feature of the SPP is that it is a highly constrained problem, all constraints being equalities. New genetic algorithm (GA) components: separate fitness and unfitness scores, adaptive mutation, matching selection and ranking replacement, are introduced to enable a GA to effectively handle such constraints. These components are generalisable to any GA for constrained problems.We present a steady-state GA in conjunction with a specialised heuristic improvement operator for solving the SPP. The performance of our algorithm is evaluated on a large set of real-world problems. Computational results show that the genetic algorithm-based heuristic is capable of producing high-quality solutions.     

Bounding the feedback vertex number of digraphs in terms of vertex degrees

The Turán bound (Turán (1941) [17]) is a famous result in graph theory, which relates the independence number of an undirected graph to its edge density. Also the Caro-Wei inequality (Caro (1979) [4] and Wei (1981) [18]), which gives a more refined bound in terms of the vertex degree sequence of a graph, might be regarded today as a classical result. We show how these statements can be generalized to directed graphs, thus yielding a bound on directed feedback vertex number in terms of vertex out-degrees and in terms of average out-degree, respectively.     

Optimization problems in multiple subtree graphs

We study various optimization problems in t-subtree graphs, the intersection graphs of t-subtrees, where a t-subtree is the union of t disjoint subtrees of some tree. This graph class generalizes both the class of chordal graphs and the class of t-interval graphs, a generalization of interval graphs that has recently been studied from a combinatorial optimization point of view. We present approximation algorithms for the Maximum Independent Set, Minimum Coloring, Minimum Vertex Cover, Minimum Dominating Set, and Maximum Clique problems.     

多因素分析的线性结构模型及其应用

线性结构模型是含有潜在因子的通径分析,该模型可以分析外生的潜在因子与内生的潜在因子之间相互影响的结构关系.研究应用线性结构模型探讨对《算法与数据结构》的学习兴趣有影响的主要因素,结果显示学生的专业基础和教学方式对学习兴趣的影响分别占各因素影响总和的49.56%和43.11%.注重专业基础课程的教学效果和采取通俗易懂的讲授方式是提高学生学习兴趣的有效途径.