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数学代写|组合优化代写Combinatorial optimization代考|Spanning Trees and Arborescences

Consider a telephone company that wants to rent a subset from an existing set of cables, each of which connects two cities. The rented cables should suffice to connect all cities and they should be as cheap as possible. It is natural to model the network by a graph: the vertices are the cities and the edges correspond to the cables. By Theorem $2.4$ the minimal connected spanning subgraphs of a given graph are its spanning trees. So we look for a spanning tree of minimum weight, where we say that a subgraph $T$ of a graph $G$ with weights $c: E(G) \rightarrow \mathbb{R}$ has weight $c(E(T))=\sum_{e \in E(T)} c(e)$. We shall refer to $c(e)$ also as the cost of $e$.

This is a simple but very important combinatorial optimization problem. It is also among the combinatorial optimization problems with the longest history; the first algorithm was given by Boruvka [1926a,1926b]; see Nešetřil, Milková and Nešetřilová [2001].

Compared to the DRILLING PROBLEM which asks for a shortest path containing all vertices of a complete graph, we now look for a shortest spanning tree. Although the number of spanning trees is even bigger than the number of paths $\left(K_n\right.$ contains $\frac{n !}{2}$ Hamiltonian paths, but as many as $n^{n-2}$ different spanning trees; cf. Theorem 6.2), the problem turns out to be much easier. In fact, a simple greedy strategy works as we shall see in Section 6.1.

Arborescences can be considered as the directed counterparts of trees; by Theorem $2.5$ they are the minimal spanning subgraphs of a digraph such that all vertices are reachable from a root. The directed version of the MINIMUM SPANNING TREE PROBLEM, the Minimum WeIGHT ARBORESCENCE Problem, is more difficult since a greedy strategy no longer works. In Section $6.2$ we show how to solve this problem.

Since there are very efficient combinatorial algorithms it is not recommended to solve these problems with Linear Programming. Nevertheless it is interesting that the corresponding polytopes (the convex hull of the incidence vectors of spanning trees or arborescences; cf. Corollary $3.33$ ) can be described in a nice way, which we shall show in Section 6.3. In Section 6.4 we prove some classical results concerning the packing of spanning trees and arborescences.

数学代写|组合优化代写Combinatorial optimization代考|Packing Spanning Trees and Arborescences

If we are looking for more than one spanning tree or arborescence, classical theorems of Tutte, Nash-Williams and Edmonds are of help. We first give a proof of ‘Iutte’s ‘Theorem on packing spanning trees which is essentially due to Mader (see Diestel [1997]) and which uses the following lemma:

Lemma 6.16. Let $G$ be an undirected graph, and let $F=\left(F_1, \ldots, F_k\right)$ be a $k$ tuple of edge-disjoint forests in $G$ such that $|E(F)|$ is maximum, where $E(F):=$ $\bigcup_{i=1}^k E\left(F_i\right)$. Let $e \in E(G) \backslash E(F)$. Then there exists a set $X \subseteq V(G)$ with $e \subseteq X$ such that $F_i[X]$ is connected for each $i \in{1, \ldots, k}$.

Proof: For two $k$-tuples $F^{\prime}=\left(F_1^{\prime}, \ldots, F_k^{\prime}\right)$ and $F^{\prime \prime}=\left(F_1^{\prime \prime}, \ldots, F_k^{\prime \prime}\right)$ of edgedisjoint forests we say that $F^{\prime \prime}$ arises from $F^{\prime}$ by exchanging $e^{\prime}$ for $e^{\prime \prime}$ if $F_j^{\prime \prime}=$ $\left(F_j^{\prime} \backslash e^{\prime}\right) \dot{\cup} e^{\prime \prime}$ for some $j$ and $F_i^{\prime \prime}=F_i^{\prime}$ for all $i \neq j$. Let $\mathcal{F}$ be the set of all $k$-tuples of edge-disjoint forests arising from $F$ by a sequence of such exchanges. Lêt $\bar{E}:=E(G) \backslash\left(\bigcap_{F^{\prime} \in \mathcal{F}} E\left(F^{\prime}\right)\right)$ and $\bar{G}:=(V(G), \bar{E})$. Wẽ hãve $F \in \mathcal{F}$ and thus $e \in \bar{E}$. Let $X$ be the vertex set of the connected component of $\bar{G}$ containing $e$. We shall prove that $F_i[X]$ is connected for each $i$.
Claim: For any $F^{\prime}=\left(F_1^{\prime}, \ldots, F_k^{\prime}\right) \in \mathcal{F}$ and any $\bar{e}={v, w} \in E(\bar{G}[X]) \backslash E\left(F^{\prime}\right)$ there exists a $v-w$-path in $F_i^{\prime}[X]$ for all $i \in{1, \ldots, k}$.

To prove this, let $i \in{1, \ldots, k}$ be fixed. Since $F^{\prime} \in \mathcal{F}$ and $\left|E\left(F^{\prime}\right)\right|=|E(F)|$ is maximum, $F_i^{\prime}+\bar{e}$ contains a circuit $C$. Now for all $e^{\prime} \in E(C) \backslash{\bar{e}}$ we have $F_{e^{\prime}}^{\prime} \in \mathcal{F}$, where $F_{e^{\prime}}^{\prime}$ arises from $F^{\prime}$ by exchanging $e^{\prime}$ for $\bar{e}$. This shows that $E(C) \subseteq$ $\bar{E}$, and so $C-\bar{e}$ is a $v-w$-path in $F_i^{\prime}[X]$. This proves the claim.

Since $\bar{G}[X]$ is connected, it suffices to prove that for each $\bar{e}={v, w} \in$ $E(\bar{G}[X])$ and each $i$ there is a $v-w$-path in $F_i[X]$.

So let $\bar{e}={v, w} \in E(\bar{G}[X])$. Since $\bar{e} \in \bar{E}$, there is some $F^{\prime}=\left(F_1^{\prime}, \ldots, F_k^{\prime}\right) \in$
$\mathcal{F}$ with $\bar{e} \notin E\left(F^{\prime}\right)$. By the claim there is a $v-w$-path in $F_i^{\prime}[X]$ for each $i$.
Now there is a sequence $F=F^{(0)}, F^{(1)} \ldots, F^{(s)}=F^{\prime}$ of elements of $\mathcal{F}$ such that $F^{(r+1)}$ arises from $F^{(r)}$ by exchanging one edge $(r=0, \ldots, s-1)$.

数学代写|组合优化代写Combinatorial optimization代考|MTH5107

组合优化代考

数学代写|组合优化代写Combinatorial optimization代考|Spanning Trees and arboresences

.


假设一家电话公司想从现有的一组电缆中租用一个子集,每条电缆连接两个城市。租用的电缆应该足以连接所有城市,而且应该尽可能便宜。用图形来建模网络是很自然的:顶点代表城市,边缘对应电缆。根据定理$2.4$,给定图的最小连通生成子图就是它的生成树。我们寻找一个权值最小的生成树,我们说权值为$c: E(G) \rightarrow \mathbb{R}$的图$G$的子图$T$的权值为$c(E(T))=\sum_{e \in E(T)} c(e)$。我们将$c(e)$也称为$e$的成本。


这是一个简单但非常重要的组合优化问题。它也是历史最悠久的组合优化问题之一;第一个算法由Boruvka [1926a,1926b]给出;参见涅捷契尔,Milková和Nešetřilová [2001].


与要求包含一个完整图的所有顶点的最短路径的钻孔问题相比,我们现在寻找的是一个最短生成树。虽然生成树的数量甚至大于路径的数量$\left(K_n\right.$包含$\frac{n !}{2}$哈密顿路径,但多达$n^{n-2}$不同的生成树;(参见定理6.2),这个问题就简单多了。事实上,我们将在第6.1节中看到一个简单的贪婪策略


Arborescences可以被认为是树木的定向对应物;根据定理$2.5$,它们是有向图的最小生成子图,这样所有顶点都可以从根处到达。最小生成树问题的有向版本,即最小权树问题,由于贪婪策略不再起作用,因此更加困难。在$6.2$部分,我们将展示如何解决这个问题


因为有非常有效的组合算法,所以不建议用线性规划来解决这些问题。然而,有趣的是,对应的多面体(生成树或树的关联向量的凸包;cf.推论$3.33$)可以用一种很好的方式描述,我们将在第6.3节中展示。在第6.4节中,我们证明了关于生成树和树生物的填充的一些经典结果

数学代写|组合优化代写Combinatorial optimization代考|Packing Spanning Trees and arboresences

.包装生成树


如果我们要寻找不止一个的生成树或树形,Tutte, Nash-Williams和Edmonds的经典定理是有帮助的。我们首先给出了关于填充生成树的Iutte定理的一个证明,该定理主要由Mader提出(见Diestel[1997]),并使用了以下引理:

引理6.16。设$G$为无向图,设$F=\left(F_1, \ldots, F_k\right)$为$G$中的边不相交林的$k$元组,使$|E(F)|$最大,其中$E(F):=$$\bigcup_{i=1}^k E\left(F_i\right)$。让$e \in E(G) \backslash E(F)$。然后存在一个带有$e \subseteq X$的集合$X \subseteq V(G)$,这样每个$i \in{1, \ldots, k}$都连接了$F_i[X]$

证明:两个 $k$-元组 $F^{\prime}=\left(F_1^{\prime}, \ldots, F_k^{\prime}\right)$ 和 $F^{\prime \prime}=\left(F_1^{\prime \prime}, \ldots, F_k^{\prime \prime}\right)$ 我们这样说边角交错的森林 $F^{\prime \prime}$ 产生于 $F^{\prime}$ 通过交换 $e^{\prime}$ 为 $e^{\prime \prime}$ 如果 $F_j^{\prime \prime}=$ $\left(F_j^{\prime} \backslash e^{\prime}\right) \dot{\cup} e^{\prime \prime}$ 对一些人来说 $j$ 和 $F_i^{\prime \prime}=F_i^{\prime}$ 为所有人 $i \neq j$。让 $\mathcal{F}$ 成为所有人的集合 $k$-边缘不相交森林的元组产生于 $F$ 通过一系列这样的交流。Lêt $\bar{E}:=E(G) \backslash\left(\bigcap_{F^{\prime} \in \mathcal{F}} E\left(F^{\prime}\right)\right)$ 和 $\bar{G}:=(V(G), \bar{E})$。Wẽ hãve $F \in \mathcal{F}$ 因此 $e \in \bar{E}$。让 $X$ 的连通分量的顶点集 $\bar{G}$ 遏制 $e$。我们将证明这一点 $F_i[X]$ 连接到每一个 $i$.
声明:对于任何 $F^{\prime}=\left(F_1^{\prime}, \ldots, F_k^{\prime}\right) \in \mathcal{F}$ 还有任何 $\bar{e}={v, w} \in E(\bar{G}[X]) \backslash E\left(F^{\prime}\right)$ 存在一个 $v-w$-路径输入 $F_i^{\prime}[X]$ 为所有人 $i \in{1, \ldots, k}$.

为了证明这一点,让$i \in{1, \ldots, k}$被固定。因为$F^{\prime} \in \mathcal{F}$和$\left|E\left(F^{\prime}\right)\right|=|E(F)|$是最大值,所以$F_i^{\prime}+\bar{e}$包含一个电路$C$。对于所有的$e^{\prime} \in E(C) \backslash{\bar{e}}$,我们有$F_{e^{\prime}}^{\prime} \in \mathcal{F}$,其中$F_{e^{\prime}}^{\prime}$由$F^{\prime}$由$e^{\prime}$变成$\bar{e}$。这表明$E(C) \subseteq$$\bar{E}$,因此$C-\bar{e}$是$F_i^{\prime}[X]$中的$v-w$ -路径。

由于$\bar{G}[X]$是连接的,它足以证明对于每个$\bar{e}={v, w} \in$$E(\bar{G}[X])$和每个$i$,在$F_i[X]$中有一个$v-w$ -路径

所以让 $\bar{e}={v, w} \in E(\bar{G}[X])$。自从 $\bar{e} \in \bar{E}$,有一些 $F^{\prime}=\left(F_1^{\prime}, \ldots, F_k^{\prime}\right) \in$
$\mathcal{F}$ 用 $\bar{e} \notin E\left(F^{\prime}\right)$。根据声明,有一个 $v-w$-路径输入 $F_i^{\prime}[X]$ 对于每一个 $i$
现在有一个序列 $F=F^{(0)}, F^{(1)} \ldots, F^{(s)}=F^{\prime}$ 的元素 $\mathcal{F}$ 如此这般 $F^{(r+1)}$ 产生于 $F^{(r)}$ 通过交换一条边 $(r=0, \ldots, s-1)$。

数学代写|组合优化代写Combinatorial optimization代考

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