# 计算机代写|组合优化代写Combinatorial optimization代考|APM6664

## 计算机代写|组合优化代写Combinatorial optimization代考|Goldberg-Tarjan Algorithm

In this section, we study a different type of incremental method for maximum network flow. In this method, a valid label will play an important role. This valid label will be on each arc to guide the incremental direction.

Consider a flow network $G=(V, E)$ with capacity $c(u, v)$ for each arc $(u, v) \in$ $E ; s$ and $t$ are source and sink, respectively. As usual, for simplicity of description, we extend capacity $c(u, v)$ to every pair of nodes $u$ and $v$ by defining $c(u, v)=0$ if $(u, v) \notin E$.
A function $f: V \times V \rightarrow R$ is called a preflow if

1. (Capacity constraint) $f(u, v) \leq c(u, v)$ for every $u, v \in V$.
2. (Skew symmetry) $f(u, v)=-f(v, u)$ for all $u, v \in V$.
3. For every $v \in V \backslash{s, t}, \sum_{v \in V \backslash{u}} f(u, v) \geq 0$, i.e., $\sum_{(u, v) \in E} f(u, v) \geq$ $\sum_{(v, w) \in E} f(v, w)$

Compared with those three conditions in the definition of flow, the first two are the same, and the third one is different. The flow conservation condition is relaxed to allow more flow coming than going out at any node other than $s$ and $t$. This difference is called the excess at node $v$ and denotes
$$e(v)=\sum_{(u, v) \in E} f(u, v) \geq \sum_{(v, w) \in E} f(v, w) .$$
A node $v$ is said to be active if $e(v)>0, v \neq s$, and $v \neq t$. In preflow-relabel algorithm, the excess will be pushed from an active node toward the sink, relying on the valid distance label $d(v)$ for $v \in V$, satisfying the following conditions.

• $d(t)=0$.
• $d(u) \leq d(v)+1$ for $(u, v) \in E$.
An $\operatorname{arc}(u, v)$ is said to be admissible if $d(u)=d(v)+1$ and $c(u, v)>0$. Note that if we consider a residual graph, then $c(u, v)$ should be considered as updated capacity.

## 计算机代写|组合优化代写Combinatorial optimization代考|Simplex Algorithm

An LP is an optimization problem with linear objective function and a constraint system of equalities and inequalities. The following is an example:
maximize $z=4 x+5 y$
subject to $2 x+3 y \leq 60$
$$x \geq 0, y \geq 0 \text {. }$$
This example can be explained in the Euclidean plane as shown in Fig. 6.1. Each of three inequalities gives a half plane. Their intersection is a triangle, which is called a feasible domain. In general, the feasible domain of an LP is the set of all points satisfying all constraints. For different value of $z, z=4 x+5 y$ gives different lines which form a family of parallel lines. When $z$ increases, line $z=4 x+5 y$ moves from left to right, and at point $(30,0)$, it is the last moment to intersect the feasible domain. Hence, $(30,0)$ is the point at which $z=4 x+5 y$ reaches its maximum value, i.e., 120 .

In general, an LP may contain a large number of variables and a large number of constraints and hence cannot be solved geometrically as above. However, above example gives us a hint to find a general method. An important observation is that the maximum value of objective function is achieved at a vertex of the feasible domain. This observation suggests an incremental method as follows: Start from a vertex of the feasible domain and move from a vertex to another vertex with improvement on objective function value.

Before we implement this idea, let us look at a standard form of LP. Every LP can be transformed into the following form:
\begin{aligned} \max & z=c x \ \text { s.t. } & A x=b \ & x \geq 0, \end{aligned}
where $c$ is an $n$-dimensional row vector; $b$ is an $m$-dimensional column vector; $A$ is an $m \times n$ coefficient matrix with rank $m$, i.e., $\operatorname{rank}(A)=m$; and $x$ is a column vector with $n$ variables as components. Thus, the above example can be transformed into the following:
$$\begin{array}{ll} \operatorname{maximize} & z=4 x+5 y \ \text { subject to } & 2 x+3 y+w=60 \ & x \geq 0, y \geq 0, w \geq 0 . \end{array}$$

# 组合优化代考

## 计算机代写|组合优化代写Combinatorial optimization代考|Goldberg-Tarjan Algorithm

1. (容量限制) $f(u, v) \leq c(u, v)$ 对于每个 $u, v \in V$.
2. (斜对称) $f(u, v)=-f(v, u)$ 对所有人 $u, v \in V$.
3. 对于每一个v $\in V \backslash s, t, \sum_{v \in V \backslash u} f(u, v) \geq 0$ ，那是， $\sum_{(u, v) \in E} f(u, v) \geq \sum_{(v, w) \in E} f(v, w)$
与流定义中的这三个条件相比，前两个相同，第三个不同。放宽了流量守恒条件，以允许在除 $s$ 和 $t$. 这种 差异称为节点处的过剩 $v$ 并表示
$$e(v)=\sum_{(u, v) \in E} f(u, v) \geq \sum_{(v, w) \in E} f(v, w) .$$
一个节点 $v$ 据说是活跃的，如果 $e(v)>0, v \neq s$ ，和 $v \neq t$. 在 preflow-relabel 算法中，多余的将从活动 节点推向汇点，依赖于有效的距离标签 $d(v)$ 为了 $v \in V$ ，满足以下条件。
• $d(t)=0$.
• $d(u) \leq d(v)+1$ 为了 $(u, v) \in E$.
一个 $\operatorname{arc}(u, v)$ 被认为是可接受的，如果 $d(u)=d(v)+1$ 和 $c(u, v)>0$. 请注意，如果我们考虑残差 图，则 $c(u, v)$ 应视为更新容量。

## 计算机代写|组合优化代写Combinatorial optimization代考|Simplex Algorithm

LP是具有线性目标函数和等式和不等式约束系统的优化问题。下面是一个例子:

$$x \geq 0, y \geq 0 .$$

$$\max z=c x \text { s.t. } A x=b x \geq 0,$$

maximize $\quad z=4 x+5 y$ subject to $\quad 2 x+3 y+w=60 \quad x \geq 0, y \geq 0, w \geq 0$.

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