最大流应用(Maximum Flow Application)

1. 二分图匹配(Bipartite Matching)

1.1 匹配(Matching)

Def. Given an undirected graph $G = (V, E)$, subset of edges $M ⊆ E$ is a matching if each node appears in at most one edge in $M$.

定义. 给定一个无向图$G = (V, E)$, 如果一个边的集合$M \subseteq E$并且每个顶点最多只出现在一条$M$中的边中, 则称$M$是一个匹配.

Max matching problem. Given a graph $G$, find a max-cardinality matching.

最大匹配问题. 给定一个无向图$G$, 找到一个集合元素最多的匹配.

1.2 二分图匹配(Bipartite matching)

Def. A graph G is bipartite if the nodes can be partitioned into two subsets $L$ and $R$ such that every edge connects a node in $L$ with a node in $R$.

Bipartite matching problem. Given a bipartite graph $G = (L ∪ R, E)$, find a max cardinality matching.

max-flow formulation. Create digraph $G' = (L ∪ R ∪ \set{s, t}, E')$:

Direct all edges from $L$ to $R$, and assign infinite (or unit) capacity.
Add unit-capacity edges from $s$ to each node in $L$.
Add unit-capacity edges from each node in $R$ to $t$.

Theorem. 1–1 correspondence between matchings of cardinality $k$ in $G$ and integral flows of value $k$ in $G'$.

1.3 二分图完美匹配(Perfect matchings in bipartite graphs)

Def. Given a graph $G = (V, E)$, a subset of edges $M ⊆ E$ is a perfect matching if each node appears in exactly one edge in $M$.

Notation. Let $S$ be a subset of nodes, and let $N(S)$ be the set of nodes adjacent to nodes in $S$.

Observation. If a bipartite graph $G = (L ∪ R, E)$ has a perfect matching, then $|N(S)|\geq|S|$ for all subsets $S \subseteq L$.

Theorem. [Frobenius 1917, Hall 1935] Let $G = (L ∪ R, E)$ be a bipartite graph with $|L| = |R|$. Then, graph G has a perfect matching iff $|N(S)|\geq|S|$ for all subsets $S \subseteq L$.

2. 不相交路径(Edge-disjoint Paths)

2.1 有向图中的不相交路径(Edge-disjoint Paths in Directed Graphs

2.1.1 问题描述(Problem Description)

Def. Two paths are edge-disjoint if they have no edge in common.

定义. 如果两个路径没有公共边, 则认为他们是不相交路径

Edge-disjoint paths problem. Given a digraph $G = (V, E)$ and two nodes $s$ and $t$, find the max number of edge-disjoint $s↝t$ paths.

不想交路径问题 给定一个有向图$G = (V, E)$和两个节点$s$, $t$, 找到其间不相交路径的最大数目.

2.1.2 解法(Solution)

Max-flow formulation. Assign unit capacity to every edge, and define it as $G^\prime$

Integral flow theorem. If each edge in a flow network has integral capacity, then there exists an integral maximal flow.

1–1 correspondence Theorem. 1–1 correspondence between k edge-disjoint $s↝t$ paths in $G$ and integral flows of value $k$ in $G'$.

Integral flow theorem. $\Rightarrow$ there exists a max flow $f^*$ in $G^\prime$ that is integral.
1–1 correspondence Theorem. $\Rightarrow$ $f^*$ corresponds to max number of edge-disjoint $s↝t$ paths in $G$.

2.1.3 补充(Addition)

Menger's Theorem. The max number of edge-disjoint $s↝t$ paths equals the min number of edges whose removal disconnects $t$ from $s$.

2.2 无向图中的不相交路径(Edge-disjoint Paths in Undirected Graphs)

2.2.1 问题描述(Problem Description)

Given a graph $G = (V, E)$ and two nodes $s$ and $t$, find the max number of edge-disjoint $s↝t$ paths.

2.2.2 解法(Solution)

Max-flow formulation. Replace each edge with two antiparallel edges and assign unit capacity to every edge.

Observation. Two paths $P1$ and $P2$ may be edge-disjoint in the digraph but not edge-disjoint in the undirected graph

Lemma. In any flow network, there exists a maximum flow $f$ in which for each pair of antiparallel edges $e$ and $e'$ : either $f(e) = 0$ or $f (e') = 0$ or both. Moreover, integrality theorem still holds.

Theorem. Max number of edge-disjoint $s↝t$ paths = value of max flow.

3. 最大流问题扩展(Extensions to Max Flow)

3.1 多源点多汇点(Multiple sources and sinks)

Def. Given a digraph $G = (V, E)$ with edge capacities $c(e) ≥ 0$ and multiple source nodes and multiple sink nodes, find max flow that can be sent from the source nodes to the sink nodes.

max-flow formulation. Define new digraph as $G'$:

Add a new source node $s$ and sink node $t$.
For each original source node $s_i$ add edge $(s, s_i)$ with capacity $\infty$.
For each original sink node $t_j$, add edge $(t_j, t)$ with capacity $\infty$.

Claim. 1–1 correspondence between flows in $G$ and $G'$.

3.2 循环(Circulation)

3.2.1 供给需求循环(Circulation with supplies and demands)

Def. Given a digraph $G = (V, E)$ with edge capacities $c(e) ≥ 0$ and node demands $d(v)$, a circulation is a function $f(e)$ that satisfies:

For each $e\in E$: $0\leq f(e)\leq c(e)$
For each $v\in V$: $\textstyle \sum_{e \space in \space to \space v}f(e) - \sum_{e \space out \space of \space v}f(e)=d(v)$

max-flow formulation. Define new digraph as $G'$:

Add new source $s$ and sink $t$.
For each $v$ with $d(v) < 0$, add $edge (s, v)$ with capacity $−d(v)$.
For each $v$ with $d(v) > 0$, add $edge (v, t)$ with capacity $d(v)$.

Claim. $G$ has circulation iff $G'$ has max flow of value $\displaystyle D = \sum_{v:d(v)>0}d(v) = \sum_{v:d(v)<0}-d(v)$

3.2.2 带下界的供给需求循环(Circulation with supplies, demands, and lower bounds

Def. Given a digraph $G = (V, E)$ with edge capacities $c(e) ≥ 0$, lower bounds ℓ(e) ≥ 0, and node demands $d(v)$, a circulation is a function $f(e)$ that satisfies:

For each $e\in E$: $\mathscr l(e)\leq f(e)\leq c(e)$
For each $v\in V$: $\textstyle \sum_{\text{e in to v}}f(e) - \sum_{\text{e out of v}}f(e)=d(v)$

Max-flow formulation. Model lower bounds as circulation with demands.

Theorem. There exists a circulation in $G$ iff there exists a circulation in $G'$.

Moreover, if all demands, capacities, and lower bounds in $G$ are integers, then there exists a circulation in $G$ that is integer-valued.

4. 调查设计问题(Survey Design)

4.1 问题描述(Problem Description)

Design survey asking $n_1$ consumers about $n_2$ products.
Can survey consumer $i$ about product $j$ only if they own it.
Ask consumer $i$ between $c_i$ and $c_i'$ questions.
Ask between $p_j$ and $p_j'$ consumers about product $j$.

Goal. Design a survey that meets these specs, if possible.

4.2 解法(Solution)

Max-flow formulation. Model as a circulation problem with lower bounds (all supplies and demands are 0):

Add edge $(i, j)$ if consumer $i$ owns product $j$.
Add edge from s to consumer $j$.
Add edge from product $i$ to $t$.
Add edge from $t$ to $s$.
All demands = 0.

Integer circulation $\Leftrightarrow$ feasible survey design.