Reduce Min-Cut to 0/1 Integer Program

Question

Given an undirected, weighted graph $G=(V,E)$ and two nodes $s,t \in V$ and weight function $w: E \rightarrow \mathbb{N}$. The weight of a (s,t)-cut $ (U, U^C)$ is given by:

$$ w(U,U^C) := \sum_{\{i,j\}\in V :i\in U \land j\in U^C} w(\{i,j\}) $$

Reduce $$ MinCUT:= \{<G,c,s,t,k>\mid \text{There is a (s,t)-cut with weight} \leq k\} $$ to $$ BinaryProgram:= \{<A,b> \mid \text{There is a vector } y\in\{0,1\}^* \text{ such that } Ay \leq b \} $$

I know i have to find a matrix $A$ and a vector $b$ which implement the following constraints:

• The weight of the edges which cross the sets of the cut must be $\leq k$
• Node $s$ must be in set $U$ and node $t$ must be in set $U^C$
• Every node must either be in $U$ or $U^C$

Can somebody give me a hint how implement these constraints into a matrix ?

Answer 1

Hint: Let's try to encode the constraints using two types of variables: $a_v$ for each vertex $v$, and $b_{uv}$ for each edge $u,v$. The variable $a_v$ encodes which part of the cut vertex $v$ belongs to. The variables $b_{uv}$ equals 1 if the edge $(u,v)$ is cut.

We want to enforce the following constraints:

• $a_s = 0$ and $a_t = 1$.
• $b_{uv} = 1$ if $a_u = 1$ and $a_v = 0$ or if $a_u = 0$ and $a_v = 1$.
• $\sum_{uv \in E} w(u,v) b_{uv} \leq k$.

Try to encode all these constraints as inequality constraints.