My problem is mainly from this lecture notes on convex optimization here page4

Consider a s-t Minimum problem, on unweighted undirected graph $G=(V,E)$,we can formalize in following linear integer programming problem

\begin{equation*} \begin{aligned} & \underset{}{\text{minimine}} & & \sum_{u,v\in E}|x_u-x_v| \\ & \text{subject to} & & x_s =1 ,x_t=0 &x_v\in \left\{ 0,1\right\} \forall v \in V \end{aligned} \end{equation*}

then we can relax to:

\begin{aligned} & \underset{}{\text{minimine}} & & \sum_{u,v\in E}|x_u-x_v| \\ & \text{subject to} & & x_s -x_t =1 \end{aligned}

for $0 \leq l \leq 1$ we define $S_l:= \left\{v|x_v \geq l\right\} $ then we have $$\sum_{u,v\in E}|x_u-x_v| \geq\int^{1}_{0}|\delta_l(S_l)|dl$$ where $\delta_l(S_l) $ denotes the crossing edge in $S_l$

how to see this inequality ?


It's important to first understand how $\delta(S_l)$ changes. When $l = 0$, $S_l$ contains all vertices, and when $l = 1$, $S_l$ contains no vertices. In either case, there is no edge going between $S_l$ and $\overline{S_l}$, so $\delta(S_l) = 0$.

For values of $l$ between $0$ and $1$, $\delta(S_l)$ contains exactly the edges with one endpoint in $S_l$ and one endpoint in $\overline{S_l}$. In particular, if we define

$$f_{ij}(l) =\left\{ \begin{array}{ll} 1 & \mbox{if } l \mbox{ is between } x_i \mbox{ and } x_j \\ 0 & \mbox{otherwise.} \end{array} \right.$$

then we have $$|\delta(S_l)| = \sum_{(i, j) \in \delta(S_l)} 1 = \sum_{\substack{(i,j) \in E \\ i \in S_l \\ j \not\in S_l}} 1 = \sum_{(i, j) \in E} f_{ij}(l).$$

Now let's consider one particular edge, from $u$ to $v$. Because $f_{uv}(\cdot)$ is just $0$ everywhere except for on one interval where it takes on the value $1$, the value of $\int_0^1 f_{uv}(l)\,\mathrm{d}l$ is just the length of the interval on which $f_{uv}(l) = 1$, that is, $|x_u - x_v|$. So

$$\sum_{(u, v) \in E} |x_u - x_v| = \sum_{(u, v) \in E} \int_0^1 f_{uv}(l)\,\mathrm{d}l = \int_0^1 \sum_{(u, v) \in E} f_{uv}(l)\,\mathrm{d}l = \int_0^1 |\delta(S_l)|\,\mathrm{d}l.$$


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