# Global min-cut with parity constraint on the edges

In a discussion with Patrick Lin, a nice problem was born.

Let $\delta(S)$ to be the set of edges with exactly one endpoint in $S$. $\delta^-(S)$ to be the set of edges with its head in $S$ and tail in $V\setminus S$. Given a non-negative weighted graph, we define the cut function $f:2^V\to \R^+$ to be $f(S) = \sum_{e\in \delta(S)} w(e)$. For directed graphs, $f(S) = \sum_{e\in \delta^-(S)} w(e)$. $f(S)$ is called the value of the cut $S$.

Let $k$ be a constant, we consider the following problem.

Give a graph and $k$ set of edges $F_1,\ldots,F_k$, $a_1,\ldots,a_k,b$. Find a cut $S$ satisfies that $|\delta(S)\cap F_i|\equiv a_i \pmod b$ for all $i$, and the value is minimized.

We will try to reduce this problem to the following

Given $T_1,\ldots,T_k$ and a submodular function $f$. Find a set $S$ such that $|T_i\cap S| \equiv a_i\pmod b_i$, and $f(S)$ is minimized.

The above problem is known as submodular minimization under congruence constraints. It is known to be solvable in polynomial time under certain conditions on the $b_i$’s [1]. We sketch the reductions.

# 1 Undirected case

In the undirected case, we only consider when $b=2$. Patrick showed a the following construction. Create a new graph $G'$ as follows. For each $uv$ in $E$, split it into edges $ux$, $xy$, $yv$, $w(ux)=w(yv)=\infty$, and $w(xy)=w(uv)$. Let $T_i$ contains the vertex $x$ and $y$ if $uv\in F_i$.

We now solve the submodular minimization under congruence constraints problem on input $f$, which is the cut function for $G'$, and same $a_1,\ldots,a_k$ and $b_1,\ldots,b_k=2$.

# 2 Directed case

In the directed case, a similar approach works. But now, instead of $\mod 2$, we can do $\mod b$ for any $b$. We consider the same approach.

$(u,v) \in E$ split into $(u,x_1),\ldots,(x_b,v)$ and $w(u,x_1)=w(u,v)$, $w(x_i,x_{i+1})=\infty$, $w(x_b,v)=\infty$. Now, let $T_i$ contain vertices $x_1,\ldots,x_b$ of $uv\in F_i$.

# References

[1] M. Nägele, B. Sudakov, R. Zenklusen, Submodular minimization under congruency constraints, in: Proceedings of the Twenty-Ninth Annual Acm-Siam Symposium on Discrete Algorithms, Society for Industrial; Applied Mathematics, Philadelphia, PA, USA, 2018: pp. 849–866.