# 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.