Max-Cut with Multiple Cardinality Constraints

We study the classic Max-Cut problem under multiple cardinality constraints, which we refer to as the Constrained Max-Cut problem. Given a graph $G=(V, E)$, a partition of the vertices into $c$ disjoint parts $V_1, \ldots, V_c$, and cardinality parameters $k_1, \ldots, k_c$, the goal is to select a set $S \subseteq V$ such that $|S \cap V_i| = k_i$ for each $i \in [c]$, maximizing the total weight of edges crossing $S$ (i.e., edges with exactly one endpoint in $S$). By designing an approximate kernel for Constrained Max-Cut and building on the correlation rounding technique of Raghavendra and Tan (2012), we present a $(0.858 - \varepsilon)$-approximation algorithm for the problem when $c = O(1)$. The algorithm runs in time $O\left(\min\{k/\varepsilon, n\}^{\poly(c/\varepsilon)} + \poly(n)\right)$, where $k = \sum_{i \in [c]} k_i$ and $n=|V|$. This improves upon the $(\frac{1}{2} + \varepsilon_0)$-approximation of Feige and Langberg (2001) for $\maxcut_k$ (the special case when $c=1, k_1 = k$), and generalizes the $(0.858 - \varepsilon)$-approximation of Raghavendra and Tan (2012), which only applies when $\min\{k,n-k\}=\Omega(n)$ and does not handle multiple constraints. We also establish that, for general values of $c$, it is NP-hard to determine whether a feasible solution exists that cuts all edges. Finally, we present a $1/2$-approximation algorithm for Max-Cut under an arbitrary matroid constraint.