Max-Cut with Multiple Cardinality Constraints
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915395101786112 |
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| author | Makarychev, Yury Pittu, Madhusudhan Reddy Vakilian, Ali |
| author_facet | Makarychev, Yury Pittu, Madhusudhan Reddy Vakilian, Ali |
| contents | 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 $\min\{k,n-k\}=Ω(n)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_12607 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Max-Cut with Multiple Cardinality Constraints Makarychev, Yury Pittu, Madhusudhan Reddy Vakilian, Ali Data Structures and Algorithms 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 $\min\{k,n-k\}=Ω(n)$ 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. |
| title | Max-Cut with Multiple Cardinality Constraints |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2507.12607 |