Max-Cut with Multiple Cardinality Constraints

2025-07-16

AI Summary

This paper studies the Maximum Cut problem under multiple cardinality constraints: given a graph $G=(V,E)$, a partition $V_1,dots,V_c$ of $V$, and cardinality bounds $k_1,dots,k_c$, find a subset $S subseteq V$ satisfying $|S cap V_i| = k_i$ for all $i$, while maximizing the total weight of cut edges. This NP-hard problem generalizes both the standard Max-Cut and the single-cardinality-constrained variant. We present the first $(0.858-varepsilon)$-approximation algorithm for constant $c = O(1)$, significantly improving prior results that only applied to single-constraint or dense-graph settings. Our approach combines correlated rounding with approximate core construction. We further extend the framework to matroid constraints, yielding a universal $1/2$-approximation algorithm. Additionally, we prove that feasibility testing—determining whether a solution satisfying all $c$ cardinality constraints exists—is NP-complete for general $c$. The algorithms are both theoretically rigorous and practically implementable.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

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 $Oleft(min{k/varepsilon, n}^{poly(c/varepsilon)} + poly(n) ight)$, 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.

Problem

• Max-Cut with multiple cardinality constraints optimization
• Approximation algorithm for Constrained Max-Cut problem
• NP-hardness of feasible solution existence proof

Innovation

• Approximate kernel for Constrained Max-Cut
• Correlation rounding technique enhancement
• Matroid constraint-based approximation algorithm