Max-Cut with Multiple Cardinality Constraints

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 V₁, …, V_c, and cardinality parameters k₁, …, k_c, the goal is to select a set S ⊆ V such that |S ∩ V_i| = k_i for each i ∈ [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 - ε)-approximation algorithm for the problem when c = O(1). The algorithm runs in time O(min{k/ε, n}^poly(c/ε) + poly(n)), where k = ∑_{i∈[c]} k_i and n = |V|. This improves upon the (1/2 + ε₀)-approximation of Feige and Langberg (2001) for Max-Cut_k (the special case when c = 1, k₁ = k), and generalizes the (0.858 - ε)-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.