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Title: 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 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).\r\nBy 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.\r\nWe 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.  more » « less
Award ID(s):
2216899 1955173
PAR ID:
10646891
Author(s) / Creator(s):
; ;
Editor(s):
Ene, Alina; Chattopadhyay, Eshan
Publisher / Repository:
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Date Published:
Volume:
353
ISSN:
1868-8969
Page Range / eLocation ID:
13:1-13:21
Subject(s) / Keyword(s):
Maxcut Semi-definite Programming Sum of Squares Hierarchy Theory of computation → Design and analysis of algorithms
Format(s):
Medium: X Size: 21 pages; 884905 bytes Other: application/pdf
Size(s):
21 pages 884905 bytes
Sponsoring Org:
National Science Foundation
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