Reference Work Entry

Encyclopedia of Algorithms

pp 1-99

Max Cut

1994; Goemans, Williamson 1995; Goemans, Williamson
  • Alantha NewmanAffiliated withDepartment of Algorithms and Complexity, Max-Planck Institute for Computer Science

Keywords and Synonyms

Maximum bipartite subgraph  

Problem Definition

Given an undirected edge-weighted graph, \( { G=(V,E) } \), the maximum cut problem (Max-Cut) is to find a bipartition of the vertices that maximizes the weight of the edges crossing the partition. If the edge weights are non-negative, then this problem is equivalent to finding a maximum weight subset of the edges that forms a bipartite subgraph, i. e. the maximum bipartite subgraph problem. All results discussed in this article assume non-negative edge weights. Max-Cut is one of Karp's original NP-complete problems [19]. In fact, it is NP-hard to approximate to within a factor better than 16/17[16,33].

For nearly twenty years, the best-known approximation factor for Max-Cut was half, which can be achieved by a very simple algorithm: Form a set S by placing each vertex in S with probability half. Since each edge crosses the cut ...

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