15-854: Approximations Algorithms Lecturer: Anupam Gupta Topic: Max-cut, Hardness of Approximations Date: 9/14

Abstract

Given an undirected graph G = (V,E), a cut in G is a subset S ⊆ V. Let S = V \S, and let E(S,S) denote the set of edges with one vertex in S and one vertex in S. We will use the term “cut ” to refer variously to the partition (S,S) of the vertices, and also the set of edges E(S,S) crossing between S and S; typically, we will use this term in a way that is unambiguous, and will disambiguate where necessary. The Max-Cut problem is to find the cut S that maximizes |E(S,S)|. In the weighted version of Max-Cut, we are also given a edge weight function w: E → R, and the problem is to find a cut with maximum weight. The Max-Cut problem is NP-hard, and is interesting to contrast with the Min-Cut problem, which is solvable in polynomial time (e.g., by reducing to the s-t min-cut problem, which is dual to the Max-Flow problem, and is solvable by algorithms such as one by Edmonds and Karp [1]. 2.1.1 Approx Max-Cut using Local Search THe general idea in local search algorithms is the same: Start with a solution, find an improving step to make a better solution, and repeat until stuck. In more detail: 1. Start with some arbitrary S0 ⊆ V