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The minimum cut problem for an undirected edge-weighted graph asks us to divide its set of nodes into two blocks while minimizing the weight sum of the cut edges. Here, we introduce a linear-time algorithm to compute near-minimum cuts. Our algorithm is based on cluster contraction using label propagation and Padberg and Rinaldi's contraction heuristics [SIAM Review, 1991]. We give both sequential and shared-memory parallel implementations of our algorithm. Extensive experiments on bothdoi:10.1137/1.9781611975055.5 dblp:conf/alenex/HenzingerN0S18 fatcat:plwjn6z36zfg3klxdrlxzhfmcq