Better Balance by Being Biased

Abstract

Recently, Raghavendra and Tan (SODA 2012) gave a 0.85-approximation algorithm for the M ax B isection problem. We improve their algorithm to a 0.8776-approximation. As M ax B isection is hard to approximate within α GW + ε ≈ 0.8786 under the Unique Games Conjecture (UGC), our algorithm is nearly optimal. We conjecture that M ax B isection is approximable within α GW − ε, that is, that the bisection constraint (essentially) does not make M ax C ut harder. We also obtain an optimal algorithm (assuming the UGC) for the analogous variant of M ax 2-S at . Our approximation ratio for this problem exactly matches the optimal approximation ratio for M ax 2-S at , that is, α LLZ + ε ≈ 0.9401, showing that the bisection constraint does not make M ax 2-S at harder. This improves on a 0.93-approximation for this problem from Raghavendra and Tan.