Simultaneous Max-Cut Is Harder to Approximate Than Max-Cut

Abstract

A systematic study of simultaneous optimization of constraint satisfaction problems was initiated by Bhangale et al. [ICALP, 2015]. The simplest such problem is the simultaneous Max-Cut. Bhangale et al. [SODA, 2018] gave a .878-minimum approximation algorithm for simultaneous Max-Cut which is almost optimal assuming the Unique Games Conjecture (UGC). For single instance Max-Cut, Goemans-Williamson [JACM, 1995] gave an ?_GW-approximation algorithm where ?_GW ? .87856720... which is optimal assuming the UGC. It was left open whether one can achieve an ?_GW-minimum approximation algorithm for simultaneous Max-Cut. We answer the question by showing that there exists an absolute constant ?? ? 10^{-5} such that it is NP-hard to get an (?_GW- ??)-minimum approximation for simultaneous Max-Cut assuming the Unique Games Conjecture