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TR19-148 | 1st November 2019 21:01

#### Simultaneous Max-Cut is harder to approximate than Max-Cut

TR19-148
Authors: Amey Bhangale, Subhash Khot
Publication: 3rd November 2019 11:17
A systematic study of simultaneous optimization of constraint satisfaction problems was initiated in [BKS15]. The simplest such problem is the simultaneous Max-Cut. [BKKST18] gave a $.878$-minimum approximation algorithm for simultaneous Max-Cut which is {\em almost optimal} assuming the Unique Games Conjecture (UGC). For a single instance Max-Cut, [GW95] gave an $\alpha_{GW}$-approximation algorithm where $\alpha_{GW}\approx .87856720...$ which is {\em optimal} assuming the UGC.
It was left open whether one can achieve an $\alpha_{GW}$-minimum approximation algorithm for simultaneous Max-Cut. We answer the question by showing that there exists an absolute constant $\varepsilon_0\geq 10^{-5}$ such that it is NP-hard to get an $(\alpha_{GW}- \varepsilon_0)$-minimum approximation for simultaneous Max-Cut assuming the UGC.