### Abstract

In the simultaneous Max-Cut problem, we are given k weighted graphs on the same set of n vertices, and the goal is to find a cut of the vertex set so that the minimum, over the k graphs, of the cut value is as large as possible. Previous work [BKS15] gave a polynomial time algorithm which achieved an approximation factor of 1-2- op1q for this problem (and an approximation factor of 1-2 + ϵk in the unweighted case, where "k Ñ 0 as k Ñ 8). In this work, we give a polynomial time approximation algorithm for simultaneous Max-Cut with an approximation factor of 0:8780 (for all constant k). The natural SDP formulation for simultaneous Max-Cut was shown to have an integrality gap of 1-2+ϵk in [BKS15]. In achieving the better approximation guarantee, we use a stronger Sum-of-Squares hierarchy SDP relaxation and a rounding algorithm based on Raghavendra-Tan [RT12], in addition to techniques from [BKS15].

Original language | English (US) |
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Title of host publication | 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 |

Publisher | Association for Computing Machinery |

Pages | 1407-1425 |

Number of pages | 19 |

ISBN (Electronic) | 9781611975031 |

DOIs | |

State | Published - Jan 1 2018 |

Event | 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 - New Orleans, United States Duration: Jan 7 2018 → Jan 10 2018 |

### Other

Other | 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 |
---|---|

Country | United States |

City | New Orleans |

Period | 1/7/18 → 1/10/18 |

### Fingerprint

### ASJC Scopus subject areas

- Software
- Mathematics(all)

### Cite this

*29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018*(pp. 1407-1425). Association for Computing Machinery. https://doi.org/10.1137/1.9781611975031.93

**Near-optimal approximation algorithm for simultaneous Max-cut.** / Bhangale, Amey; Khot, Subhash; Kopparty, Swastik; Sachdeva, Sushant; Thiruvenkatachari, Devanathan.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018.*Association for Computing Machinery, pp. 1407-1425, 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, United States, 1/7/18. https://doi.org/10.1137/1.9781611975031.93

}

TY - GEN

T1 - Near-optimal approximation algorithm for simultaneous Max-cut

AU - Bhangale, Amey

AU - Khot, Subhash

AU - Kopparty, Swastik

AU - Sachdeva, Sushant

AU - Thiruvenkatachari, Devanathan

PY - 2018/1/1

Y1 - 2018/1/1

N2 - In the simultaneous Max-Cut problem, we are given k weighted graphs on the same set of n vertices, and the goal is to find a cut of the vertex set so that the minimum, over the k graphs, of the cut value is as large as possible. Previous work [BKS15] gave a polynomial time algorithm which achieved an approximation factor of 1-2- op1q for this problem (and an approximation factor of 1-2 + ϵk in the unweighted case, where "k Ñ 0 as k Ñ 8). In this work, we give a polynomial time approximation algorithm for simultaneous Max-Cut with an approximation factor of 0:8780 (for all constant k). The natural SDP formulation for simultaneous Max-Cut was shown to have an integrality gap of 1-2+ϵk in [BKS15]. In achieving the better approximation guarantee, we use a stronger Sum-of-Squares hierarchy SDP relaxation and a rounding algorithm based on Raghavendra-Tan [RT12], in addition to techniques from [BKS15].

AB - In the simultaneous Max-Cut problem, we are given k weighted graphs on the same set of n vertices, and the goal is to find a cut of the vertex set so that the minimum, over the k graphs, of the cut value is as large as possible. Previous work [BKS15] gave a polynomial time algorithm which achieved an approximation factor of 1-2- op1q for this problem (and an approximation factor of 1-2 + ϵk in the unweighted case, where "k Ñ 0 as k Ñ 8). In this work, we give a polynomial time approximation algorithm for simultaneous Max-Cut with an approximation factor of 0:8780 (for all constant k). The natural SDP formulation for simultaneous Max-Cut was shown to have an integrality gap of 1-2+ϵk in [BKS15]. In achieving the better approximation guarantee, we use a stronger Sum-of-Squares hierarchy SDP relaxation and a rounding algorithm based on Raghavendra-Tan [RT12], in addition to techniques from [BKS15].

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U2 - 10.1137/1.9781611975031.93

DO - 10.1137/1.9781611975031.93

M3 - Conference contribution

AN - SCOPUS:85045562479

SP - 1407

EP - 1425

BT - 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018

PB - Association for Computing Machinery

ER -