### Abstract

Given a graph with maximum cut of (fractional) size c, the Goemans-Williamson [GW95] semidefinite programming (SDP) algorithm is guaranteed to find a cut of size .878 &· c. However this guarantee becomes trivial when c is near 1/2, since a random cut has expected size 1/2. Recently, Charikar and Worth [CW04] (analyzing an algorithm of Feige and Langberg [FL01]) showed that given a graph with maximum cut 1/2 + ε, one can find a cut of size 1/2 + ω (ε / log(1/ε)). The main contribution of our paper is twofold: 1. We give a natural 1/2 + ε vs. 1/2 + O(ε/ log(1/ε)) SDP gap for MAXCUT in Gaussian space. This shows that the SDP-rounding algorithm of Charikar-Worth is essentially best possible. Further, the "s-linear rounding functions" used in [CW04, FL01] arise as optimizers in our analysis, somewhat confirming a suggestion of [FL01]. 2. We show how this SDP gap can be translated into a Long Code test with the same parameters. This implies that beating the Charikar-Worth guarantee with any efficient algorithm is NP-hard, assuming the Unique Games Conjecture (UGC) [Kho02]. We view this result as essentially settling the approximability of MAXCUT, assuming UGC. Building on (1) we show how "randomness reduction" on related SDP gaps for the QUADRATICPRO-GRAMMlNG programming problem lets us make the Ω(log(1/ε)) gap as large as Ω(log n) for n-vertex graphs. In addition to optimally answering an open question of [AMMN06], this technique may prove useful for other SDP gap problems. Finally, illustrating the generality of our technique in (2), we also show how to translate Reeds's [Ree93] SDP gap for the Grothendieck Inequality into a UGC-hardness result for computing the ∥·∥ _{∞→1} norm of a matrix.

Original language | English (US) |
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Title of host publication | 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2006 |

Pages | 217-226 |

Number of pages | 10 |

DOIs | |

State | Published - 2006 |

Event | 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2006 - Berkeley, CA, United States Duration: Oct 21 2006 → Oct 24 2006 |

### Other

Other | 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2006 |
---|---|

Country | United States |

City | Berkeley, CA |

Period | 10/21/06 → 10/24/06 |

### Fingerprint

### ASJC Scopus subject areas

- Engineering(all)

### Cite this

*47th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2006*(pp. 217-226). [4031358] https://doi.org/10.1109/FOCS.2006.67

**SDP gaps and UGC-hardness for MAXCUTGAIN.** / Khot, Subhash; O'Donneil, Ryan.

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

*47th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2006.*, 4031358, pp. 217-226, 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2006, Berkeley, CA, United States, 10/21/06. https://doi.org/10.1109/FOCS.2006.67

}

TY - GEN

T1 - SDP gaps and UGC-hardness for MAXCUTGAIN

AU - Khot, Subhash

AU - O'Donneil, Ryan

PY - 2006

Y1 - 2006

N2 - Given a graph with maximum cut of (fractional) size c, the Goemans-Williamson [GW95] semidefinite programming (SDP) algorithm is guaranteed to find a cut of size .878 &· c. However this guarantee becomes trivial when c is near 1/2, since a random cut has expected size 1/2. Recently, Charikar and Worth [CW04] (analyzing an algorithm of Feige and Langberg [FL01]) showed that given a graph with maximum cut 1/2 + ε, one can find a cut of size 1/2 + ω (ε / log(1/ε)). The main contribution of our paper is twofold: 1. We give a natural 1/2 + ε vs. 1/2 + O(ε/ log(1/ε)) SDP gap for MAXCUT in Gaussian space. This shows that the SDP-rounding algorithm of Charikar-Worth is essentially best possible. Further, the "s-linear rounding functions" used in [CW04, FL01] arise as optimizers in our analysis, somewhat confirming a suggestion of [FL01]. 2. We show how this SDP gap can be translated into a Long Code test with the same parameters. This implies that beating the Charikar-Worth guarantee with any efficient algorithm is NP-hard, assuming the Unique Games Conjecture (UGC) [Kho02]. We view this result as essentially settling the approximability of MAXCUT, assuming UGC. Building on (1) we show how "randomness reduction" on related SDP gaps for the QUADRATICPRO-GRAMMlNG programming problem lets us make the Ω(log(1/ε)) gap as large as Ω(log n) for n-vertex graphs. In addition to optimally answering an open question of [AMMN06], this technique may prove useful for other SDP gap problems. Finally, illustrating the generality of our technique in (2), we also show how to translate Reeds's [Ree93] SDP gap for the Grothendieck Inequality into a UGC-hardness result for computing the ∥·∥ ∞→1 norm of a matrix.

AB - Given a graph with maximum cut of (fractional) size c, the Goemans-Williamson [GW95] semidefinite programming (SDP) algorithm is guaranteed to find a cut of size .878 &· c. However this guarantee becomes trivial when c is near 1/2, since a random cut has expected size 1/2. Recently, Charikar and Worth [CW04] (analyzing an algorithm of Feige and Langberg [FL01]) showed that given a graph with maximum cut 1/2 + ε, one can find a cut of size 1/2 + ω (ε / log(1/ε)). The main contribution of our paper is twofold: 1. We give a natural 1/2 + ε vs. 1/2 + O(ε/ log(1/ε)) SDP gap for MAXCUT in Gaussian space. This shows that the SDP-rounding algorithm of Charikar-Worth is essentially best possible. Further, the "s-linear rounding functions" used in [CW04, FL01] arise as optimizers in our analysis, somewhat confirming a suggestion of [FL01]. 2. We show how this SDP gap can be translated into a Long Code test with the same parameters. This implies that beating the Charikar-Worth guarantee with any efficient algorithm is NP-hard, assuming the Unique Games Conjecture (UGC) [Kho02]. We view this result as essentially settling the approximability of MAXCUT, assuming UGC. Building on (1) we show how "randomness reduction" on related SDP gaps for the QUADRATICPRO-GRAMMlNG programming problem lets us make the Ω(log(1/ε)) gap as large as Ω(log n) for n-vertex graphs. In addition to optimally answering an open question of [AMMN06], this technique may prove useful for other SDP gap problems. Finally, illustrating the generality of our technique in (2), we also show how to translate Reeds's [Ree93] SDP gap for the Grothendieck Inequality into a UGC-hardness result for computing the ∥·∥ ∞→1 norm of a matrix.

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U2 - 10.1109/FOCS.2006.67

DO - 10.1109/FOCS.2006.67

M3 - Conference contribution

AN - SCOPUS:35448938376

SN - 0769527205

SN - 9780769527208

SP - 217

EP - 226

BT - 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2006

ER -