research-article

The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative-Type Metrics into ℓ₁

Authors:

Subhash A. Khot,

Nisheeth K. VishnoiAuthors Info & Claims

Journal of the ACM (JACM), Volume 62, Issue 1

Article No.: 8, Pages 1 - 39

https://doi.org/10.1145/2629614

Published: 02 March 2015 Publication History

Abstract

In this article, we disprove a conjecture of Goemans and Linial; namely, that every negative type metric embeds into ℓ₁ with constant distortion. We show that for an arbitrarily small constant δ > 0, for all large enough n, there is an n-point negative type metric which requires distortion at least (log log n)^1/6-δ to embed into ℓ₁. Surprisingly, our construction is inspired by the Unique Games Conjecture (UGC), establishing a previously unsuspected connection between probabilistically checkable proof systems (PCPs) and the theory of metric embeddings. We first prove that the UGC implies a super-constant hardness result for the (nonuniform) SPARSESTCUT problem. Though this hardness result relies on the UGC, we demonstrate, nevertheless, that the corresponding PCP reduction can be used to construct an “integrality gap instance” for SPARSESTCUT. Towards this, we first construct an integrality gap instance for a natural SDP relaxation of UNIQUEGAMES. Then we “simulate” the PCP reduction and “translate” the integrality gap instance of UNIQUEGAMES to an integrality gap instance of SPARSESTCUT. This enables us to prove a (log log n)^1/6-δ integrality gap for SPARSESTCUT, which is known to be equivalent to the metric embedding lower bound.

References

[1]

Amit Agarwal, Moses Charikar, Konstantin Makarychev, and Yury Makarychev. 2005. O(√ log n) approximation algorithms for min-UnCut, min-2CNF deletion, and directed cut problems. In Proceedings of the ACM Symposium on the Theory of Computing. 573--581.

Digital Library

[2]

Sanjeev Arora, Boaz Barak, and David Steurer. 2010. Subexponential algorithms for unique games and related problems. In Proceedings of FOCS. 563--572.

Digital Library

[3]

Sanjeev Arora, Eli Berger, Elad Hazan, Guy Kindler, and Muli Safra. 2005. On non-approximability for quadratic programs. In Proceedings of FOCS. 206--215.

Digital Library

[4]

Sanjeev Arora, Subhash Khot, Alexandra Kolla, David Steurer, Madhur Tulsiani, and Nisheeth K. Vishnoi. 2008. Unique games on expanding constraint graphs are easy. In Proceedings of STOC. 21--28.

Digital Library

[5]

Sanjeev Arora, James R. Lee, and Assaf Naor. 2007. Fréchet embeddings of negative type metrics. Disc. Computat. Geom. 38, 4, 726--739.

Digital Library

[6]

Sanjeev Arora, Satish Rao, and Umesh V. Vazirani. 2009. Expander flows, geometric embeddings and graph partitioning. J. ACM 56, 2.

Digital Library

[7]

Yonatan Aumann and Yuval Rabani. 1998. An O(log k) approximate min-cut max-flow theorem and approximation algorithm. SIAM J. Comput. 27, 1, 291--301.

Digital Library

[8]

Tim Austin, Assaf Naor, and Alain Valette. 2010. The Euclidean distortion of the lamplighter group. Disc. Computat. Geom. 44, 1, 55--74.

Digital Library

[9]

Boaz Barak, Parikshit Gopalan, Johan Håstad, Raghu Meka, Prasad Raghavendra, and David Steurer. 2012. Making the long code shorter. In Proceedings of FOCS. 370--379.

Digital Library

[10]

Mihir Bellare, Oded Goldreich, and Madhu Sudan. 1998. Free bits, PCPs, and non-approximability -- towards tight results. In SIAM J. Comput. 422--431.

Digital Library

[11]

Jean Bourgain. 1985. On Lipschitz embeddings of finite metric spaces in Hilbert space. Israel J. Math. 52, 46--52.

[12]

Jean Bourgain. 2002. On the distribution of the Fourier spectrum of Boolean functions. Isr. J. Math. 131, 269--276.

[13]

Shuchi Chawla, Anupam Gupta, and Harald Räcke. 2008. Embeddings of negative-type metrics and an improved approximation to generalized sparsest cut. ACM Trans. Algor. 4, 2.

Digital Library

[14]

Shuchi Chawla, Robert Krauthgamer, Ravi Kumar, Yuval Rabani, and D. Sivakumar. 2006. On the hardness of approximating multicut and sparsest-cut. Computat. Complex. 15, 2, 94--114.

Digital Library

[15]

Jeff Cheeger and Bruce Kleiner. 2010. Differentiating maps into L₁, and the geometry of BV functions. Ann. Math. 171, 1347--1385.

[16]

Jeff Cheeger, Bruce Kleiner, and Assaf Naor. 2009. A (log n)^Ω(1) integrality gap for the sparsest cut SDP. In Proceedings of FOCS. 555--564.

Digital Library

[17]

Nikhil R. Devanur, Subhash Khot, Rishi Saket, and Nisheeth K. Vishnoi. 2006. Integrality gaps for sparsest cut and minimum linear arrangement problems. In Proceedings of STOC. 537--546.

Digital Library

[18]

M. Deza and Monique Laurent. 1997. Geometry of Cuts and Metrics. Springer-Verlag, New York.

Digital Library

[19]

Peter Enflo. 1969. On the non-existence of uniform homeomorphism between L_p spaces. Arkiv. Mat. 8, 103--105.

[20]

Uriel Feige, MohammadTaghi Hajiaghayi, and James R. Lee. 2008. Improved approximation algorithms for minimum weight vertex separators. SIAM J. Comput. 38, 2, 629--657.

Digital Library

[21]

Uriel Feige and László Lovász. 2002. Two-prover one-round proof systems, their power and their problems. In Proceedings of the ACM Symposium on the Theory of Computing. 733--744.

Digital Library

[22]

Uriel Feige and Gideon Schechtman. 2002. On the optimality of the random hyperplane rounding technique for MAX CUT. Random Struct. Algor. 20, 3, 403--440.

Digital Library

[23]

Michel X. Goemans. 1997. Semidefinite programming in combinatorial optimization. Math. Program. 79, 143--161.

Digital Library

[24]

Michel X. Goemans and David P. Williamson. 1995. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 6, 1115--1145.

Digital Library

[25]

Jeff Kahn, Gil Kalai, and Nathan Linial. 1988. The influence of variables on Boolean functions (extended abstract). In Proceedings of FOCS. 68--80.

Digital Library

[26]

Daniel M. Kane and Raghu Meka. 2013. A PRG for Lipschitz functions of polynomials with applications to sparsest cut. In Proceedings of the ACM Symposium on the Theory of Computing.

Digital Library

[27]

George Karakostas. 2009. A better approximation ratio for the vertex cover problem. ACM Trans. Algor. 5, 4.

Digital Library

[28]

Subhash Khot. 2002. On the power of unique 2-prover 1-round games. In Proceedings of the ACM Symposium on the Theory of Computing. 767--775.

Digital Library

[29]

Subhash Khot and Assaf Naor. 2006. Nonembeddability theorems via Fourier analysis. Math. Ann. 334, 4, 821--852.

[30]

Subhash Khot and Ryan O'Donnell. 2009. SDP gaps and UGC-hardness for Max-Cut-Gain. Theory Comput. 5, 1, 83--117.

[31]

Subhash Khot and Rishi Saket. 2009. SDP integrality gaps with local &ell;₁-embeddability. In Proceedings of FOCS. 565--574.

Digital Library

[32]

Subhash Khot and Nisheeth K. Vishnoi. 2005. The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into &ell;₁. In Proceedings of FOCS. 53--62.

Digital Library

[33]

Robert Krauthgamer and Yuval Rabani. 2009. Improved lower bounds for embeddings into L₁. SIAM J. Comput. 38, 6, 2487--2498.

Digital Library

[34]

James R. Lee and Assaf Naor. 2006. L_p metrics on the Heisenberg group and the Goemans-Linial conjecture. In Proceedings of FOCS. 99--108.

Digital Library

[35]

Frank Thomson Leighton and Satish Rao. 1999. Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. J. ACM 46, 6, 787--832.

Digital Library

[36]

Nathan Linial. 2002. Finite metric spaces: Combinatorics, geometry and algorithms. In Proceedings of the International Congress of Mathematicians. 573--586.

Digital Library

[37]

Nathan Linial, Eran London, and Yuri Rabinovich. 1995. The geometry of graphs and some of its algorithmic applications. Combinatorica 15, 2, 215--245.

[38]

Jiri Matousek. 2002. Lectures on Discrete Geometry. Graduate Texts in Mathematics, vol. 212. Springer-Verlag.

Digital Library

[39]

Assaf Naor, Yuval Rabani, and Alistair Sinclair. 2005. Quasisymmetric embeddings, the observable diameter, and expansion properties of graphs. J. Funct. Anal. 227, 2, 273--303.

[40]

R. O'Donnell. 2004. Computational aspects of noise sensitivity. Ph.D. Dissertation. MIT.

Digital Library

[41]

Prasad Raghavendra. 2008. Optimal algorithms and inapproximability results for every CSP&quest; In Proceedings of STOC. 245--254.

Digital Library

[42]

Prasad Raghavendra and David Steurer. 2009. Integrality gaps for strong SDP relaxations of UNIQUE GAMES. In Proceedings of FOCS. 575--585.

Digital Library

[43]

Prasad Raghavendra and David Steurer. 2010. Graph expansion and the unique games conjecture. In Proceedings of STOC. 755--764.

Digital Library

[44]

Gideon Schechtman. 2003. Handbook of the Geometry of Banach Spaces. Vol. 2, North Holland, Chapter Concentration results and applications.

[45]

Nisheeth K. Vishnoi. 2013. Lx = b. Foundations and Trends® in Theoretical Computer Science. Vol. 8, 1--2, 1--141. http://dx.doi.org/10.1561/0400000054.

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Cohen-Addad VMömke TVerdugo V(2024)A 2-approximation for the bounded treewidth sparsest cut problem in TimeMathematical Programming: Series A and B10.1007/s10107-023-02044-1206:1-2(479-495)Online publication date: 1-Jul-2024
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Index Terms

The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative-Type Metrics into ℓ₁
1. Theory of computation
  1. Design and analysis of algorithms

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Published In

cover image Journal of the ACM

Journal of the ACM Volume 62, Issue 1

February 2015

264 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/2742144

Editor:
Victor Vianu
University of California, San Diego

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Copyright © 2015 ACM.

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Association for Computing Machinery

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Publication History

Published: 02 March 2015

Accepted: 01 February 2014

Revised: 01 July 2013

Received: 01 January 2006

Published in JACM Volume 62, Issue 1

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https://dl.acm.org/doi/10.1145/3618260.3649747
Vijendran VDas AKoh DAssad SLam P(2024)An expressive ansatz for low-depth quantum approximate optimisationQuantum Science and Technology10.1088/2058-9565/ad200a9:2(025010)Online publication date: 13-Feb-2024
https://doi.org/10.1088/2058-9565/ad200a
Cohen-Addad VMömke TVerdugo V(2024)A 2-approximation for the bounded treewidth sparsest cut problem in TimeMathematical Programming: Series A and B10.1007/s10107-023-02044-1206:1-2(479-495)Online publication date: 1-Jul-2024
https://dl.acm.org/doi/10.1007/s10107-023-02044-1
Jones CPotechin ARajendran GXu JSaha BServedio R(2023)Sum-of-Squares Lower Bounds for Densest k-SubgraphProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585221(84-95)Online publication date: 2-Jun-2023
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