# The integrality gap of the goemans-linial SDP relaxation for sparsest cut is at least a constant multiple of √logn

Assaf Naor, Robert Young

Research output: Chapter in Book/Report/Conference proceedingConference contribution

### Abstract

We prove that the integrality gap of the Goemans-Linial semi-definite programming relaxation for the Sparsest Cut Problem is Ω(√logn) on inputs with n vertices, thus matching the previously best known upper bound (log n)1/2+o(1) up to lower-order factors. This statement is a consequence of the following new isoperimetric-type inequality. Consider the 8-regular graph whose vertex set is the 5-dimensional integer grid ℤ5 and where each vertex (a, b, c, d, e) ∈ ℤ5 is connected to the 8 vertices (a ± 1, b, c, d, e), (a, b ± 1, c, d, e), (a, b, c ± 1, d, e ± a), (a, b, c, d ± 1, e ± b). This graph is known as the Cayley graph of the 5-dimensional discrete Heisenberg group. Given Ω ⊆ ℤ5, denote the size of its edge boundary in this graph (a.k.a. the horizontal perimeter of Ω) by |∂hΩ|. For t ∈ N, denote by |∂t vΩ| the number of (a, b, c, d, e) ∈ ℤ5 such that exactly one of the two vectors (a, b, c, d, e), (a, b, c, d, e + f) is in ω. The vertical perimeter of Ω is defined to be |∂vΩ| = √Σ t=1 |∂t vΩ|2/t2. We show that every subset Ω ⊆ ℤ5 satisfies |∂vΩ| = O(|∂hΩ|). This vertical-versus-horizontal isoperimetric inequality yields the above-stated integrality gap for Sparsest Cut and answers several geometric and analytic questions of independent interest. The theorem stated above is the culmination of a program whose aim is to understand the performance of the Goemans-Linial semi-definite program through the embeddability properties of Heisenberg groups. These investigations have mathematical significance even beyond their established relevance to approximation algorithms and combinatorial optimization. In particular they contribute to a range of mathematical disciplines including functional analysis, geometric group theory, harmonic analysis, sub-Riemannian geometry, geometric measure theory, ergodic theory, group representations, and metric differentiation. This article builds on the above cited works, with the "twist" that while those works were equally valid for any finite dimensional Heisenberg group, our result holds for the Heisenberg group of dimension 5 (or higher) but fails for the 3-dimensional Heisenberg group. This insight leads to our core contribution, which is a deduction of an endpoint L1-boundedness of a certain singular integral on ℝ5 from the (local) L2-boundedness of the corresponding singular integral on ℝ3. To do this, we devise a corona-type decomposition of subsets of a Heisenberg group, in the spirit of the construction that David and Semmes performed in ℝn, but with two main conceptual differences (in addition to more technical differences that arise from the peculiarities of the geometry of Heisenberg group). Firstly, the "atoms" of our decomposition are perturbations of intrinsic Lipschitz graphs in the sense of Franchi, Serapioni, and Serra Cassano (plus the requisite "wild" regions that satisfy a Carleson packing condition). Secondly, we control the local overlap of our corona decomposition by using quantitative monotonicity rather than Jones-type β-numbers.

Original language English (US) STOC 2017 - Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing Association for Computing Machinery 564-575 12 Part F128415 9781450345286 https://doi.org/10.1145/3055399.3055413 Published - Jun 19 2017 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017 - Montreal, CanadaDuration: Jun 19 2017 → Jun 23 2017

### Other

Other 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017 Canada Montreal 6/19/17 → 6/23/17

### Fingerprint

Group theory
Decomposition
Functional analysis
Harmonic analysis
Geometry
Combinatorial optimization
Approximation algorithms
Atoms

### Keywords

• Approximation algorithms
• Metric embeddings
• Semidefinite programming
• Sparsest cut problem

• Software

### Cite this

Naor, A., & Young, R. (2017). The integrality gap of the goemans-linial SDP relaxation for sparsest cut is at least a constant multiple of √logn. In STOC 2017 - Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing (Vol. Part F128415, pp. 564-575). Association for Computing Machinery. https://doi.org/10.1145/3055399.3055413
STOC 2017 - Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing. Vol. Part F128415 Association for Computing Machinery, 2017. p. 564-575.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Naor, A & Young, R 2017, The integrality gap of the goemans-linial SDP relaxation for sparsest cut is at least a constant multiple of √logn. in STOC 2017 - Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing. vol. Part F128415, Association for Computing Machinery, pp. 564-575, 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, Canada, 6/19/17. https://doi.org/10.1145/3055399.3055413
Naor A, Young R. The integrality gap of the goemans-linial SDP relaxation for sparsest cut is at least a constant multiple of √logn. In STOC 2017 - Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing. Vol. Part F128415. Association for Computing Machinery. 2017. p. 564-575 https://doi.org/10.1145/3055399.3055413
Naor, Assaf ; Young, Robert. / The integrality gap of the goemans-linial SDP relaxation for sparsest cut is at least a constant multiple of √logn. STOC 2017 - Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing. Vol. Part F128415 Association for Computing Machinery, 2017. pp. 564-575
title = "The integrality gap of the goemans-linial SDP relaxation for sparsest cut is at least a constant multiple of √logn",
abstract = "We prove that the integrality gap of the Goemans-Linial semi-definite programming relaxation for the Sparsest Cut Problem is Ω(√logn) on inputs with n vertices, thus matching the previously best known upper bound (log n)1/2+o(1) up to lower-order factors. This statement is a consequence of the following new isoperimetric-type inequality. Consider the 8-regular graph whose vertex set is the 5-dimensional integer grid ℤ5 and where each vertex (a, b, c, d, e) ∈ ℤ5 is connected to the 8 vertices (a ± 1, b, c, d, e), (a, b ± 1, c, d, e), (a, b, c ± 1, d, e ± a), (a, b, c, d ± 1, e ± b). This graph is known as the Cayley graph of the 5-dimensional discrete Heisenberg group. Given Ω ⊆ ℤ5, denote the size of its edge boundary in this graph (a.k.a. the horizontal perimeter of Ω) by |∂hΩ|. For t ∈ N, denote by |∂t vΩ| the number of (a, b, c, d, e) ∈ ℤ5 such that exactly one of the two vectors (a, b, c, d, e), (a, b, c, d, e + f) is in ω. The vertical perimeter of Ω is defined to be |∂vΩ| = √Σ∞ t=1 |∂t vΩ|2/t2. We show that every subset Ω ⊆ ℤ5 satisfies |∂vΩ| = O(|∂hΩ|). This vertical-versus-horizontal isoperimetric inequality yields the above-stated integrality gap for Sparsest Cut and answers several geometric and analytic questions of independent interest. The theorem stated above is the culmination of a program whose aim is to understand the performance of the Goemans-Linial semi-definite program through the embeddability properties of Heisenberg groups. These investigations have mathematical significance even beyond their established relevance to approximation algorithms and combinatorial optimization. In particular they contribute to a range of mathematical disciplines including functional analysis, geometric group theory, harmonic analysis, sub-Riemannian geometry, geometric measure theory, ergodic theory, group representations, and metric differentiation. This article builds on the above cited works, with the {"}twist{"} that while those works were equally valid for any finite dimensional Heisenberg group, our result holds for the Heisenberg group of dimension 5 (or higher) but fails for the 3-dimensional Heisenberg group. This insight leads to our core contribution, which is a deduction of an endpoint L1-boundedness of a certain singular integral on ℝ5 from the (local) L2-boundedness of the corresponding singular integral on ℝ3. To do this, we devise a corona-type decomposition of subsets of a Heisenberg group, in the spirit of the construction that David and Semmes performed in ℝn, but with two main conceptual differences (in addition to more technical differences that arise from the peculiarities of the geometry of Heisenberg group). Firstly, the {"}atoms{"} of our decomposition are perturbations of intrinsic Lipschitz graphs in the sense of Franchi, Serapioni, and Serra Cassano (plus the requisite {"}wild{"} regions that satisfy a Carleson packing condition). Secondly, we control the local overlap of our corona decomposition by using quantitative monotonicity rather than Jones-type β-numbers.",
keywords = "Approximation algorithms, Metric embeddings, Semidefinite programming, Sparsest cut problem",
author = "Assaf Naor and Robert Young",
year = "2017",
month = "6",
day = "19",
doi = "10.1145/3055399.3055413",
language = "English (US)",
volume = "Part F128415",
pages = "564--575",
booktitle = "STOC 2017 - Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing",
publisher = "Association for Computing Machinery",

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AU - Naor, Assaf

AU - Young, Robert

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N2 - We prove that the integrality gap of the Goemans-Linial semi-definite programming relaxation for the Sparsest Cut Problem is Ω(√logn) on inputs with n vertices, thus matching the previously best known upper bound (log n)1/2+o(1) up to lower-order factors. This statement is a consequence of the following new isoperimetric-type inequality. Consider the 8-regular graph whose vertex set is the 5-dimensional integer grid ℤ5 and where each vertex (a, b, c, d, e) ∈ ℤ5 is connected to the 8 vertices (a ± 1, b, c, d, e), (a, b ± 1, c, d, e), (a, b, c ± 1, d, e ± a), (a, b, c, d ± 1, e ± b). This graph is known as the Cayley graph of the 5-dimensional discrete Heisenberg group. Given Ω ⊆ ℤ5, denote the size of its edge boundary in this graph (a.k.a. the horizontal perimeter of Ω) by |∂hΩ|. For t ∈ N, denote by |∂t vΩ| the number of (a, b, c, d, e) ∈ ℤ5 such that exactly one of the two vectors (a, b, c, d, e), (a, b, c, d, e + f) is in ω. The vertical perimeter of Ω is defined to be |∂vΩ| = √Σ∞ t=1 |∂t vΩ|2/t2. We show that every subset Ω ⊆ ℤ5 satisfies |∂vΩ| = O(|∂hΩ|). This vertical-versus-horizontal isoperimetric inequality yields the above-stated integrality gap for Sparsest Cut and answers several geometric and analytic questions of independent interest. The theorem stated above is the culmination of a program whose aim is to understand the performance of the Goemans-Linial semi-definite program through the embeddability properties of Heisenberg groups. These investigations have mathematical significance even beyond their established relevance to approximation algorithms and combinatorial optimization. In particular they contribute to a range of mathematical disciplines including functional analysis, geometric group theory, harmonic analysis, sub-Riemannian geometry, geometric measure theory, ergodic theory, group representations, and metric differentiation. This article builds on the above cited works, with the "twist" that while those works were equally valid for any finite dimensional Heisenberg group, our result holds for the Heisenberg group of dimension 5 (or higher) but fails for the 3-dimensional Heisenberg group. This insight leads to our core contribution, which is a deduction of an endpoint L1-boundedness of a certain singular integral on ℝ5 from the (local) L2-boundedness of the corresponding singular integral on ℝ3. To do this, we devise a corona-type decomposition of subsets of a Heisenberg group, in the spirit of the construction that David and Semmes performed in ℝn, but with two main conceptual differences (in addition to more technical differences that arise from the peculiarities of the geometry of Heisenberg group). Firstly, the "atoms" of our decomposition are perturbations of intrinsic Lipschitz graphs in the sense of Franchi, Serapioni, and Serra Cassano (plus the requisite "wild" regions that satisfy a Carleson packing condition). Secondly, we control the local overlap of our corona decomposition by using quantitative monotonicity rather than Jones-type β-numbers.

AB - We prove that the integrality gap of the Goemans-Linial semi-definite programming relaxation for the Sparsest Cut Problem is Ω(√logn) on inputs with n vertices, thus matching the previously best known upper bound (log n)1/2+o(1) up to lower-order factors. This statement is a consequence of the following new isoperimetric-type inequality. Consider the 8-regular graph whose vertex set is the 5-dimensional integer grid ℤ5 and where each vertex (a, b, c, d, e) ∈ ℤ5 is connected to the 8 vertices (a ± 1, b, c, d, e), (a, b ± 1, c, d, e), (a, b, c ± 1, d, e ± a), (a, b, c, d ± 1, e ± b). This graph is known as the Cayley graph of the 5-dimensional discrete Heisenberg group. Given Ω ⊆ ℤ5, denote the size of its edge boundary in this graph (a.k.a. the horizontal perimeter of Ω) by |∂hΩ|. For t ∈ N, denote by |∂t vΩ| the number of (a, b, c, d, e) ∈ ℤ5 such that exactly one of the two vectors (a, b, c, d, e), (a, b, c, d, e + f) is in ω. The vertical perimeter of Ω is defined to be |∂vΩ| = √Σ∞ t=1 |∂t vΩ|2/t2. We show that every subset Ω ⊆ ℤ5 satisfies |∂vΩ| = O(|∂hΩ|). This vertical-versus-horizontal isoperimetric inequality yields the above-stated integrality gap for Sparsest Cut and answers several geometric and analytic questions of independent interest. The theorem stated above is the culmination of a program whose aim is to understand the performance of the Goemans-Linial semi-definite program through the embeddability properties of Heisenberg groups. These investigations have mathematical significance even beyond their established relevance to approximation algorithms and combinatorial optimization. In particular they contribute to a range of mathematical disciplines including functional analysis, geometric group theory, harmonic analysis, sub-Riemannian geometry, geometric measure theory, ergodic theory, group representations, and metric differentiation. This article builds on the above cited works, with the "twist" that while those works were equally valid for any finite dimensional Heisenberg group, our result holds for the Heisenberg group of dimension 5 (or higher) but fails for the 3-dimensional Heisenberg group. This insight leads to our core contribution, which is a deduction of an endpoint L1-boundedness of a certain singular integral on ℝ5 from the (local) L2-boundedness of the corresponding singular integral on ℝ3. To do this, we devise a corona-type decomposition of subsets of a Heisenberg group, in the spirit of the construction that David and Semmes performed in ℝn, but with two main conceptual differences (in addition to more technical differences that arise from the peculiarities of the geometry of Heisenberg group). Firstly, the "atoms" of our decomposition are perturbations of intrinsic Lipschitz graphs in the sense of Franchi, Serapioni, and Serra Cassano (plus the requisite "wild" regions that satisfy a Carleson packing condition). Secondly, we control the local overlap of our corona decomposition by using quantitative monotonicity rather than Jones-type β-numbers.

KW - Approximation algorithms

KW - Metric embeddings

KW - Semidefinite programming

KW - Sparsest cut problem

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