[Paper Review] Constant Factor Lasserre Integrality Gaps for Graph Partitioning Problems

Venkatesan Guruswami, Ali Kemal Sinop+1 more|arXiv (Cornell University)|Feb 27, 2012

Complexity and Algorithms in Graphs4 citations

TL;DR

This paper establishes constant-factor integrality gaps for the Lasserre hierarchy in the context of Balanced Separator and Uniform Sparsest Cut problems, demonstrating that even Ω(n) levels of the hierarchy fail to achieve a constant-factor approximation. The authors construct explicit integrality gap instances using expander graphs and structural constraints, showing that the hierarchy cannot certify constant-approximate solutions for these fundamental graph partitioning problems.

ABSTRACT

Partitioning the vertices of a graph into two roughly equal parts while minimizing the number of edges crossing the cut is a fundamental problem (called Balanced Separator) that arises in many settings. For this problem, and variants such as the Uniform Sparsest Cut problem where the goal is to minimize the fraction of pairs on opposite sides of the cut that are connected by an edge, there are large gaps between the known approximation algorithms and non-approximability results. While no constant factor approximation algorithms are known, even APX-hardness is not known either. In this work we prove that for balanced separator and uniform sparsest cut, semidefinite programs from the Lasserre hierarchy (which are the most powerful relaxations studied in the literature) have an integrality gap bounded away from $1$, even for $Ω(n)$ levels of the hierarchy. This complements recent algorithmic results in Guruswami and Sinop (2011) which used the Lasserre hierarchy to give an approximation scheme for these problems (with runtime depending on the spectrum of the graph). Along the way, we make an observation that simplifies the task of lifting "polynomial constraints" (such as the global balance constraint in balanced separator) to higher levels of the Lasserre hierarchy.

Motivation & Objective

• To close the gap between known approximation algorithms and inapproximability results for Balanced Separator and Uniform Sparsest Cut.
• To investigate whether the Lasserre hierarchy—among the most powerful SDP hierarchies—can achieve constant-factor approximation for these problems.
• To construct explicit integrality gap instances that remain hard even for Ω(n) levels of the Lasserre hierarchy.
• To simplify the lifting of polynomial constraints (e.g., global balance) into higher levels of the Lasserre hierarchy.
• To provide evidence that the Lasserre hierarchy may not yield a PTAS for these problems, despite recent algorithmic progress.

Proposed method

• Constructs a family of graph instances using expanders and structured components (LΦ, RΦ, Zr, Dl, Dr) to simulate hard cases for partitioning.
• Uses expansion properties of subgraphs to bound edge counts across cuts, ensuring sparsity remains above a threshold unless balance constraints are violated.
• Applies a probabilistic argument to show that any cut with sparsity below a threshold γ must be nearly balanced and separate Dl′ and Dr′ into opposite sides.
• Leverages the structure of the Lasserre hierarchy to lift global balance and cut constraints into higher-level relaxations, simplifying the analysis of polynomial constraints.
• Employs a chain of inequalities to bound the sparsity of any candidate cut, showing it exceeds γ unless the cut is both balanced and separates key components.
• Uses spectral properties and eigenvalue bounds (λr) to relate integrality gaps to the graph’s normalized Laplacian, linking to prior algorithmic results.

Experimental results

Research questions

• RQ1 Can the Lasserre hierarchy achieve a constant-factor approximation for Balanced Separator and Uniform Sparsest Cut?
• RQ2 Do integrality gaps for the Lasserre hierarchy remain bounded away from 1 even at Ω(n) levels?
• RQ3 Can polynomial constraints like global balance be efficiently lifted into higher levels of the Lasserre hierarchy?
• RQ4 Is there a super-constant integrality gap for the Lasserre hierarchy in these problems, despite strong algorithmic results at lower levels?
• RQ5 Does the Lasserre hierarchy fail to yield a PTAS for these problems, even when weaker hierarchies like Sherali-Adams fail?

Key findings

• The Lasserre hierarchy has a constant-factor integrality gap for Balanced Separator and Uniform Sparsest Cut, even at Ω(n) levels.
• The integrality gap is bounded away from 1, meaning the hierarchy cannot certify constant-approximate solutions.
• The gap instance uses expanders and structured components to force high sparsity unless cuts are both balanced and separate key parts.
• The analysis shows that any cut with sparsity below a threshold γ must be nearly balanced and must separate Dl′ and Dr′ into opposite sides.
• The construction demonstrates that even with O(n) levels, the Lasserre hierarchy cannot achieve a (1+ε)-approximation for these problems.
• The paper provides a simplified method for lifting global balance constraints into higher levels of the Lasserre hierarchy, enabling cleaner analysis of integrality gaps.