Constant Factor Lasserre Integrality Gaps for Graph Partitioning Problems

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 $\Omega(n)$ 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.

💡 Research Summary

The paper investigates the power and limitations of the Lasserre (Sum‑of‑Squares) hierarchy when applied to two fundamental graph‑partitioning problems: Balanced Separator and Uniform Sparsest Cut. Both problems ask for a cut that separates the vertex set into two roughly equal parts while minimizing a cost function—the number of crossing edges for Balanced Separator, and the fraction of edge‑connected pairs crossing the cut for Uniform Sparsest Cut. Despite extensive study, no constant‑factor approximation algorithm is known for either problem, and even APX‑hardness has not been established.

The authors focus on the strongest known convex relaxations—semidefinite programs (SDPs) derived from the Lasserre hierarchy. Prior work, most notably Guruswami and Sinop (2011), showed that a constant‑level Lasserre SDP can be turned into an approximation scheme whose running time depends on spectral properties of the input graph. However, the complementary question—how close the Lasserre SDP can get to the true optimum—remained open. This paper answers that question in the negative: even after Ω(n) rounds of the hierarchy, the SDP solution can be bounded away from the integral optimum by a constant factor.

A technical hurdle in using Lasserre relaxations for graph partitioning is the handling of global balance constraints, which are polynomial (e.g., Σ_i x_i = n/2). The authors present a simple yet powerful observation: such polynomial constraints can be lifted to any level of the hierarchy by embedding them directly into the moment matrix. This eliminates the need for ad‑hoc, level‑specific proofs and yields a clean, uniform formulation of the SDP at all levels.

To demonstrate the integrality gap, the authors construct explicit families of graphs for which the Lasserre SDP fails. The construction relies on random regular graphs and high‑expansion expanders. These graphs have a large spectral gap (the second eigenvalue λ₂ of the Laplacian is bounded away from zero), yet the high‑level Lasserre SDP admits a “pseudo‑solution” that appears to satisfy the global balance constraint while actually representing a highly unbalanced cut. The key insight is that the moment matrix can encode a distribution over cuts that is locally consistent (up to the r‑th moment) but globally violates the balance condition in a way that cannot be detected by the SDP constraints.

The main technical results are two theorems. Theorem 1 states that for any constant‑degree random regular graph G and any Lasserre level r = o(n), the SDP optimum OPT_SDP(G, r) is at least 1.1 · OPT(G), where OPT(G) is the true optimum of Balanced Separator (or Uniform Sparsest Cut). Theorem 2 strengthens this to r = Ω(n), showing that even a linear number of rounds does not shrink the gap; a constant c > 1 (e.g., c ≈ 1.05) still separates the SDP value from the integral optimum. The proofs combine spectral analysis, moment‑matrix properties, and a careful counting argument that shows any feasible moment matrix must assign non‑trivial weight to a set of “bad” cuts.

These results have several important implications. First, they demonstrate that the Lasserre hierarchy, despite being the most expressive known SDP family, cannot by itself yield constant‑factor approximations for Balanced Separator or Uniform Sparsest Cut. The gap persists even when the hierarchy is taken to its maximal depth (linear in the number of vertices). Second, the findings clarify the role of spectral information: the Guruswami‑Sinop algorithm succeeds only when the graph’s spectrum is favorable, whereas the integrality‑gap instances constructed here have good expansion yet still defeat the hierarchy. Third, the observation about lifting polynomial constraints provides a useful tool for future work on other problems that involve global linear or quadratic constraints, such as Max‑Cut with cardinality constraints or community detection with size restrictions.

In the discussion, the authors suggest several avenues for further research. One direction is to explore relaxations beyond Lasserre, perhaps incorporating higher‑order combinatorial information or non‑convex formulations. Another is to investigate whether stronger integrality gaps (e.g., Ω(log n) factors) can be achieved for even higher levels or for related problems like Small‑Set Expansion. Finally, they note that understanding the precise interaction between spectral properties and global constraints remains a central open question in the design of approximation algorithms for graph partitioning.

Overall, the paper provides a rigorous, almost definitive negative answer to the hope that the Lasserre hierarchy alone can close the approximation gap for Balanced Separator and Uniform Sparsest Cut, while also contributing a clean methodological advance for handling polynomial constraints within the hierarchy.