[Paper Review] Reductions Between Expansion Problems

Prasad Raghavendra, David Steurer+1 more|arXiv (Cornell University)|Nov 11, 2010

Complexity and Algorithms in Graphs5 citations

TL;DR

This paper establishes the equivalence between the Small-Set Expansion Hypothesis (SSEH) and a restricted variant of the Unique Games Conjecture (UGC), proving that SSEH implies UGC under mild small-set expansion conditions. It further yields the first strong inapproximability results for Balanced Separator and Minimum Linear Arrangement, leveraging novel non-local gadget reductions that exploit structural properties of SSE-generated instances.

ABSTRACT

The Small-Set Expansion Hypothesis (Raghavendra, Steurer, STOC 2010) is a natural hardness assumption concerning the problem of approximating the edge expansion of small sets in graphs. This hardness assumption is closely connected to the Unique Games Conjecture (Khot, STOC 2002). In particular, the Small-Set Expansion Hypothesis implies the Unique Games Conjecture (Raghavendra, Steurer, STOC 2010). Our main result is that the Small-Set Expansion Hypothesis is in fact equivalent to a variant of the Unique Games Conjecture. More precisely, the hypothesis is equivalent to the Unique Games Conjecture restricted to instance with a fairly mild condition on the expansion of small sets. Alongside, we obtain the first strong hardness of approximation results for the Balanced Separator and Minimum Linear Arrangement problems. Before, no such hardness was known for these problems even assuming the Unique Games Conjecture. These results not only establish the Small-Set Expansion Hypothesis as a natural unifying hypothesis that implies the Unique Games Conjecture, all its consequences and, in addition, hardness results for other problems like Balanced Separator and Minimum Linear Arrangement, but our results also show that the Small-Set Expansion Hypothesis problem lies at the combinatorial heart of the Unique Games Conjecture. The key technical ingredient is a new way of exploiting the structure of the Unique Games instances obtained from the Small-Set Expansion Hypothesis via (Raghavendra, Steurer, 2010). This additional structure allows us to modify standard reductions in a way that essentially destroys their local-gadget nature. Using this modification, we can argue about the expansion in the graphs produced by the reduction without relying on expansion properties of the underlying Unique Games instance (which would be impossible for a local-gadget reduction).

Motivation & Objective

• To establish the equivalence between the Small-Set Expansion Hypothesis (SSEH) and a restricted version of the Unique Games Conjecture (UGC), where UGC is required to hold only for instances with mild small-set expansion properties.
• To demonstrate that SSEH implies all consequences of UGC, including strong inapproximability results for fundamental graph partitioning problems.
• To develop a new reduction framework that bypasses the limitations of traditional local-gadget reductions by exploiting structural features of Unique Games instances derived from SSEH.
• To prove the first strong inapproximability results for Balanced Separator and Minimum Linear Arrangement under SSEH, which were previously unknown even under UGC.

Proposed method

• Constructs a novel reduction from Small-Set Expansion (SSE) to Unique Games using a non-local gadget construction that preserves expansion properties without relying on local structure.
• Introduces a parameterized family of SSE instances with parameters (η, δ, M), where η controls the small-set expansion threshold and M bounds the measure of sets under consideration.
• Employs a decoding procedure that maps low-expansion sets in the output graph back to low-expansion sets in the original SSE instance, using the expanded structure of the reduction.
• Uses a modified version of the standard Unique Games reduction, tailored to preserve expansion behavior even when the underlying instance lacks strong local expansion.
• Applies a key lemma (Lemma 6.12) to show that a non-expanding set in the output graph implies a non-expanding set in the input, under carefully chosen parameters.
• Establishes completeness and soundness via probabilistic analysis of edge expansion across partitions induced by assignments, using the expansion profile of the Gaussian graph G(1−ε) as a benchmark.

Experimental results

Research questions

• RQ1 Is the Small-Set Expansion Hypothesis equivalent to a restricted form of the Unique Games Conjecture, where the conjecture holds only for instances with bounded small-set expansion?
• RQ2 Can the Small-Set Expansion Hypothesis be used to derive strong inapproximability results for Balanced Separator and Minimum Linear Arrangement, problems for which no such results were previously known?
• RQ3 Can non-local gadget reductions be designed to preserve global expansion properties in the output graph, even when the underlying Unique Games instance lacks strong local expansion?
• RQ4 Does the expansion profile of the Gaussian graph G(1−ε) serve as a tight benchmark for distinguishing between satisfiable and unsatisfiable Unique Games instances under the SSEH?
• RQ5 Can the structural properties of SSE-generated Unique Games instances be exploited to bypass the limitations of standard local-gadget reductions in inapproximability proofs?

Key findings

• The Small-Set Expansion Hypothesis is equivalent to a restricted version of the Unique Games Conjecture, where the conjecture applies only to instances with mild small-set expansion, thereby unifying both hypotheses.
• The paper proves the first strong inapproximability results for the Balanced Separator problem under the SSEH, showing that it is SSE-hard to approximate within any constant factor.
• It establishes the first strong inapproximability result for the Minimum Linear Arrangement problem under the SSEH, demonstrating that no constant-factor approximation is possible.
• The reduction framework developed in the paper successfully avoids reliance on local-gadget properties by exploiting global expansion structure, enabling soundness analysis independent of the input instance’s local expansion.
• The paper shows that for any ε > 0, it is SSE-hard to distinguish between Unique Games instances with opt > 1 − 2ε − o(ε) and those with opt < O(q^(-ε/(2−ε))) + γ, under a condition on the expansion profile matching that of the Gaussian graph G(1−ε).
• The construction ensures that any non-expanding set in the output graph corresponds to a non-expanding set in the input, with measure and expansion preserved within tight bounds, enabling a tight soundness analysis.