Finding Large Independent Sets in One-Sided Expander Graphs: New Algorithms and Hardness Results

Core Concepts

This paper presents new algorithms for finding large independent sets in 3-colorable or almost-bipartite one-sided expander graphs, based on a novel clustering property of their independent sets, while also proving that the problem becomes NP-hard for almost 4-colorable one-sided expanders (assuming the Unique Games Conjecture).

Abstract

• Bibliographic Information: Bafna, M., Hsieh, J., & Kothari, P. K. (2024). Rounding Large Independent Sets on Expanders. arXiv preprint arXiv:2405.10238v2.

• Research Objective: This paper investigates the computational complexity of finding large independent sets in one-sided expander graphs, particularly focusing on the impact of almost-k-colorability on the problem's tractability.

• Methodology: The authors develop new algorithms based on rounding sum-of-squares relaxations, leveraging a novel clustering property of independent sets in expander graphs. They also provide hardness results by reduction from the Unique Games Conjecture.

• Key Findings:

  • A new polynomial-time algorithm finds linear-sized independent sets in one-sided expanders that are either almost 3-colorable or contain an independent set of size (1/2 - ε)n.
  • The algorithm extends to graphs with certifiable small-set vertex expansion, a weaker notion than spectral expansion.
  • Finding a linear-sized independent set in almost 4-colorable one-sided expanders is NP-hard, assuming the Unique Games Conjecture.

• Main Conclusions: The paper demonstrates a stark difference in the complexity of finding large independent sets between almost 3-colorable and almost 4-colorable one-sided expanders. The new algorithms and the clustering property contribute significantly to the understanding of independent set approximation in structured graph families.

• Significance: This work advances the field of approximation algorithms by providing new techniques for tackling a fundamental graph problem under natural structural assumptions. The results shed light on the power and limitations of spectral expansion in the context of independent set approximation.

• Limitations and Future Research: The algorithms' dependence on the specific expansion and almost-colorability parameters leaves room for exploring tighter bounds and potential generalizations. Investigating the clustering property in other graph families could lead to further algorithmic advancements.

Stats

The algorithm finds independent sets of size at least 10^-4n in 10^-4-almost 3-colorable one-sided spectral expanders with λ2 ≤ 10^-4.
For graphs with an independent set of size (1/2 - ε)n and λ2 ≤ 1 - 40ε, the algorithm outputs an independent set of size at least 10^-3n.

Quotes

"All prior algorithms that beat the worst-case guarantees for this problem rely on bottom eigenspace enumeration techniques... and require two-sided expansion."
"Our rounding scheme builds on the method of simulating multiple samples from a pseudo-distribution introduced in [BBK+21] for rounding Unique Games instances."
"The key to our analysis is a new clustering property of large independent sets in expanding graphs — every large independent set has a larger-than-expected intersection with some member of a small list — and its formalization in the low-degree sum-of-squares proof system."

Key Insights Distilled From

Rounding Large Independent Sets on Expanders

by Mitali Bafna... at arxiv.org 11-07-2024

https://arxiv.org/pdf/2405.10238.pdf

Deeper Inquiries

Can the clustering property of independent sets be leveraged to design efficient algorithms for other graph problems, such as graph coloring or finding maximum cliques?

It's certainly possible that the clustering property, or variations of it, could be useful for other graph problems on expanders. Here's a breakdown of its potential applicability to graph coloring and maximum cliques:
Graph Coloring:

Promise Problems: The paper already demonstrates success with almost 3-colorable expanders. Extending this to almost k-colorable graphs for k > 3 is unlikely using the same techniques due to the inherent hardness results (Proposition 1.1). However, exploring variations of the clustering property, perhaps with weaker agreement guarantees or different rounding schemes, might be fruitful for specific values of k.
Beyond Promise Problems: For general graph coloring without a promise on the chromatic number, directly applying the clustering property seems challenging. The property relies on the existence of large independent sets, which might not be present in a general k-colorable graph.
New Algorithmic Ideas: The success with 3-colorable expanders suggests that focusing on the "solution space geometry" could inspire new coloring algorithms. Instead of traditional methods like finding large independent sets and iterating, exploring algorithms that directly exploit the clustering of valid colorings might be a promising direction.
Finding Maximum Cliques:

Complementarity: Finding a maximum clique in a graph is equivalent to finding a maximum independent set in its complement graph. However, the complement of an expander graph is generally not an expander. This means the clustering property, as is, doesn't directly apply.
Structural Insights: While not directly applicable, the clustering property might offer insights into the structure of independent sets in expanders, which could indirectly inform algorithms for cliques. For instance, understanding how independent sets cluster might shed light on the distribution of edges in the complement graph, potentially aiding clique-finding algorithms.
Overall: The clustering property is a powerful tool that has proven useful for independent sets in expanders. Its direct applicability to other problems like general graph coloring and maximum clique finding seems limited due to the specific nature of the property and the problems themselves. However, it can serve as inspiration for developing new algorithmic ideas that exploit the geometry of the solution space in these domains.

Could there be alternative algorithmic approaches, beyond spectral methods and sum-of-squares relaxations, that circumvent the NP-hardness barrier for almost 4-colorable one-sided expanders?

While the paper highlights the limitations of spectral methods and the success of sum-of-squares relaxations for specific cases, exploring alternative approaches is always valuable. Here are some potential avenues:

Combinatorial Algorithms: Developing purely combinatorial algorithms, avoiding spectral or SDP-based techniques altogether, might offer a way around the current barriers. This could involve exploiting specific structural properties of one-sided expanders beyond spectral expansion, or leveraging the almost-4-colorability in novel ways.
Local Search and Message Passing: Algorithms based on local search or message passing, often used in constraint satisfaction problems, could be investigated. These methods operate by iteratively improving a candidate solution based on local information. Adapting such techniques to exploit the structure of one-sided expanders and the almost-4-colorability constraint might lead to new results.
Approximation Algorithms with Weaker Guarantees: Instead of aiming for linear-sized independent sets, focusing on approximation algorithms with weaker guarantees might be fruitful. For instance, could we design algorithms that find sub-linear but still significantly larger-than-trivial independent sets in almost-4-colorable one-sided expanders?
Quantum Algorithms: Quantum algorithms have shown promise for certain graph problems. Investigating whether quantum techniques can exploit the specific structure of almost-4-colorable one-sided expanders to circumvent the classical NP-hardness barrier is an intriguing direction.
Challenges and Considerations:

Hardness Results: The NP-hardness result (Proposition 1.1) poses a significant challenge. Any successful alternative approach must somehow circumvent this barrier, perhaps by exploiting additional structural properties or relaxing the problem's requirements.
One-Sided Expansion: The restriction to one-sided expanders limits the tools available compared to two-sided expanders. Algorithms must be carefully designed to handle the potential asymmetry in the graph structure.
Overall: While the NP-hardness result presents a hurdle, exploring alternative algorithmic approaches beyond spectral methods and sum-of-squares relaxations is crucial. Focusing on combinatorial algorithms, local search techniques, weaker approximation guarantees, or even quantum algorithms might uncover new possibilities for tackling this problem.

How does the geometry of the solution space for finding large independent sets in one-sided expanders relate to the geometry of solutions in other combinatorial optimization problems?

The concept of "solution space geometry" is a powerful lens for understanding the behavior of algorithms in various combinatorial optimization problems. Here's how it connects to the findings in the paper and relates to other problems:
Key Idea: The paper reveals that in one-sided expanders, the solution space of large independent sets exhibits a clustering phenomenon. This means that feasible solutions (large independent sets) aren't randomly scattered but tend to be clustered together, implying a certain structure in the solution space.
Connections to Other Problems:

Clustering in Random Instances: This clustering phenomenon is reminiscent of similar observations in the study of random constraint satisfaction problems (CSPs). For instance, in random k-SAT or graph coloring on random graphs, as the density of constraints increases, the solution space undergoes a "phase transition." Beyond a certain threshold, solutions tend to cluster into well-separated regions.
Low-Degree Polynomials and Rounding: The success of sum-of-squares relaxations in the paper stems from the ability of low-degree polynomials to capture this clustering property. This connection between solution space geometry and the power of low-degree polynomials is a recurring theme in the analysis of SDP and SoS hierarchies for various problems.
Landscape Analysis: The notion of solution space geometry is closely related to the concept of "landscape analysis" in optimization. A "smooth" landscape with clustered solutions often implies the existence of efficient algorithms, while a "rugged" landscape with scattered solutions suggests computational hardness.
Examples:

Unique Games on Expanders: The paper draws inspiration from techniques used for Unique Games on expanders. In that context, the solution space geometry (under certain expansion properties) allows for rounding procedures based on finding a small number of "influential coordinates."
Community Detection: In network analysis, community detection aims to partition a graph into densely connected clusters. Algorithms for community detection often implicitly or explicitly exploit the geometry of the solution space, searching for clusterings that optimize certain objective functions.
Broader Implications:

Algorithmic Design: Understanding the solution space geometry can guide the design of more effective algorithms. If we know solutions cluster, we can tailor algorithms to exploit this structure, as demonstrated by the sum-of-squares rounding in the paper.
Hardness of Approximation: Conversely, if the solution space is highly unstructured or solutions are scattered, it suggests inherent limitations on our ability to find good solutions efficiently. This insight can point towards proving hardness of approximation results.
Overall: The clustering property in one-sided expanders highlights the importance of solution space geometry in combinatorial optimization. Similar geometric insights have proven valuable in understanding the performance of algorithms and the computational complexity of various problems, including CSPs, Unique Games, and community detection. Exploring the solution space geometry in other domains might reveal hidden structures that can be exploited for algorithmic development or provide evidence for computational hardness.