Computing a dense subgraph is a fundamental problem in graph mining, with a diverse set of applications ranging from electronic commerce to community detection in social networks. In many of these applications, the underlying context is better modelled as a weighted hypergraph that keeps evolving with time.
This motivates the problem of maintaining the densest subhypergraph of a weighted hypergraph in a dynamic setting, where the input keeps changing via a sequence of updates (hyperedge insertions/deletions). Previously, the only known algorithm for this problem was due to Hu et al. [19]. This algorithm worked only on unweighted hypergraphs, and had an approximation ratio of (1 +ϵ)r2 and an update time of O(poly(r, log n)), where r denotes the maximum rank of the input across all the updates.
We obtain a new algorithm for this problem, which works even when the input hypergraph is weighted. Our algorithm has a significantly improved (near-optimal) approximation ratio of (1 +ϵ) that is independent of r, and a similar update time of O(poly(r, log n)). It is the first (1 +ϵ)-approximation algorithm even for the special case of weighted simple graphs.
To complement our theoretical analysis, we perform experiments with our dynamic algorithm on large-scale, real-world data-sets. Our algorithm significantly outperforms the state of the art [19] both in terms of accuracy and efficiency.
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This content will become publicly available on June 2, 2024
Sum-of-Squares Lower Bounds for Densest k-Subgraph
Given a graph and an integer k, Densest k-Subgraph is the algorithmic task of finding the subgraph on k vertices with the maximum number of edges. This is a fundamental problem that has been subject to intense study for decades, with applications spanning a wide variety of fields. The state-of-the-art algorithm is an O(n^{1/4+ϵ})-factor approximation (for any ϵ>0) due to Bhaskara et al. [STOC '10]. Moreover, the so-called log-density framework predicts that this is optimal, i.e. it is impossible for an efficient algorithm to achieve an O(n^{1/4−ϵ})-factor approximation. In the average case, Densest k-Subgraph is a prototypical noisy inference task which is conjectured to exhibit a statistical-computational gap.
In this work, we provide the strongest evidence yet of hardness for Densest k-Subgraph by showing matching lower bounds against the powerful Sum-of-Squares (SoS) algorithm, a meta-algorithm based on convex programming that achieves state-of-art algorithmic guarantees for many optimization and inference problems. For k ≤ n^1/2, we obtain a degree n^δ SoS lower bound for the hard regime as predicted by the log-density framework.
To show this, we utilize the modern framework for proving SoS lower bounds on average-case problems pioneered by Barak et al. [FOCS '16]. A key issue is that small denser-than-average subgraphs in the input will greatly affect the value of the candidate pseudo-expectation operator around the subgraph. To handle this challenge, we devise a novel matrix factorization scheme based on the positive minimum vertex separator. We then prove an intersection tradeoff lemma to show that the error terms when using this separator are indeed small.
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- Award ID(s):
- 2008920
- NSF-PAR ID:
- 10483143
- Publisher / Repository:
- ACM
- Date Published:
- Journal Name:
- STOC 2023: Proceedings of the 55th Annual ACM Symposium on Theory of Computing
- Page Range / eLocation ID:
- 84 to 95
- Format(s):
- Medium: X
- Location:
- Orlando FL USA
- Sponsoring Org:
- National Science Foundation
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