Algorithms Approaching the Threshold for Semi-random Planted Clique
METADATA ONLY
Loading...
Author / Producer
Date
2023-06-02
Publication Type
Conference Paper
ETH Bibliography
yes
Citations
Altmetric
METADATA ONLY
Data
Rights / License
Abstract
We design new polynomial-time algorithms for recovering planted cliques in the semi-random graph model introduced by Feige and Kilian. The previous best algorithms for this model succeed if the planted clique has size at least n2/3 in a graph with n vertices. Our algorithms work for planted-clique sizes approaching n1/2 — the information-theoretic threshold in the semi-random model and a conjectured computational threshold even in the easier fully-random model. This result comes close to resolving open questions by Feige and Steinhardt. To generate a graph in the semi-random planted-clique model, we first 1) plant a clique of size k in an n-vertex –graph with edge probability 1/2 and then adversarially add or delete an arbitrary number edges not touching the planted clique and delete any subset of edges going out of the planted clique. For every є>0, we give an nO(1/є)-time algorithm that recovers a clique of size k in this model whenever k ≥ n1/2+є. In fact, our algorithm computes, with high probability, a list of about n/k cliques of size k that contains the planted clique. Our algorithms also extend to arbitrary edge probabilities p and improve on the previous best guarantee whenever p ≤ 1−n−0.001. Our algorithms rely on a new conceptual connection that translates certificates of upper bounds on biclique numbers in unbalanced bipartite –random graphs into algorithms for semi-random planted clique. Analogous to the (conjecturally) optimal algorithms for the fully-random model, the previous best guarantees for semi-random planted clique correspond to spectral relaxations of biclique numbers based on eigenvalues of adjacency matrices. We construct an SDP lower bound that shows that the n2/3 threshold in prior works is an inherent limitation of these spectral relaxations. We go beyond this limitation by using higher-order sum-of-squares relaxations for biclique numbers.
We also provide some evidence that the information-computation trade-off of our current algorithms may be inherent by proving an average-case lower bound for unbalanced bicliques in the low-degree polynomial model.
Permanent link
Publication status
published
External links
Book title
STOC 2023: Proceedings of the 55th Annual ACM Symposium on Theory of Computing
Journal / series
Volume
Pages / Article No.
1918 - 1926
Publisher
Association for Computing Machinery
Event
55th Annual ACM Symposium on Theory of Computing (STOC '23)
Edition / version
Methods
Software
Geographic location
Date collected
Date created
Subject
planted clique; semi-random; semidefinite programming; sum-of-squares hierarchy
Organisational unit
09622 - Steurer, David / Steurer, David
Notes
Funding
815464 - Unified Theory of Efficient Optimization and Estimation (EC)