Statistics of small subgraph counts such as triangles, four-cycles, and s-t paths of short lengths reveal important structural properties of the underlying graph. These problems have been widely studied in social network analysis. In most relevant applications, the graphs are not only massive but also change dynamically over time. Most of these problems become hard in the dynamic setting when considering the worst case. In this paper, we ask whether the question of small subgraph counting over dynamic graphs is hard also in the average case.
We consider the simplest possible average case model where the updates follow an Erdős-Rényi graph: each update selects a pair of vertices (u, v) uniformly at random and flips the existence of the edge (u, v). We develop new lower bounds and matching algorithms in this model for counting four-cycles, counting triangles through a specified point s, or a random queried point, and st paths of length 3, 4 and 5. Our results indicate while computing st paths of length 3, and 4 are easy in the average case with O(1) update time (note that they are hard in the worst case), it becomes hard when considering st paths of length 5.
We introduce new techniques which allow us to get average-case hardness for these graph problems from the worst-case hardness of the Online Matrix vector problem (OMv). Our techniques rely on recent advances in fine-grained average-case complexity. Our techniques advance this literature, giving the ability to prove new lower bounds on average-case dynamic algorithms.
Read More: https://epubs.siam.org/doi/abs/10.1137/1.9781611977073.23
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Leibniz International Proceedings in Informatics (LIPIcs):15th Innovations in Theoretical Computer Science Conference (ITCS 2024)
Algorithms with predictions is a new research direction that leverages machine learned predictions for algorithm design. So far a plethora of recent works have incorporated predictions to improve on worst-case bounds for online problems. In this paper, we initiate the study of complexity of dynamic data structures with predictions, including dynamic graph algorithms. Unlike online algorithms, the goal in dynamic data structures is to maintain the solution efficiently with every update.
We investigate three natural models of prediction: (1) δ-accurate predictions where each predicted request matches the true request with probability δ, (2) list-accurate predictions where a true request comes from a list of possible requests, and (3) bounded delay predictions where the true requests are a permutation of the predicted requests. We give general reductions among the prediction models, showing that bounded delay is the strongest prediction model, followed by list-accurate, and δ-accurate.
Further, we identify two broad problem classes based on lower bounds due to the Online Matrix Vector (OMv) conjecture. Specifically, we show that locally correctable dynamic problems have strong conditional lower bounds for list-accurate predictions that are equivalent to the non-prediction setting, unless list-accurate predictions are perfect. Moreover, we show that locally reducible dynamic problems have time complexity that degrades gracefully with the quality of bounded delay predictions. We categorize problems with known OMv lower bounds accordingly and give several upper bounds in the delay model that show that our lower bounds are almost tight.
We note that concurrent work by v.d.Brand et al. [SODA '24] and Liu and Srinivas [arXiv:2307.08890] independently study dynamic graph algorithms with predictions, but their work is mostly focused on showing upper bounds.
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- NSF-PAR ID:
- 10496598
- Editor(s):
- Guruswami, Venkatesan
- Publisher / Repository:
- Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- Date Published:
- Journal Name:
- 15th Innovations in Theoretical Computer Science Conference, {ITCS}
- ISSN:
- 1608-1218
- Subject(s) / Keyword(s):
- ["Dynamic Graph Algorithms","Algorithms with Predictions","Theory of computation → Dynamic graph algorithms"]
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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