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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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Counting Trees in Graphs
Counting paths and walks in graphs have numerous applications, such as finding bounds on the spectral radius of the graph and the energy of a graph, as well as in standard graph theoretic reductions of combinatorial problems such as counting words over alphabets with forbidden patterns. In this paper, the authors strengthen and generalize Erdős and Simonovits's results on counting paths to counting isomorphic copies of trees in a graph of average degree d.
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- Award ID(s):
- 1800832
- PAR ID:
- 10094062
- Date Published:
- Journal Name:
- The Electronic journal of combinatorics
- Volume:
- 23
- Issue:
- 3
- ISSN:
- 1077-8926
- Page Range / eLocation ID:
- P3.39
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
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