Solving tall dense linear programs in nearly linear time
- Award ID(s):
- 1844855
- PAR ID:
- 10211394
- Date Published:
- Journal Name:
- Proccedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020
- Page Range / eLocation ID:
- 775 to 788
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract A classical result of Erdős and, independently, of Bondy and Simonovits [3] says that the maximum number of edges in ann-vertex graph not containingC2k, the cycle of length 2k, isO(n1+1/k). Simonovits established a corresponding supersaturation result forC2k’s, showing that there exist positive constantsC,cdepending only onksuch that everyn-vertex graphGwithe(G)⩾Cn1+1/kcontains at leastc(e(G)/v(G))2kcopies ofC2k, this number of copies tightly achieved by the random graph (up to a multiplicative constant). In this paper we extend Simonovits' result to a supersaturation result ofr-uniform linear cycles of even length inr-uniform linear hypergraphs. Our proof is self-contained and includes ther= 2 case. As an auxiliary tool, we develop a reduction lemma from general host graphs to almost-regular host graphs that can be used for other supersaturation problems, and may therefore be of independent interest.more » « less
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