Random Tensors and Planted Cliques

  • Authors:

  • S. Charles Brubaker;Santosh S. Vempala

  • Affiliations:

  • Georgia Institute of Technology, Atlanta 30332;Georgia Institute of Technology, Atlanta 30332

  • Venue:
  • APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
  • Year:
  • 2009

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Abstract

The r -parity tensor of a graph is a generalization of the adjacency matrix, where the tensor's entries denote the parity of the number of edges in subgraphs induced by r distinct vertices. For r = 2, it is the adjacency matrix with 1's for edges and *** 1's for nonedges. It is well-known that the 2-norm of the adjacency matrix of a random graph is $O(\sqrt{n})$. Here we show that the 2-norm of the r -parity tensor is at most $f(r)\sqrt{n}\log^{O(r)}n$, answering a question of Frieze and Kannan [1] who proved this for r = 3. As a consequence, we get a tight connection between the planted clique problem and the problem of finding a vector that approximates the 2-norm of the r -parity tensor of a random graph. Our proof method is based on an inductive application of concentration of measure.