TY - JOUR
T1 - A tight bound for the clique query problem in two rounds
AU - Feige, Uriel
AU - Ferster, Tom
PY - 2021/12/11
Y1 - 2021/12/11
N2 - We consider a problem introduced by Feige, Gamarnik, Neeman, Rácz and Tetali [2020], that of finding a large clique in a random graph G∼G(n,½), where the graph G is accessible by queries to entries of its adjacency matrix. The query model allows some limited adaptivity, with a constant number of rounds of queries, and nδ queries in each round. With high probability, the maximum clique in G is of size roughly 2 log n, and the goal is to find cliques of size α log n, for α as large as possible. We prove that no two-rounds algorithm is likely to find a clique larger than 4⁄3 δlog n, which is a tight upper bound when 1≤δ≤6⁄5. For other ranges of parameters, namely, two-rounds with 6⁄5<δ<2, and three-rounds with 1≤δ<2, we improve over the previously known upper bounds on α, but our upper bounds are not tight. If early rounds are restricted to have fewer queries than the last round, then for some such restrictions we do prove tight upper bounds.
AB - We consider a problem introduced by Feige, Gamarnik, Neeman, Rácz and Tetali [2020], that of finding a large clique in a random graph G∼G(n,½), where the graph G is accessible by queries to entries of its adjacency matrix. The query model allows some limited adaptivity, with a constant number of rounds of queries, and nδ queries in each round. With high probability, the maximum clique in G is of size roughly 2 log n, and the goal is to find cliques of size α log n, for α as large as possible. We prove that no two-rounds algorithm is likely to find a clique larger than 4⁄3 δlog n, which is a tight upper bound when 1≤δ≤6⁄5. For other ranges of parameters, namely, two-rounds with 6⁄5<δ<2, and three-rounds with 1≤δ<2, we improve over the previously known upper bounds on α, but our upper bounds are not tight. If early rounds are restricted to have fewer queries than the last round, then for some such restrictions we do prove tight upper bounds.
U2 - https://doi.org/10.48550/arXiv.2112.06072
DO - https://doi.org/10.48550/arXiv.2112.06072
M3 - مقالة
SN - 2331-8422
JO - arxiv.org
JF - arxiv.org
ER -