591,783 research outputs found
On Approximating the Number of -cliques in Sublinear Time
We study the problem of approximating the number of -cliques in a graph
when given query access to the graph.
We consider the standard query model for general graphs via (1) degree
queries, (2) neighbor queries and (3) pair queries. Let denote the number
of vertices in the graph, the number of edges, and the number of
-cliques. We design an algorithm that outputs a
-approximation (with high probability) for , whose
expected query complexity and running time are
O\left(\frac{n}{C_k^{1/k}}+\frac{m^{k/2}}{C_k}\right)\poly(\log
n,1/\varepsilon,k).
Hence, the complexity of the algorithm is sublinear in the size of the graph
for . Furthermore, we prove a lower bound showing that
the query complexity of our algorithm is essentially optimal (up to the
dependence on , and ).
The previous results in this vein are by Feige (SICOMP 06) and by Goldreich
and Ron (RSA 08) for edge counting () and by Eden et al. (FOCS 2015) for
triangle counting (). Our result matches the complexities of these
results.
The previous result by Eden et al. hinges on a certain amortization technique
that works only for triangle counting, and does not generalize for larger
cliques. We obtain a general algorithm that works for any by
designing a procedure that samples each -clique incident to a given set
of vertices with approximately equal probability. The primary difficulty is in
finding cliques incident to purely high-degree vertices, since random sampling
within neighbors has a low success probability. This is achieved by an
algorithm that samples uniform random high degree vertices and a careful
tradeoff between estimating cliques incident purely to high-degree vertices and
those that include a low-degree vertex
New results on lower bounds for the number of k-facets
In this paper we present three different results dealing with the number of (≤ k)- facets of a set of points: (i) We give structural properties of sets in the plane that achieve the optimal lower bound 3_k+2 2 _ of (≤ k)-edges for a fixed k ≤ [n/3 ]− 1; (ii) We show that the new lower bound 3((k+2) 2 ) + 3((k−(n/ 3)+2) 2 ) for the number of (≤ k)-edges of a planar point set is optimal in the range [n/3] ≤ k ≤ [5n/12] − 1; (iii) We show that for k < n/4 the number of (≤ k)-facets of a set of n points in R3 in general position is at least 4((k+3 )3 ), and that this bound is tight in that range
On lower bounds for the matching number of subcubic graphs
We give a complete description of the set of triples (a,b,c) of real numbers
with the following property. There exists a constant K such that a n_3 + b n_2
+ c n_1 - K is a lower bound for the matching number of every connected
subcubic graph G, where n_i denotes the number of vertices of degree i for each
i
On the Signed -independence Number of Graphs
In this paper, we study the signed 2-independence number in graphs and give new sharp upper and lower bounds on the signed 2-independence number of a graph by a simple uniform approach. In this way, we can improve and generalize some known results in this area
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