51 research outputs found
Limits on Counting Triangles using Bipartite Independent Set Queries
Beame et al. [ITCS 2018 & TALG 2021] introduced and used the Bipartite
Independent Set (BIS) and Independent Set (IS) oracle access to an unknown,
simple, unweighted and undirected graph and solved the edge estimation problem.
The introduction of this oracle set forth a series of works in a short span of
time that either solved open questions mentioned by Beame et al. or were
generalizations of their work as in Dell and Lapinskas [STOC 2018], Dell,
Lapinskas and Meeks [SODA 2020], Bhattacharya et al. [ISAAC 2019 & Theory
Comput. Syst. 2021], and Chen et al. [SODA 2020]. Edge estimation using BIS can
be done using polylogarithmic queries, while IS queries need sub-linear but
more than polylogarithmic queries. Chen et al. improved Beame et al.'s upper
bound result for edge estimation using IS and also showed an almost matching
lower bound. Beame et al. in their introductory work asked a few open questions
out of which one was on estimating structures of higher order than edges, like
triangles and cliques, using BIS queries. Motivated by this question, we
completely resolve the query complexity of estimating triangles using BIS
oracle. While doing so, we prove a lower bound for an even stronger query
oracle called Edge Emptiness (EE) oracle, recently introduced by Assadi,
Chakrabarty and Khanna [ESA 2021] to test graph connectivity.Comment: 30 page
Distance Estimation Between Unknown Matrices Using Sublinear Projections on Hamming Cube
Using geometric techniques like projection and dimensionality reduction, we
show that there exists a randomized sub-linear time algorithm that can estimate
the Hamming distance between two matrices. Consider two matrices and
of size whose dimensions are known to the algorithm but
the entries are not. The entries of the matrix are real numbers. The access to
any matrix is through an oracle that computes the projection of a row (or a
column) of the matrix on a vector in . We call this query oracle to
be an {\sc Inner Product} oracle (shortened as {\sc IP}). We show that our
algorithm returns a approximation to with high probability by making {\cal
O}\left(\frac{n}{\sqrt{{{\bf D}}_{\bf M} ({\bf A},{\bf
B})}}\mbox{poly}\left(\log n, \frac{1}{\epsilon}\right)\right) oracle queries,
where denotes the Hamming distance (the
number of corresponding entries in which and differ)
between two matrices and of size . We also show
a matching lower bound on the number of such {\sc IP} queries needed. Though
our main result is on estimating using
{\sc IP}, we also compare our results with other query models.Comment: 30 pages. Accepted in RANDOM'2
Triangle Estimation Using Tripartite Independent Set Queries
Estimating the number of triangles in a graph is one of the most fundamental problems in sublinear algorithms. In this work, we provide an approximate triangle counting algorithm using only polylogarithmic queries when the number of triangles on any edge in the graph is polylogarithmically bounded. Our query oracle Tripartite Independent Set (TIS) takes three disjoint sets of vertices A, B and C as input, and answers whether there exists a triangle having one endpoint in each of these three sets. Our query model generally belongs to the class of group queries (Ron and Tsur, ACM ToCT, 2016; Dell and Lapinskas, STOC 2018) and in particular is inspired by the Bipartite Independent Set (BIS) query oracle of Beame et al. (ITCS 2018). We extend the algorithmic framework of Beame et al., with TIS replacing BIS, for triangle counting using ideas from color coding due to Alon et al. (J. ACM, 1995) and a concentration inequality for sums of random variables with bounded dependency (Janson, Rand. Struct. Alg., 2004)
-set problem in graphs
A subset of a graph is a -set if every
vertex is adjacent to at least but not more than
vertices in D. The cardinality of a minimum -set of , denoted as
, is called the -domination number of . Given a
graph and an integer , the decision version of the -set
problem is to decide whether has a -set of cardinality at most .
In this paper, we first obtain an upper bound on using
probabilistic methods, for bounded minimum and maximum degree graphs. Our bound
is constructive, by the randomized algorithm of Moser and Tardos [MT10], We
also show that the - set problem is NP-complete for chordal graphs.
Finally, we design two algorithms for finding of a tree
and a split graph, for any fixed , which answers an open question posed in
[CHHM13]
Query Complexity of Global Minimum Cut
In this work, we resolve the query complexity of global minimum cut problem for a graph by designing a randomized algorithm for approximating the size of minimum cut in a graph, where the graph can be accessed through local queries like Degree, Neighbor, and Adjacency queries.
Given ? ? (0,1), the algorithm with high probability outputs an estimate t? satisfying the following (1-?) t ? t? ? (1+?) t, where t is the size of minimum cut in the graph. The expected number of local queries used by our algorithm is min{m+n,m/t}poly(log n,1/(?)) where n and m are the number of vertices and edges in the graph, respectively. Eden and Rosenbaum showed that ?(m/t) local queries are required for approximating the size of minimum cut in graphs, {but no local query based algorithm was known. Our algorithmic result coupled with the lower bound of Eden and Rosenbaum [APPROX 2018] resolve the query complexity of the problem of estimating the size of minimum cut in graphs using local queries.}
Building on the lower bound of Eden and Rosenbaum, we show that, for all t ? ?, ?(m) local queries are required to decide if the size of the minimum cut in the graph is t or t-2. Also, we show that, for any t ? ?, ?(m) local queries are required to find all the minimum cut edges even if it is promised that the input graph has a minimum cut of size t. Both of our lower bound results are randomized, and hold even if we can make Random Edge queries in addition to local queries
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