2,087 research outputs found
Large-girth roots of graphs
We study the problem of recognizing graph powers and computing roots of
graphs. We provide a polynomial time recognition algorithm for r-th powers of
graphs of girth at least 2r+3, thus improving a bound conjectured by Farzad et
al. (STACS 2009). Our algorithm also finds all r-th roots of a given graph that
have girth at least 2r+3 and no degree one vertices, which is a step towards a
recent conjecture of Levenshtein that such root should be unique. On the
negative side, we prove that recognition becomes an NP-complete problem when
the bound on girth is about twice smaller. Similar results have so far only
been attempted for r=2,3.Comment: 14 pages, 4 figure
Approximating the Largest Root and Applications to Interlacing Families
We study the problem of approximating the largest root of a real-rooted
polynomial of degree using its top coefficients and give nearly
matching upper and lower bounds. We present algorithms with running time
polynomial in that use the top coefficients to approximate the maximum
root within a factor of and when and respectively. We also prove corresponding
information-theoretic lower bounds of and
, and show strong lower
bounds for noisy version of the problem in which one is given access to
approximate coefficients.
This problem has applications in the context of the method of interlacing
families of polynomials, which was used for proving the existence of Ramanujan
graphs of all degrees, the solution of the Kadison-Singer problem, and bounding
the integrality gap of the asymmetric traveling salesman problem. All of these
involve computing the maximum root of certain real-rooted polynomials for which
the top few coefficients are accessible in subexponential time. Our results
yield an algorithm with the running time of for all
of them
Computing Graph Roots Without Short Cycles
Graph G is the square of graph H if two vertices x, y have an edge in G if
and only if x, y are of distance at most two in H. Given H it is easy to
compute its square H2, however Motwani and Sudan proved that it is NP-complete
to determine if a given graph G is the square of some graph H (of girth 3). In
this paper we consider the characterization and recognition problems of graphs
that are squares of graphs of small girth, i.e. to determine if G = H2 for some
graph H of small girth. The main results are the following. - There is a graph
theoretical characterization for graphs that are squares of some graph of girth
at least 7. A corollary is that if a graph G has a square root H of girth at
least 7 then H is unique up to isomorphism. - There is a polynomial time
algorithm to recognize if G = H2 for some graph H of girth at least 6. - It is
NP-complete to recognize if G = H2 for some graph H of girth 4. These results
almost provide a dichotomy theorem for the complexity of the recognition
problem in terms of girth of the square roots. The algorithmic and graph
theoretical results generalize previous results on tree square roots, and
provide polynomial time algorithms to compute a graph square root of small
girth if it exists. Some open questions and conjectures will also be discussed
Square-Root Finding Problem In Graphs, A Complete Dichotomy Theorem
Graph G is the square of graph H if two vertices x,y have an edge in G if and
only if x,y are of distance at most two in H. Given H it is easy to compute its
square H^2. Determining if a given graph G is the square of some graph is not
easy in general. Motwani and Sudan proved that it is NP-complete to determine
if a given graph G is the square of some graph. The graph introduced in their
reduction is a graph that contains many triangles and is relatively dense.
Farzad et al. proved the NP-completeness for finding a square root for girth 4
while they gave a polynomial time algorithm for computing a square root of
girth at least six. Adamaszek and Adamaszek proved that if a graph has a square
root of girth six then this square root is unique up to isomorphism. In this
paper we consider the characterization and recognition problem of graphs that
are square of graphs of girth at least five. We introduce a family of graphs
with exponentially many non-isomorphic square roots, and as the main result of
this paper we prove that the square root finding problem is NP-complete for
square roots of girth five. This proof is providing the complete dichotomy
theorem for square root problem in terms of the girth of the square roots
An Alon-Boppana Type Bound for Weighted Graphs and Lowerbounds for Spectral Sparsification
We prove the following Alon-Boppana type theorem for general (not necessarily
regular) weighted graphs: if is an -node weighted undirected graph of
average combinatorial degree (that is, has edges) and girth , and if are the
eigenvalues of the (non-normalized) Laplacian of , then (The Alon-Boppana theorem implies that if is unweighted and
-regular, then if the diameter is at least .)
Our result implies a lower bound for spectral sparsifiers. A graph is a
spectral -sparsifier of a graph if where is the Laplacian matrix of and is
the Laplacian matrix of . Batson, Spielman and Srivastava proved that for
every there is an -sparsifier of average degree where
and the edges of are a
(weighted) subset of the edges of . Batson, Spielman and Srivastava also
show that the bound on cannot be reduced below when is a clique; our Alon-Boppana-type result implies that
cannot be reduced below when comes
from a family of expanders of super-constant degree and super-constant girth.
The method of Batson, Spielman and Srivastava proves a more general result,
about sparsifying sums of rank-one matrices, and their method applies to an
"online" setting. We show that for the online matrix setting the bound is tight, up to lower order terms
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