129,964 research outputs found
The degree-diameter problem for sparse graph classes
The degree-diameter problem asks for the maximum number of vertices in a
graph with maximum degree and diameter . For fixed , the answer
is . We consider the degree-diameter problem for particular
classes of sparse graphs, and establish the following results. For graphs of
bounded average degree the answer is , and for graphs of
bounded arboricity the answer is \Theta(\Delta^{\floor{k/2}}), in both cases
for fixed . For graphs of given treewidth, we determine the the maximum
number of vertices up to a constant factor. More precise bounds are given for
graphs of given treewidth, graphs embeddable on a given surface, and
apex-minor-free graphs
Sparse geometric graphs with small dilation
Given a set S of n points in R^D, and an integer k such that 0 <= k < n, we
show that a geometric graph with vertex set S, at most n - 1 + k edges, maximum
degree five, and dilation O(n / (k+1)) can be computed in time O(n log n). For
any k, we also construct planar n-point sets for which any geometric graph with
n-1+k edges has dilation Omega(n/(k+1)); a slightly weaker statement holds if
the points of S are required to be in convex position
Undirected Connectivity of Sparse Yao Graphs
Given a finite set S of points in the plane and a real value d > 0, the
d-radius disk graph G^d contains all edges connecting pairs of points in S that
are within distance d of each other. For a given graph G with vertex set S, the
Yao subgraph Y_k[G] with integer parameter k > 0 contains, for each point p in
S, a shortest edge pq from G (if any) in each of the k sectors defined by k
equally-spaced rays with origin p. Motivated by communication issues in mobile
networks with directional antennas, we study the connectivity properties of
Y_k[G^d], for small values of k and d. In particular, we derive lower and upper
bounds on the minimum radius d that renders Y_k[G^d] connected, relative to the
unit radius assumed to render G^d connected. We show that d=sqrt(2) is
necessary and sufficient for the connectivity of Y_4[G^d]. We also show that,
for d =
2/sqrt(3), Y_3[G^d] is always connected. Finally, we show that Y_2[G^d] can be
disconnected, for any d >= 1.Comment: 7 pages, 11 figure
On k-Column Sparse Packing Programs
We consider the class of packing integer programs (PIPs) that are column
sparse, i.e. there is a specified upper bound k on the number of constraints
that each variable appears in. We give an (ek+o(k))-approximation algorithm for
k-column sparse PIPs, improving on recent results of and
. We also show that the integrality gap of our linear programming
relaxation is at least 2k-1; it is known that k-column sparse PIPs are
-hard to approximate. We also extend our result (at the loss
of a small constant factor) to the more general case of maximizing a submodular
objective over k-column sparse packing constraints.Comment: 19 pages, v3: additional detail
Index statistical properties of sparse random graphs
Using the replica method, we develop an analytical approach to compute the
characteristic function for the probability that a
large adjacency matrix of sparse random graphs has eigenvalues
below a threshold . The method allows to determine, in principle, all
moments of , from which the typical sample to sample
fluctuations can be fully characterized. For random graph models with localized
eigenvectors, we show that the index variance scales linearly with
for , with a model-dependent prefactor that can be exactly
calculated. Explicit results are discussed for Erd\"os-R\'enyi and regular
random graphs, both exhibiting a prefactor with a non-monotonic behavior as a
function of . These results contrast with rotationally invariant
random matrices, where the index variance scales only as , with an
universal prefactor that is independent of . Numerical diagonalization
results confirm the exactness of our approach and, in addition, strongly
support the Gaussian nature of the index fluctuations.Comment: 10 pages, 5 figure
Sparse square roots.
We show that it can be decided in polynomial time whether a graph of maximum degree 6 has a square root; if a square root exists, then our algorithm finds one with minimum number of edges. We also show that it is FPT to decide whether a connected n-vertex graph has a square root with at most n − 1 + k edges when this problem is parameterized by k. Finally, we give an exact exponential time algorithm for the problem of finding a square root with maximum number of edges
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