58 research outputs found
Polynomial Time Algorithm for Min-Ranks of Graphs with Simple Tree Structures
The min-rank of a graph was introduced by Haemers (1978) to bound the Shannon
capacity of a graph. This parameter of a graph has recently gained much more
attention from the research community after the work of Bar-Yossef et al.
(2006). In their paper, it was shown that the min-rank of a graph G
characterizes the optimal scalar linear solution of an instance of the Index
Coding with Side Information (ICSI) problem described by the graph G. It was
shown by Peeters (1996) that computing the min-rank of a general graph is an
NP-hard problem. There are very few known families of graphs whose min-ranks
can be found in polynomial time. In this work, we introduce a new family of
graphs with efficiently computed min-ranks. Specifically, we establish a
polynomial time dynamic programming algorithm to compute the min-ranks of
graphs having simple tree structures. Intuitively, such graphs are obtained by
gluing together, in a tree-like structure, any set of graphs for which the
min-ranks can be determined in polynomial time. A polynomial time algorithm to
recognize such graphs is also proposed.Comment: Accepted by Algorithmica, 30 page
Caching Piggyback Information for Efficient Index Code Transmission
The index coding problem is a fundamental transmission problem arising in
content distribution and wireless networks. Traditional approach to solve this
problem is to find heuristic/ approximation minimum clique partition solution
on an appropriately mapped graph of the index coding problem. In this paper we
study index code for unicast data flow for which we propose updated clique
index coding (UCIC) scheme, UCIC piggybacks additional information in the coded
symbol such that an unsatisfied client can update its cache. We show that UCIC
has higher coding gain than previously proposed index coding schemes, and it is
optimal for those instances where index code of minimum length is known.Comment: This paper has been accepted for publication in the 39th IEEE
Conference on Local Computer Networks (LCN) to be held in Edmonton, Canada,
Sep. 8-11, 201
Geometry of large Boltzmann outerplanar maps
We study the phase diagram of random outerplanar maps sampled according to
non-negative Boltzmann weights that are assigned to each face of a map. We
prove that for certain choices of weights the map looks like a rescaled version
of its boundary when its number of vertices tends to infinity. The Boltzmann
outerplanar maps are then shown to converge in the Gromov-Hausdorff sense
towards the -stable looptree introduced by Curien and Kortchemski
(2014), with the parameter depending on the specific weight-sequence.
This allows us to describe the transition of the asymptotic geometric shape
from a deterministic circle to the Brownian tree
On Index coding for Complementary Graphs with focus on Circular Perfect Graphs
Circular perfect graphs are those undirected graphs such that the circular
clique number is equal to the circular chromatic number for each induced
subgraph. They form a strict superclass of the perfect graphs, whose index
coding broadcast rates are well known. We present the broadcast rate of index
coding for side-information graphs whose complements are circular perfect,
along with an optimal achievable scheme. We thus enlarge the known classes of
graphs for which the broadcast rate is exactly characterized. In an attempt to
understand the broadcast rate of a graph given that of its complement, we
obtain upper and lower bounds for the product and sum of the vector linear
broadcast rates of a graph and its complement. We show that these bounds are
satisfied with equality even for some perfect graphs. Curating prior results,
we show that there are circular perfect but imperfect graphs which satisfy the
lower bound on the product of the broadcast rate of the complementary graphs
with equality
Linear Programming Approximations for Index Coding
Index coding, a source coding problem over broadcast channels, has been a
subject of both theoretical and practical interest since its introduction (by
Birk and Kol, 1998). In short, the problem can be defined as follows: there is
an input , a set of
clients who each desire a single symbol of the input, and a
broadcaster whose goal is to send as few messages as possible to all clients so
that each one can recover its desired symbol. Additionally, each client has
some predetermined "side information," corresponding to certain symbols of the
input , which we represent as the "side information graph"
. The graph has a vertex for each client and a
directed edge indicating that client knows the th symbol of
the input. Given a fixed side information graph , we are
interested in determining or approximating the "broadcast rate" of index coding
on the graph, i.e. the fewest number of messages the broadcaster can transmit
so that every client gets their desired information.
Using index coding schemes based on linear programs (LPs), we take a
two-pronged approach to approximating the broadcast rate. First, extending
earlier work on planar graphs, we focus on approximating the broadcast rate for
special graph families such as graphs with small chromatic number and disk
graphs. In certain cases, we are able to show that simple LP-based schemes give
constant-factor approximations of the broadcast rate, which seem extremely
difficult to obtain in the general case. Second, we provide several LP-based
schemes for the general case which are not constant-factor approximations, but
which strictly improve on the prior best-known schemes.Comment: To appear in IEEE Transactions on Information Theory, 201
Approximately Counting Embeddings into Random Graphs
Let H be a graph, and let C_H(G) be the number of (subgraph isomorphic)
copies of H contained in a graph G. We investigate the fundamental problem of
estimating C_H(G). Previous results cover only a few specific instances of this
general problem, for example, the case when H has degree at most one
(monomer-dimer problem). In this paper, we present the first general subcase of
the subgraph isomorphism counting problem which is almost always efficiently
approximable. The results rely on a new graph decomposition technique.
Informally, the decomposition is a labeling of the vertices such that every
edge is between vertices with different labels and for every vertex all
neighbors with a higher label have identical labels. The labeling implicitly
generates a sequence of bipartite graphs which permits us to break the problem
of counting embeddings of large subgraphs into that of counting embeddings of
small subgraphs. Using this method, we present a simple randomized algorithm
for the counting problem. For all decomposable graphs H and all graphs G, the
algorithm is an unbiased estimator. Furthermore, for all graphs H having a
decomposition where each of the bipartite graphs generated is small and almost
all graphs G, the algorithm is a fully polynomial randomized approximation
scheme.
We show that the graph classes of H for which we obtain a fully polynomial
randomized approximation scheme for almost all G includes graphs of degree at
most two, bounded-degree forests, bounded-length grid graphs, subdivision of
bounded-degree graphs, and major subclasses of outerplanar graphs,
series-parallel graphs and planar graphs, whereas unbounded-length grid graphs
are excluded.Comment: Earlier version appeared in Random 2008. Fixed an typo in Definition
3.
Algorithms to Exploit Data Sparsity
While data in the real world is very high-dimensional, it generally has some underlying structure; for instance, if we think of an image as a set of pixels with associated color values, most possible settings of color values correspond to something more like random noise than what we typically think of as a picture. With an appropriate transformation of basis, this underlying structure can often be converted into sparsity in data, giving an equivalent representation of the data where the magnitude is large in only a few directions relative to the ambient dimension. This motivates a variety of theoretical questions around designing algorithms that can exploit this data sparsity to achieve better performance than what would be possible naively, and in this thesis we tackle several such questions.We first examine the question of simply approximating the level of sparsity of a signal under several different measurement models, a natural first step if the sparsity is to be exploited by other algorithms. Second, we look at a particular sparse signal recovery problem called nonadaptive probabilistic group testing, and investigate the question of exactly how sparse the signal needs to be before the methods used for recovering sparse signals outperform those used for non-sparse signals. Third, we prove novel upper bounds on the number of measurements needed to recover a sparse signal in the universal one-bit compressed sensing model of sparse signal recovery. Fourth, we give some approximations of an information-theoretic quantity called the index coding rate of a network modeled by a graph, in the special case that the graph is sparse or otherwise highly structured. For each of the problems considered, we also discuss some remaining open questions and conjectures, as well as possible directions towards their solutions
Near-Optimal Induced Universal Graphs for Bounded Degree Graphs
A graph U is an induced universal graph for a family F of graphs if every graph in F is a vertex-induced subgraph of U.
We give upper and lower bounds for the size of induced universal graphs for the family of graphs with n vertices of maximum degree D. Our new bounds improve several previous results except for the special cases where D is either near-constant or almost n/2. For constant even D Butler [Graphs and Combinatorics 2009] has shown O(n^(D/2)) and recently Alon and Nenadov [SODA 2017] showed the same bound for constant odd D. For constant D Butler also gave a matching lower bound. For generals graphs, which corresponds to D = n, Alon [Geometric and Functional Analysis, to appear] proved the existence of an induced universal graph with (1+o(1)) cdot 2^((n-1)/2) vertices, leading to a smaller constant than in the previously best known bound of 16 * 2^(n/2) by Alstrup, Kaplan, Thorup, and Zwick [STOC 2015].
In this paper we give the following lower and upper bound of
binom(floor(n/2))(floor(D/2)) * n^(-O(1))
and
binom(floor(n/2))(floor(D/2)) * 2^(O(sqrt(D log D) * log(n/D))),
respectively, where the upper bound is the main contribution. The proof that it is an induced universal graph relies on a randomized argument. We also give a deterministic upper bound of O(n^k / (k-1)!). These upper bounds are the best known when D <= n/2 - tilde-Omega(n^(3/4)) and either D is even and D = omega(1) or D is odd and D = omega(log n/log log n). In this range we improve asymptotically on the previous best known results by Butler [Graphs and Combinatorics 2009], Esperet, Arnaud and Ochem [IPL 2008], Adjiashvili and Rotbart [ICALP 2014], Alon and Nenadov [SODA 2017], and Alon [Geometric and Functional Analysis, to appear]
Linear Index Coding via Graph Homomorphism
It is known that the minimum broadcast rate of a linear index code over
is equal to the of the underlying digraph. In [3] it
is proved that for and any positive integer ,
iff there exists a homomorphism from the complement of the
graph to the complement of a particular undirected graph family called
"graph family ". As observed in [2], by combining these two results
one can relate the linear index coding problem of undirected graphs to the
graph homomorphism problem. In [4], a direct connection between linear index
coding problem and graph homomorphism problem is introduced. In contrast to the
former approach, the direct connection holds for digraphs as well and applies
to any field size. More precisely, in [4], a graph family is
introduced and shown that whether or not the scalar linear index of a digraph
is less than or equal to is equivalent to the existence of a graph
homomorphism from the complement of to the complement of .
Here, we first study the structure of the digraphs . Analogous to the
result of [2] about undirected graphs, we prove that 's are vertex
transitive digraphs. Using this, and by applying a lemma of Hell and Nesetril
[5], we derive a class of necessary conditions for digraphs to satisfy
. Particularly, we obtain new lower bounds on .
Our next result is about the computational complexity of scalar linear index
of a digraph. It is known that deciding whether the scalar linear index of an
undirected graph is equal to or not is NP-complete for and is
polynomially decidable for [3]. For digraphs, it is shown in [6] that
for the binary alphabet, the decision problem for is NP-complete. We use
graph homomorphism framework to extend this result to arbitrary alphabet.Comment: 10 pages, to appear in the 2nd International Conference on Control,
Decision and Information Technologies (CoDIT'14
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