129 research outputs found
Quantum Query Complexity of Subgraph Isomorphism and Homomorphism
Let be a fixed graph on vertices. Let iff the input
graph on vertices contains as a (not necessarily induced) subgraph.
Let denote the cardinality of a maximum independent set of . In
this paper we show:
where
denotes the quantum query complexity of .
As a consequence we obtain a lower bounds for in terms of several
other parameters of such as the average degree, minimum vertex cover,
chromatic number, and the critical probability.
We also use the above bound to show that for any
, improving on the previously best known bound of . Until
very recently, it was believed that the quantum query complexity is at least
square root of the randomized one. Our bound for
matches the square root of the current best known bound for the randomized
query complexity of , which is due to Gr\"oger.
Interestingly, the randomized bound of for
still remains open.
We also study the Subgraph Homomorphism Problem, denoted by , and
show that .
Finally we extend our results to the -uniform hypergraphs. In particular,
we show an bound for quantum query complexity of the Subgraph
Isomorphism, improving on the previously known bound. For the
Subgraph Homomorphism, we obtain an bound for the same.Comment: 16 pages, 2 figure
Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs
We present a deterministic way of assigning small (log bit) weights to the
edges of a bipartite planar graph so that the minimum weight perfect matching
becomes unique. The isolation lemma as described in (Mulmuley et al. 1987)
achieves the same for general graphs using a randomized weighting scheme,
whereas we can do it deterministically when restricted to bipartite planar
graphs. As a consequence, we reduce both decision and construction versions of
the matching problem to testing whether a matrix is singular, under the promise
that its determinant is 0 or 1, thus obtaining a highly parallel SPL algorithm
for bipartite planar graphs. This improves the earlier known bounds of
non-uniform SPL by (Allender et al. 1999) and by (Miller and Naor 1995,
Mahajan and Varadarajan 2000). It also rekindles the hope of obtaining a
deterministic parallel algorithm for constructing a perfect matching in
non-bipartite planar graphs, which has been open for a long time. Our
techniques are elementary and simple
Efficient Compression Technique for Sparse Sets
Recent technological advancements have led to the generation of huge amounts
of data over the web, such as text, image, audio and video. Most of this data
is high dimensional and sparse, for e.g., the bag-of-words representation used
for representing text. Often, an efficient search for similar data points needs
to be performed in many applications like clustering, nearest neighbour search,
ranking and indexing. Even though there have been significant increases in
computational power, a simple brute-force similarity-search on such datasets is
inefficient and at times impossible. Thus, it is desirable to get a compressed
representation which preserves the similarity between data points. In this
work, we consider the data points as sets and use Jaccard similarity as the
similarity measure. Compression techniques are generally evaluated on the
following parameters --1) Randomness required for compression, 2) Time required
for compression, 3) Dimension of the data after compression, and 4) Space
required to store the compressed data. Ideally, the compressed representation
of the data should be such, that the similarity between each pair of data
points is preserved, while keeping the time and the randomness required for
compression as low as possible.
We show that the compression technique suggested by Pratap and Kulkarni also
works well for Jaccard similarity. We present a theoretical proof of the same
and complement it with rigorous experimentations on synthetic as well as
real-world datasets. We also compare our results with the state-of-the-art
"min-wise independent permutation", and show that our compression algorithm
achieves almost equal accuracy while significantly reducing the compression
time and the randomness
Evasiveness and the Distribution of Prime Numbers
We confirm the eventual evasiveness of several classes of monotone graph
properties under widely accepted number theoretic hypotheses. In particular we
show that Chowla's conjecture on Dirichlet primes implies that (a) for any
graph , "forbidden subgraph " is eventually evasive and (b) all
nontrivial monotone properties of graphs with edges are
eventually evasive. ( is the number of vertices.)
While Chowla's conjecture is not known to follow from the Extended Riemann
Hypothesis (ERH, the Riemann Hypothesis for Dirichlet's functions), we show
(b) with the bound under ERH.
We also prove unconditional results: (a) for any graph , the query
complexity of "forbidden subgraph " is ; (b) for
some constant , all nontrivial monotone properties of graphs with edges are eventually evasive.
Even these weaker, unconditional results rely on deep results from number
theory such as Vinogradov's theorem on the Goldbach conjecture.
Our technical contribution consists in connecting the topological framework
of Kahn, Saks, and Sturtevant (1984), as further developed by Chakrabarti,
Khot, and Shi (2002), with a deeper analysis of the orbital structure of
permutation groups and their connection to the distribution of prime numbers.
Our unconditional results include stronger versions and generalizations of some
result of Chakrabarti et al.Comment: 12 pages (conference version for STACS 2010
Space Complexity of Perfect Matching in Bounded Genus Bipartite Graphs
We investigate the space complexity of certain perfect matching problems over
bipartite graphs embedded on surfaces of constant genus (orientable or
non-orientable). We show that the problems of deciding whether such graphs have
(1) a perfect matching or not and (2) a unique perfect matching or not, are in
the logspace complexity class \SPL. Since \SPL\ is contained in the logspace
counting classes \oplus\L (in fact in \modk\ for all ), \CeqL, and
\PL, our upper bound places the above-mentioned matching problems in these
counting classes as well. We also show that the search version, computing a
perfect matching, for this class of graphs is in \FL^{\SPL}. Our results
extend the same upper bounds for these problems over bipartite planar graphs
known earlier. As our main technical result, we design a logspace computable
and polynomially bounded weight function which isolates a minimum weight
perfect matching in bipartite graphs embedded on surfaces of constant genus. We
use results from algebraic topology for proving the correctness of the weight
function.Comment: 23 pages, 13 figure
Minwise-Independent Permutations with Insertion and Deletion of Features
In their seminal work, Broder \textit{et. al.}~\citep{BroderCFM98} introduces
the algorithm that computes a low-dimensional sketch of
high-dimensional binary data that closely approximates pairwise Jaccard
similarity. Since its invention, has been commonly used by
practitioners in various big data applications. Further, the data is dynamic in
many real-life scenarios, and their feature sets evolve over time. We consider
the case when features are dynamically inserted and deleted in the dataset. We
note that a naive solution to this problem is to repeatedly recompute
with respect to the updated dimension. However, this is an
expensive task as it requires generating fresh random permutations. To the best
of our knowledge, no systematic study of is recorded in the
context of dynamic insertion and deletion of features. In this work, we
initiate this study and suggest algorithms that make the
sketches adaptable to the dynamic insertion and deletion of features. We show a
rigorous theoretical analysis of our algorithms and complement it with
extensive experiments on several real-world datasets. Empirically we observe a
significant speed-up in the running time while simultaneously offering
comparable performance with respect to running from scratch.
Our proposal is efficient, accurate, and easy to implement in practice
Trust in Hospitals-Evidence from India
Various explanations have been offered for outbursts of violence against doctors and other staff in India, drawing attention to growing supply-demand imbalance in healthcare, quality deterioration, overburdened doctors, weak security for medical staff, high expectations of patients who come in advanced stages of chronic and other illnesses, overcrowding of public hospitals with limited sanitary facilities. But underlying all these explanations is lack of trust in doctors and hospitals-especially public. Our focus here is on trust and its covariates over the period 2005-2012. The motivation stems from the fact that the existing evidence is patchy and scattered. Our aim, therefore, is to build on the empirical evidence through a systematic state-of-art analysis of trust in public and private hospitals and doctors. Combining our analysis with other evidence, we identify specific challenges to build patient-hospital trust and how these could be overcome
Poverty Transitions, Health, and Socio-Economic Disparities in India
SDGs offer an inclusive and just vision for 2030, in which the interrelationships between (near) elimination of poverty, health reforms and elimination of socio-economic disparities play an important role. The present study focuses on the associations between poverty transitions over a period, and health indicators such as NCDs, disabilities, socio-economic disparities, state affluence and inequality in income distribution. These health indicators reflect their growing importance in recent years. We have used a Multinomial Probit specification which is an improvement on the methodologies used in earlier research. The analysis is based on panel data from the India Human Development Survey 2015. What our analysis emphasises is that changes in the prevalence of poverty/headcount ratio over time do not throw light on how poverty has evolved: whether there were escapes from poverty, whether there were descents into poverty, whether segments persisted in poverty, and whether (the relatively) affluent remained largely unaffected. A significant contribution of this study is to explore the relationships between such poverty transitions and NCDs and disabilities, socio-economic disparities and other covariates. The analysis confirms these linkages. Drawing upon this analysis and other relevant research, policy challenges in achieving the SDG vision of an inclusive and fair economy are delineated
Persistence of Non-Communicable Diseases, Affluence and Inequality in India
This study builds on the extant literature by highlighting the persistence of non-communicable diseases (NCDs), their cross-associations, and how these diseases are linked to different forms of inequality-socio-economic, gaps in affluence measured by asset quartile, and in the overall economic environment, based on a nation-wide panel survey, India Human Development Survey 2015. A multinomial probit specification is used to analyse NCD outcomes. Those at the bottom of the caste hierarchy and least wealthy exhibit lowest vulnerability to NCDs despite their deprivation and limited access to healthcare facilities while those at the higher end of the caste hierarchy and the wealthiest are most vulnerable. However, overall economic inequality, using Piketty’s (2013) measure, is insidious as it corrodes social cohesion and support, and the capability to live a healthy and productive life. New light is thrown on whether social networks are associated with better NCD outcomes. So policy interventions have to be not just medical but much broader in scope
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