Quantum Query Complexity of Subgraph Isomorphism and Homomorphism

Let $H$ be a fixed graph on $n$ vertices. Let $f_H(G) = 1$ iff the input graph $G$ on $n$ vertices contains $H$ as a (not necessarily induced) subgraph. Let $\alpha_H$ denote the cardinality of a maximum independent set of $H$. In this paper we show: $$Q(f_H) = \Omega\left(\sqrt{\alpha_H \cdot n}\right),$$ where $Q(f_H)$ denotes the quantum query complexity of $f_H$. As a consequence we obtain a lower bounds for $Q(f_H)$ in terms of several other parameters of $H$ such as the average degree, minimum vertex cover, chromatic number, and the critical probability. We also use the above bound to show that $Q(f_H) = \Omega(n^{3/4})$ for any $H$, improving on the previously best known bound of $\Omega(n^{2/3})$. Until very recently, it was believed that the quantum query complexity is at least square root of the randomized one. Our $\Omega(n^{3/4})$ bound for $Q(f_H)$ matches the square root of the current best known bound for the randomized query complexity of $f_H$, which is $\Omega(n^{3/2})$ due to Gr\"oger. Interestingly, the randomized bound of $\Omega(\alpha_H \cdot n)$ for $f_H$ still remains open. We also study the Subgraph Homomorphism Problem, denoted by $f_{[H]}$, and show that $Q(f_{[H]}) = \Omega(n)$. Finally we extend our results to the $3$-uniform hypergraphs. In particular, we show an $\Omega(n^{4/5})$ bound for quantum query complexity of the Subgraph Isomorphism, improving on the previously known $\Omega(n^{3/4})$ bound. For the Subgraph Homomorphism, we obtain an $\Omega(n^{3/2})$ 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 $NC^2$ 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 $H$, "forbidden subgraph $H$" is eventually evasive and (b) all nontrivial monotone properties of graphs with $\le n^{3/2-\epsilon}$ edges are eventually evasive. ($n$ 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 $L$ functions), we show (b) with the bound $O(n^{5/4-\epsilon})$ under ERH. We also prove unconditional results: (a$'$) for any graph $H$, the query complexity of "forbidden subgraph $H$" is $\binom{n}{2} - O(1)$; (b$'$) for some constant $c>0$, all nontrivial monotone properties of graphs with $\le cn\log n+O(1)$ 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 $k\geq 2$), \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 $\mathrm{minHash}$ algorithm that computes a low-dimensional sketch of high-dimensional binary data that closely approximates pairwise Jaccard similarity. Since its invention, $\mathrm{minHash}$ 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 $\mathrm{minHash}$ 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 $\mathrm{minHash}$ 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 $\mathrm{minHash}$ 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 $\mathrm{minHash}$ from scratch. Our proposal is efficient, accurate, and easy to implement in practice

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

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