On Index Coding and Graph Homomorphism

In this work, we study the problem of index coding from graph homomorphism perspective. We show that the minimum broadcast rate of an index coding problem for different variations of the problem such as non-linear, scalar, and vector index code, can be upper bounded by the minimum broadcast rate of another index coding problem when there exists a homomorphism from the complement of the side information graph of the first problem to that of the second problem. As a result, we show that several upper bounds on scalar and vector index code problem are special cases of one of our main theorems. For the linear scalar index coding problem, it has been shown in [1] that the binary linear index of a graph is equal to a graph theoretical parameter called minrank of the graph. For undirected graphs, in [2] it is shown that $\mathrm{minrank}(G) = k$ if and only if there exists a homomorphism from $\bar{G}$ to a predefined graph $\bar{G}_k$. Combining these two results, it follows that for undirected graphs, all the digraphs with linear index of at most k coincide with the graphs $G$ for which there exists a homomorphism from $\bar{G}$ to $\bar{G}_k$. In this paper, we give a direct proof to this result that works for digraphs as well. We show how to use this classification result to generate lower bounds on scalar and vector index. In particular, we provide a lower bound for the scalar index of a digraph in terms of the chromatic number of its complement. Using our framework, we show that by changing the field size, linear index of a digraph can be at most increased by a factor that is independent from the number of the nodes.Comment: 5 pages, to appear in "IEEE Information Theory Workshop", 201

Subdeterminant Maximization via Nonconvex Relaxations and Anti-concentration

Several fundamental problems that arise in optimization and computer science can be cast as follows: Given vectors $v_1,\ldots,v_m \in \mathbb{R}^d$ and a constraint family ${\cal B}\subseteq 2^{[m]}$, find a set $S \in \cal{B}$ that maximizes the squared volume of the simplex spanned by the vectors in $S$. A motivating example is the data-summarization problem in machine learning where one is given a collection of vectors that represent data such as documents or images. The volume of a set of vectors is used as a measure of their diversity, and partition or matroid constraints over $[m]$ are imposed in order to ensure resource or fairness constraints. Recently, Nikolov and Singh presented a convex program and showed how it can be used to estimate the value of the most diverse set when ${\cal B}$ corresponds to a partition matroid. This result was recently extended to regular matroids in works of Straszak and Vishnoi, and Anari and Oveis Gharan. The question of whether these estimation algorithms can be converted into the more useful approximation algorithms -- that also output a set -- remained open. The main contribution of this paper is to give the first approximation algorithms for both partition and regular matroids. We present novel formulations for the subdeterminant maximization problem for these matroids; this reduces them to the problem of finding a point that maximizes the absolute value of a nonconvex function over a Cartesian product of probability simplices. The technical core of our results is a new anti-concentration inequality for dependent random variables that allows us to relate the optimal value of these nonconvex functions to their value at a random point. Unlike prior work on the constrained subdeterminant maximization problem, our proofs do not rely on real-stability or convexity and could be of independent interest both in algorithms and complexity.Comment: in FOCS 201

Fractional forcing number of graphs

The notion of forcing sets for perfect matchings was introduced by Harary, Klein, and \v{Z}ivkovi\'{c}. The application of this problem in chemistry, as well as its interesting theoretical aspects, made this subject very active. In this work, we introduce the notion of the forcing function of fractional perfect matchings which is continuous analogous to forcing sets defined over the perfect matching polytope of graphs. We show that our defined object is a continuous and concave function extension of the integral forcing set. Then, we use our results about this extension to conclude new bounds and results about the integral case of forcing sets for the family of edge and vertex-transitive graphs and in particular, hypercube graphs

Differentially Private All-Pairs Shortest Distances for Low Tree-Width Graphs

In this paper, we present a polynomial time algorithm for the problem of differentially private all pair shortest distances over the class of low tree-width graphs. Our result generalizes the result of Sealfon 2016 for the case of trees to a much larger family of graphs. Furthermore, if we restrict to the class of low tree-width graphs, the additive error of our algorithm is significantly smaller than that of the best known algorithm for this problem, proposed by Chen et. al. 2023

On the sum of two largest eigenvalues of a symmetric matrix

D. Gernert conjectured that the sum of two largest eigenvalues of the adjacency matrix of any simple graph is at most the number of vertices of the graph. This can be proved, in particular, for all regular graphs. Gernert’s conjecture was recently disproved by one of the authors [4], who also provided a nontrivial upper bound for the sum of two largest eigenvalues. In this paper we improve the lower and upper bounds to near-optimal ones, and extend results from graphs to general non-negative matrices

Optimal Binary Differential Privacy via Graphs

We present the notion of \emph{reasonable utility} for binary mechanisms, which applies to all utility functions in the literature. This notion induces a partial ordering on the performance of all binary differentially private (DP) mechanisms. DP mechanisms that are maximal elements of this ordering are optimal DP mechanisms for every reasonable utility. By looking at differential privacy as a randomized graph coloring, we characterize these optimal DP in terms of their behavior on a certain subset of the boundary datasets we call a boundary hitting set. In the process of establishing our results, we also introduce a useful notion that generalizes DP conditions for binary-valued queries, which we coin as suitable pairs. Suitable pairs abstract away the algebraic roles of $\varepsilon,\delta$ in the DP framework, making the derivations and understanding of our proofs simpler. Additionally, the notion of a suitable pair can potentially capture privacy conditions in frameworks other than DP and may be of independent interest

Network Simplification: The Gaussian diamond network with multiple antennas

We consider the N-relay Gaussian diamond network when the source and the destination have n(s) >= 2 and n(d) >= 2 antennas respectively. We show that when n(s) = n(d) = 2 and when the individual MISO channels from the source to each relay and the SIMO channels from each relay to the destination have the same capacity, there exists a two relay sub-network that achieves approximately all the capacity of the network. To prove this result, we establish a simple relation between the joint entropies of three Gaussian random variables, which is not implied by standard Shannon-type entropy inequalities.(1

The clinical and genetic spectrum of autosomal-recessive TOR1A-related disorders.

In the field of rare diseases, progress in molecular diagnostics led to the recognition that variants linked to autosomal-dominant neurodegenerative diseases of later onset can, in the context of biallelic inheritance, cause devastating neurodevelopmental disorders and infantile or childhood-onset neurodegeneration. TOR1A-associated arthrogryposis multiplex congenita 5 (AMC5) is a rare neurodevelopmental disorder arising from biallelic variants in TOR1A, a gene that in the heterozygous state is associated to torsion dystonia-1 (DYT1 or DYT-TOR1A), an early-onset dystonia with reduced penetrance. While 15 individuals with TOR1A-AMC5 have been reported (less than 10 in detail), a systematic investigation of the full disease-associated spectrum has not been conducted. Here, we assess the clinical, radiological and molecular characteristics of 57 individuals from 40 families with biallelic variants in TOR1A. Median age at last follow-up was 3 years (0-24 years). Most individuals presented with severe congenital flexion contractures (95%) and variable developmental delay (79%). Motor symptoms were reported in 79% and included lower limb spasticity and pyramidal signs, as well as gait disturbances. Facial dysmorphism was an integral part of the phenotype, with key features being a broad/full nasal tip, narrowing of the forehead and full cheeks. Analysis of disease-associated manifestations delineated a phenotypic spectrum ranging from normal cognition and mild gait disturbance to congenital arthrogryposis, global developmental delay, intellectual disability, absent speech and inability to walk. In a subset, the presentation was consistent with fetal akinesia deformation sequence with severe intrauterine abnormalities. Survival was 71% with higher mortality in males. Death occurred at a median age of 1.2 months (1 week - 9 years) due to respiratory failure, cardiac arrest, or sepsis. Analysis of brain MRI studies identified non-specific neuroimaging features, including a hypoplastic corpus callosum (72%), foci of signal abnormality in the subcortical and periventricular white matter (55%), diffuse white matter volume loss (45%), mega cisterna magna (36%) and arachnoid cysts (27%). The molecular spectrum included 22 distinct variants, defining a mutational hotspot in the C-terminal domain of the Torsin-1A protein. Genotype-phenotype analysis revealed an association of missense variants in the 3-helix bundle domain to an attenuated phenotype, while missense variants near the Walker A/B motif as well as biallelic truncating variants were linked to early death. In summary, this systematic cross-sectional analysis of a large cohort of individuals with biallelic TOR1A variants across a wide age-range delineates the clinical and genetic spectrum of TOR1A-related autosomal-recessive disease and highlights potential predictors for disease severity and survival

Mapping 123 million neonatal, infant and child deaths between 2000 and 2017

Since 2000, many countries have achieved considerable success in improving child survival, but localized progress remains unclear. To inform efforts towards United Nations Sustainable Development Goal 3.2—to end preventable child deaths by 2030—we need consistently estimated data at the subnational level regarding child mortality rates and trends. Here we quantified, for the period 2000–2017, the subnational variation in mortality rates and number of deaths of neonates, infants and children under 5 years of age within 99 low- and middle-income countries using a geostatistical survival model. We estimated that 32% of children under 5 in these countries lived in districts that had attained rates of 25 or fewer child deaths per 1,000 live births by 2017, and that 58% of child deaths between 2000 and 2017 in these countries could have been averted in the absence of geographical inequality. This study enables the identification of high-mortality clusters, patterns of progress and geographical inequalities to inform appropriate investments and implementations that will help to improve the health of all populations