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In this paper, we introduce and study the problem of \textit{binary stretch embedding} of edge-weighted graph. This problem is closely related to the well-known \textit{addressing problem} of Graham and Pollak. Addressing problem is the problem of assigning the shortest possible length strings (called ``addresses") over the alphabet $\{0,1,*\}$ to the vertices of an input graph $G$ with the following property. For every pair $u,v$ of vertices, the number of positions in which one of their addresses is $1$, and the other is $0$ is exactly equal to the distance of $u,v$ in graph $G$. When the addresses do not contain the symbol $*$, the problem is called \textit{isometric hypercube embedding}. As far as we know, the isometric hypercube embedding was introduced by Firsov in 1965. It is known that such addresses do not exist for general graphs. Inspired by the addressing problem, in this paper, we introduce the \textit{binary stretch embedding problem}, or BSEP for short, for the edge-weighted undirected graphs. We also argue how this problem is related to other graph embedding problems in the literature. Using tools and techniques such as Hadamard codes and the theory of linear programming, several upper and lower bounds as well as exact solutions for certain classes of graphs will be discovered. As an application of the results in this paper, we derive improved upper bounds or exact values for the maximum size of Lee metric codes of certain parameters.

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.

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.

It is known that the entropy function over a set of jointly distributed random variables is a submodular set function. However, not any submodular function is of this form. In this paper, we consider a family of submodular set functions, called weighted rank functions of matroids, and study the necessary or sufficient conditions under which they are entropic. We prove that weighted rank functions are located on the boundary of the submodularity cone. For the representable matroids over a characteristic 2 field, we show that the integer valued weighted rank functions are entropic. We derive a necessary condition for constant weight rank functions to be entropic and show that for the case of graphic matroids, this condition is indeed sufficient. Since these functions generalize the rank of a matroid, our findings generalize some of the results of Abbe et. al. about entropic properties of the rank function of matroids.

We generalize a previous framework for designing utility-optimal differentially private (DP) mechanisms via graphs, where datasets are vertices in the graph and edges represent dataset neighborhood. The boundary set contains datasets where an individual's response changes the binary-valued query compared to its neighbors. Previous work was limited to the homogeneous case where the privacy parameter $\varepsilon$ across all datasets was the same and the mechanism at boundary datasets was identical. In our work, the mechanism can take different distributions at the boundary and the privacy parameter $\varepsilon$ is a function of neighboring datasets, which recovers an earlier definition of personalized DP as special case. The problem is how to extend the mechanism, which is only defined at the boundary set, to other datasets in the graph in a computationally efficient and utility optimal manner. Using the concept of strongest induced DP condition we solve this problem efficiently in polynomial time (in the size of the graph).

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., Comment: 23 pages

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 2017

Group testing with inhibitors (GTI) introduced by Farach at al. is studied in this paper. There are three types of items, $d$ defectives, $r$ inhibitors and $n-d-r$ normal items in a population of $n$ items. The presence of any inhibitor in a test can prevent the expression of a defective. For this model, we propose a probabilistic non-adaptive pooling design with a low complexity decoding algorithm. We show that the sample complexity of the number of tests required for guaranteed recovery with vanishing error probability using the proposed algorithm scales as $T=O(d \log n)$ and $T=O(\frac{r^2}{d}\log n)$ in the regimes $r=O(d)$ and $d=o(r)$ respectively. In the former regime, the number of tests meets the lower bound order while in the latter regime, the number of tests is shown to exceed the lower bound order by a $\log \frac{r}{d}$ multiplicative factor. When only upper bounds on the number of defectives $D$ and the number of inhibitors $R$ are given instead of their exact values, the sample complexity of the number of tests using the proposed algorithm scales as $T=O(D \log n)$ and $T=O(R^2 \log n)$ in the regimes $R^2=O(D)$ and $D=o(R^2)$ respectively. In the former regime, the number of tests meets the lower bound order while in the latter regime, the number of tests exceeds the lower bound order by a $\log R$ multiplicative factor. The time complexity of the proposed decoding algorithms scale as $O(nT)$., Comment: Updated with results for the case of knowledge of only upper bounds on no. of defectives and inhibitors; 11 pages, 2 figures

It is known that the minimum broadcast rate of a linear index code over $\mathbb{F}_q$ is equal to the $minrank_q$ of the underlying digraph. In [3] it is proved that for $\mathbb{F}_2$ and any positive integer $k$, $minrank_q(G)\leq k$ iff there exists a homomorphism from the complement of the graph $G$ to the complement of a particular undirected graph family called "graph family $\{G_k\}$". 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 $\{H_k^q\}$ is introduced and shown that whether or not the scalar linear index of a digraph $G$ is less than or equal to $k$ is equivalent to the existence of a graph homomorphism from the complement of $G$ to the complement of $H_k^q$. Here, we first study the structure of the digraphs $H_k^q$. Analogous to the result of [2] about undirected graphs, we prove that $H_k^q$'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 $G$ to satisfy $lind_q(G)\leq k$. Particularly, we obtain new lower bounds on $lind_q(G)$. 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 $k$ or not is NP-complete for $k\ge 3$ and is polynomially decidable for $k=1,2$ [3]. For digraphs, it is shown in [6] that for the binary alphabet, the decision problem for $k=2$ 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)

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", 2014