Related papers: Binomial Random Matroids
Let $M$ be a random $m \times n$ matrix with binary entries and i.i.d. rows. The weight (i.e., number of ones) of a row has a specified probability distribution, with the row chosen uniformly at random given its weight. Let $N(n,m)$ denote…
Matrix completion is a classical problem that has received recurring interest across a wide range of fields. In this paper, we revisit this problem in an ultra-sparse sampling regime, where each entry of an unknown, $n\times d$ matrix $M$…
We consider the problem of finding an independent set of maximum weight simultaneously contained in $k$ matroids over a common ground set. This $k$-matroid intersection problem appears naturally in many contexts, for example in generalizing…
Motivated by several applications, we consider the problem of randomly rounding a fractional solution in a matroid (base) polytope to an integral one. We consider the pipage rounding technique and also present a new technique, randomized…
We study budgeted variants of well known maximization problems with multiple matroid constraints. Given an $\ell$-matchoid $\cm$ on a ground set $E$, a profit function $p:E \rightarrow \mathbb{R}_{\geq 0}$, a cost function $c:E \rightarrow…
In a stochastic probing problem we are given a universe $E$, where each element $e \in E$ is active independently with probability $p_e$, and only a probe of e can tell us whether it is active or not. On this universe we execute a process…
Motivated by the Koml\'os conjecture in combinatorial discrepancy, we study the discrepancy of random matrices with $m$ rows and $n$ independent columns drawn from a bounded lattice random variable. It is known that for $n$ tending to…
A natural way to deal with multiple, partially conflicting objectives is turning all the objectives but one into budget constraints. Some classical polynomial-time optimization problems, such as spanning tree and forest, shortest path,…
For a matroid of rank $r$ and a non-negative integer $k$, an element is called $k$-loose if every circuit containing it has size greater than $r-k$. Zaslavsky and the author characterized all binary matroids with a $1$-loose element. In…
A subset $S$ of $\mathbb R^d$ has the Borsuk property if it can be decomposed into at most $d+1$ parts of diameter smaller than $S$. This is an important geometric property, inspired by a conjecture of Borsuk from the 1930s, which has…
In the matroid partitioning problem, we are given $k$ matroids $\mathcal{M}_1 = (V, \mathcal{I}_1), \dots , \mathcal{M}_k = (V, \mathcal{I}_k)$ defined over a common ground set $V$ of $n$ elements, and we need to find a partitionable set $S…
We study a family of matroid optimization problems with a linear constraint (MOL). In these problems, we seek a subset of elements which optimizes (i.e., maximizes or minimizes) a linear objective function subject to (i) a matroid…
For a simplicial complex ${\mathcal C}$ denote by $\beta({\mathcal C})$ the minimal number of edges from ${\mathcal C}$ needed to cover the ground set. If ${\mathcal C}$ is a matroid then for every partition $A_1, \ldots, A_m$ of the ground…
We study the following problem: Given a variable of interest, we would like to find a best linear predictor for it by choosing a subset of $k$ relevant variables obeying a matroid constraint. This problem is a natural generalization of…
We consider random mappings on n = kr nodes with preimage sizes restricted to a set of the form {0,k}, where k = k(r) is greater than 1. We prove that T, the least common multiple of the cycle lengths, and B= the product of the cycle…
Given a symmetric $n\times n$ matrix $P$ with $0 \le P(u, v)\le 1$, we define a random graph $G_{n, P}$ on $[n]$ by independently including any edge $\{u, v\}$ with probability $P(u, v)$. For $k\ge 1$ let $\mathcal{A}_k$ be the property of…
We present an optimal, combinatorial 1-1/e approximation algorithm for monotone submodular optimization over a matroid constraint. Compared to the continuous greedy algorithm (Calinescu, Chekuri, Pal and Vondrak, 2008), our algorithm is…
Given vectors $v_1,\dots,v_n\in\mathbb{R}^d$ and a matroid $M=([n],I)$, we study the problem of finding a basis $S$ of $M$ such that $\det(\sum_{i \in S}v_i v_i^\top)$ is maximized. This problem appears in a diverse set of areas such as…
We define a notion of isotropy for discrete set distributions. If $\mu$ is a distribution over subsets $S$ of a ground set $[n]$, we say that $\mu$ is in isotropic position if $P[e \in S]$ is the same for all $e\in [n]$. We design a new…
Despite a surge of interest in submodular maximization in the data stream model, there remain significant gaps in our knowledge about what can be achieved in this setting, especially when dealing with multiple constraints. In this work, we…