Related papers: Breaking the Quadratic Barrier for Matroid Interse…
Motivated by the growing interest in correlation-robust stochastic optimization, we investigate stochastic selection problems beyond independence. Specifically, we consider the instructive case of pairwise-independent priors and matroid…
Matroids are a fundamental object of study in combinatorial optimization. Three closely related and important problems involving matroids are maximizing the size of the union of $k$ independent sets (that is, $k$-fold matroid union),…
We consider the problem of finding a basis of a matroid with weight exactly equal to a given target. Here weights can be discrete values from $\{-\Delta,\ldots,\Delta\}$ or more generally $m$-dimensional vectors of such discrete values. We…
We solve an open problem of Diaconis that asks what are the largest orders of $p_n$ and $q_n$ such that $Z_n,$ the $p_n\times q_n$ upper left block of a random matrix $\boldsymbol{\Gamma}_n$ which is uniformly distributed on the orthogonal…
One of the most intriguing unsolved questions of matroid optimization is the characterization of the existence of $k$ disjoint common bases of two matroids. The significance of the problem is well-illustrated by the long list of conjectures…
We study the forrelation problem: given a pair of $n$-bit Boolean functions $f$ and $g$, estimate the correlation between $f$ and the Fourier transform of $g$. This problem is known to provide the largest possible quantum speedup in terms…
In the minimum $k$-cut problem, we want to find the minimum number of edges whose deletion breaks the input graph into at least $k$ connected components. The classic algorithm of Karger and Stein runs in $\tilde O(n^{2k-2})$ time, and…
In algorithmic graph theory, a classic open question is to determine the complexity of the Maximum Independent Set problem on $P_t$-free graphs, that is, on graphs not containing any induced path on $t$ vertices. So far, polynomial-time…
A matrix framework is presented for the solution of ODEs, including initial-, boundary and inner-value problems. The framework enables the solution of the ODEs for arbitrary nodes. There are four key issues involved in the formulation of…
Traditional orthogonal range problems allow queries over a static set of points, each with some value. Dynamic variants allow points to be added or removed, one at a time. To support more powerful updates, we introduce the Grid Range class…
We consider the problem of preprocessing an $n\times n$ matrix $\mathbf{M}$, and supporting queries that, for any vector $v$, returns the matrix-vector product $\mathbf{M} v$. This problem has been extensively studied in both theory and…
The control and sensing of large-scale systems results in combinatorial problems not only for sensor and actuator placement but also for scheduling or observability/controllability. Such combinatorial constraints in system design and…
Given two points in the plane, a set of obstacles defined by closed curves, and an integer $k$, does there exist a path between the two designated points intersecting at most $k$ of the obstacles? This is a fundamental and well-studied…
This paper considers the matrix completion problem. We show that it is not necessary to assume joint incoherence, which is a standard but unintuitive and restrictive condition that is imposed by previous studies. This leads to a sample…
The secretary problem became one of the most prominent online selection problems due to its numerous applications in online mechanism design. The task is to select a maximum weight subset of elements subject to given constraints, where…
We study the computational complexity of approximately computing the partition function of a spin system. Techniques based on standard counting-to-sampling reductions yield $\tilde{O}(n^2)$-time algorithms, where $n$ is the size of the…
We study the tractability of the maximum independent set problem from the viewpoint of graph width parameters, with the goal of defining a width parameter that is as general as possible and allows to solve independent set in polynomial-time…
Seymour's decomposition theorem for regular matroids is a fundamental result with a number of combinatorial and algorithmic applications. In this work we demonstrate how this theorem can be used in the design of parameterized algorithms on…
Given two point sets S and T, we study the many-to-many matching with demands problem (MMD problem) which is a generalization of the many-to-many matching problem (MM problem). In an MMD, each point of one set must be matched to a given…
We revisit the algorithmic problem of finding a triangle in a graph: We give a randomized combinatorial algorithm for triangle detection in a given $n$-vertex graph with $m$ edges running in $O(n^{7/3})$ time, or alternatively in…