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We present a framework for deterministically rounding a dynamic fractional matching. Applying our framework in a black-box manner on top of existing fractional matching algorithms, we derive the following new results: (1) The first…
In this paper we study the problem of dynamically maintaining graph properties under batches of edge insertions and deletions in the massively parallel model of computation. In this setting, the graph is stored on a number of machines, each…
Many robotic exploration algorithms rely on graph structures for frontier-based exploration and dynamic path planning. However, these graphs grow rapidly, accumulating redundant information and impacting performance. We present a…
We introduce a new approach to spectral sparsification that approximates the quadratic form of the pseudoinverse of a graph Laplacian restricted to a subspace. We show that sparsifiers with a near-linear number of edges in the dimension of…
In this paper we study the Steiner tree problem over a dynamic set of terminals. We consider the model where we are given an $n$-vertex graph $G=(V,E,w)$ with positive real edge weights, and our goal is to maintain a tree which is a good…
We study two fundamental decremental dynamic graph problems. In both problems, we need to maintain a vertex-weighted forest of size $n$ under edge deletions, weight updates, and a certain information-retrieval query. Both problems can be…
The (non-uniform) sparsest cut problem is the following graph-partitioning problem: given a "supply" graph, and demands on pairs of vertices, delete some subset of supply edges to minimize the ratio of the supply edges cut to the total…
We present a polynomial time dynamic programming algorithm for optimal partitions in the shortest path metric induced by a tree. This resolves, among other things, the exact complexity status of the optimal partition problems in one…
Structured sparse optimization is an important and challenging problem for analyzing high-dimensional data in a variety of applications such as bioinformatics, medical imaging, social networks, and astronomy. Although a number of structured…
We present an algorithm for a fault tolerant Depth First Search (DFS) Tree in an undirected graph. This algorithm is drastically simpler than the current state-of-the-art algorithms for this problem, uses optimal space and optimal…
A hypergraph spectral sparsifier of a hypergraph $G$ is a weighted subgraph $H$ that approximates the Laplacian of $G$ to a specified precision. Recent work has shown that similar to ordinary graphs, there exist $\widetilde{O}(n)$-size…
We give a simple algorithm for the dynamic approximate All-Pairs Shortest Paths (APSP) problem. Given a graph $G = (V, E, l)$ with polynomially bounded edge lengths, our data structure processes $|E|$ edge insertions and deletions in total…
This paper considers the problem of distributed optimization over time-varying graphs. For the case of undirected graphs, we introduce a distributed algorithm, referred to as DIGing, based on a combination of a distributed inexact gradient…
We propose Deep Hierarchical Machine (DHM), a model inspired from the divide-and-conquer strategy while emphasizing representation learning ability and flexibility. A stochastic routing framework as used by recent deep neural…
Dynamic programming on various graph decompositions is one of the most fundamental techniques used in parameterized complexity. Unfortunately, even if we consider concepts as simple as path or tree decompositions, such dynamic programming…
A geometric graph associated with a set of points $P= \{x_1, x_2, \cdots, x_n \} \subset \mathbb{R}^d$ and a fixed kernel function $\mathsf{K}:\mathbb{R}^d\times \mathbb{R}^d\to\mathbb{R}_{\geq 0}$ is a complete graph on $P$ such that the…
In recent years, hypergraph generalizations of many graph cut problems have been introduced and analyzed as a way to better explore and understand complex systems and datasets characterized by multiway relationships. Recent work has made…
We introduce top trees as a design of a new simpler interface for data structures maintaining information in a fully-dynamic forest. We demonstrate how easy and versatile they are to use on a host of different applications. For example, we…
Hierarchical clustering over graphs is a fundamental task in data mining and machine learning with applications in domains such as phylogenetics, social network analysis, and information retrieval. Specifically, we consider the recently…
This paper proposes a local search algorithm for a specific combinatorial optimisation problem in graph theory: the Hamiltonian Completion Problem (HCP) on undirected graphs. In this problem, the objective is to add as few edges as possible…