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Dynamic trees are a well-studied and fundamental building block of dynamic graph algorithms dating back to the seminal work of Sleator and Tarjan [STOC'81, (1981), pp. 114-122]. The problem is to maintain a tree subject to online edge…

Truss decomposition is a method used to analyze large sparse graphs in order to identify successively better connected subgraphs. Since in many domains the underlying graph changes over time, its associated truss decomposition needs to be…

Graph-based clustering plays an important role in the clustering area. Recent studies about graph convolution neural networks have achieved impressive success on graph type data. However, in general clustering tasks, the graph structure of…

In this paper, we discuss how to design the graph topology to reduce the communication complexity of certain algorithms for decentralized optimization. Our goal is to minimize the total communication needed to achieve a prescribed accuracy.…

We study finite-sum nonlinear programs with localized variable coupling encoded by a (hyper)graph. We introduce a graph-compliant decomposition framework that brings message passing into continuous optimization in a rigorous, implementable,…

Denoising-based models, including diffusion and flow matching, have led to substantial advances in graph generation. Despite this progress, such models remain constrained by two fundamental limitations: a computational cost that scales…

Answering connectivity queries is fundamental to fully dynamic graphs where edges and vertices are inserted and deleted frequently. Existing work proposes data structures and algorithms with worst-case guarantees. We propose a new data…

In 1996, Karger [Kar96] gave a startling randomized algorithm that finds a minimum-cut in a (weighted) graph in time $O(m\log^3n)$ which he termed near-linear time meaning linear (in the size of the input) times a polylogarthmic factor. In…

In this paper, we consider tree decompositions, branch decompositions, and clique decompositions. We improve the running time of dynamic programming algorithms on these graph decompositions for a large number of problems as a function of…

Depth first search (DFS) tree is a fundamental data structure for solving various problems in graphs. It is well known that it takes $O(m+n)$ time to build a DFS tree for a given undirected graph $G=(V,E)$ on $n$ vertices and $m$ edges. We…

Two fundamental algorithm-design paradigms are Tree Search and Dynamic Programming. The techniques used therein have been shown to complement one another when solving the complete set partitioning problem, also known as the coalition…

Computing a dense subgraph is a fundamental problem in graph mining, with a diverse set of applications ranging from electronic commerce to community detection in social networks. In many of these applications, the underlying context is…

Graph clustering is a fundamental computational problem with a number of applications in algorithm design, machine learning, data mining, and analysis of social networks. Over the past decades, researchers have proposed a number of…

Given a large graph, the densest-subgraph problem asks to find a subgraph with maximum average degree. When considering the top-$k$ version of this problem, a na\"ive solution is to iteratively find the densest subgraph and remove it in…

We study the problem of maintaining robust and sparse overlay networks in fully distributed settings where nodes continuously join and leave the system. This scenario closely models real-world unstructured peer-to-peer networks, where…

A recent paper by Abboud and Wallheimer [ITCS 2023] presents self-reductions for various fundamental graph problems, which transform worst-case instances to expanders, thus proving that the complexity remains unchanged if the input is…

This paper considers fully dynamic graph algorithms with both faster worst case update time and sublinear space. The fully dynamic graph connectivity problem is the following: given a graph on a fixed set of n nodes, process an online…

A tree-packing is a collection of spanning trees of a graph. It has been a useful tool for computing the minimum cut in static, dynamic, and distributed settings. In particular, [Thorup, Comb. 2007] used them to obtain his dynamic min-cut…

Graph clustering (or community detection) has long drawn enormous attention from the research on web mining and information networks. Recent literature on this topic has reached a consensus that node contents and link structures should be…

This paper leverages the framework of algorithms-with-predictions to design data structures for two fundamental dynamic graph problems: incremental topological ordering and cycle detection. In these problems, the input is a directed graph…