Related papers: Sample compression schemes for balls in graphs
We prove the existence of optimal separators for intersection graphs of balls and spheres in any dimension $d$. One of our results is that if an intersection graph of $n$ spheres in $\mathbb{R}^d$ has $m$ edges, then it contains a balanced…
There is a graph reduction system so that every optimal 1-planar graph can be reduced to an irreducible extended wheel graph, provided the reductions are applied such that the given graph class is preserved. A graph is optimal 1-planar if…
Graph Neural Networks (GNNs) have demonstrated significant achievements in processing graph data, yet scalability remains a substantial challenge. To address this, numerous graph coarsening methods have been developed. However, most…
We give a new $(1+\epsilon)$-approximation for sparsest cut problem on graphs where small sets expand significantly more than the sparsest cut (sets of size $n/r$ expand by a factor $\sqrt{\log n\log r}$ bigger, for some small $r$; this…
Over the past decade, we witness an increasing amount of interest in the design of exact exponential-time and parameterized algorithms for problems in Graph Drawing. Unfortunately, we still lack knowledge of general methods to develop such…
Edge connectivity of a graph is one of the most fundamental graph-theoretic concepts. The celebrated tree packing theorem of Tutte and Nash-Williams from 1961 states that every $k$-edge connected graph $G$ contains a collection $\cal{T}$ of…
Graphs have been extensively used to represent data from various domains. In the era of Big Data, information is being generated at a fast pace, and analyzing the same is a challenge. Various methods have been proposed to speed up the…
Sampling of signals defined over the nodes of a graph is one of the crucial problems in graph signal processing. While in classical signal processing sampling is a well defined operation, when we consider a graph signal many new challenges…
Given a metric space $(X,d_X)$, a $(\beta,s,\Delta)$-sparse cover is a collection of clusters $\mathcal{C}\subseteq P(X)$ with diameter at most $\Delta$, such that for every point $x\in X$, the ball $B_X(x,\frac\Delta\beta)$ is fully…
3D Point clouds (PCs) are commonly used to represent 3D scenes. They can have millions of points, making subsequent downstream tasks such as compression and streaming computationally expensive. PC sampling (selecting a subset of points) can…
We study the problem of sampling k-bandlimited signals on graphs. We propose two sampling strategies that consist in selecting a small subset of nodes at random. The first strategy is non-adaptive, i.e., independent of the graph structure,…
Many applications require randomly sampling bipartite graphs with fixed degrees, or randomly sampling incidence matrices with fixed row and column sums. Although several sampling algorithms exist, the ``curveball'' algorithm is the most…
We consider the problem of estimating the value of MAX-CUT in a graph in the streaming model of computation. At one extreme, there is a trivial $2$-approximation for this problem that uses only $O(\log n)$ space, namely, count the number of…
Can we use machine learning to compress graph data? The absence of ordering in graphs poses a significant challenge to conventional compression algorithms, limiting their attainable gains as well as their ability to discover relevant…
We prove the following theorem. Given a planar graph $G$ and an integer $k$, it is possible in polynomial time to randomly sample a subset $A$ of vertices of $G$ with the following properties: (i) $A$ induces a subgraph of $G$ of treewidth…
Can we analyze data without decompressing it? As our data keeps growing, understanding the time complexity of problems on compressed inputs, rather than in convenient uncompressed forms, becomes more and more relevant. Suppose we are given…
Compressed indexing is a powerful technique that enables efficient querying over data stored in compressed form, significantly reducing memory usage and often accelerating computation. While extensive progress has been made for…
Tensor network contraction is central to problems ranging from many-body physics to computer science. We describe how to approximate tensor network contraction through bond compression on arbitrary graphs. In particular, we introduce a…
Given a graph $G$, we define ${\bf bcg}(G)$ as the minimum $k$ for which $G$ can be contracted to the uniformly triangulated grid $\Gamma_{k}$. A graph class ${\cal G}$ has the SQG${\bf C}$ property if every graph $G\in{\cal G}$ has…
Graph compression or sparsification is a basic information-theoretic and computational question. A major open problem in this research area is whether $(1+\epsilon)$-approximate cut-preserving vertex sparsifiers with size close to the…