**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Person# Mikhail Kapralov

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related units

Loading

Courses taught by this person

Loading

Related research domains

Loading

Related publications

Loading

People doing similar research

Loading

People doing similar research (108)

Related research domains (12)

Algorithm

In mathematics and computer science, an algorithm (ˈælɡərɪðəm) is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algo

Hypergraph

In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices.
Formally, a

Time complexity

In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the

Courses taught by this person (2)

CS-250: Algorithms

The students learn the theory and practice of basic concepts and techniques in algorithms. The course covers mathematical induction, techniques for analyzing algorithms, elementary data structures, major algorithmic paradigms such as dynamic programming, sorting and searching, and graph algorithms.

CS-450: Advanced algorithms

A first graduate course in algorithms, this course assumes minimal background, but moves rapidly. The objective is to learn the main techniques of algorithm analysis and design, while building a repertory of basic algorithmic solutions to problems in many domains.

Related publications (17)

Loading

Loading

Loading

Related units (4)

Mikhail Kapralov, Jakab Tardos

Graph sparsification has been studied extensively over the past two decades, culminating in spectral sparsifiers of optimal size (up to constant factors). Spectral hypergraph sparsification is a natural analogue of this problem, for which optimal bounds on the sparsifier size are not known, mainly because the hypergraph Laplacian is non-linear, and thus lacks the linear-algebraic structure and tools that have been so effective for graphs. Our main contribution is the first algorithm for constructing epsilon-spectral sparsifiers for hyper-graphs with O*(n) hyperedges, where O* suppresses (epsilon(-1) log n)(O(1)) factors. This bound is independent of the rank r (maximum cardinality of a hyperedge), and is essentially best possible due to a recent bit complexity lower bound of Omega(nr) for hypergraph sparsification. This result is obtained by introducing two new tools. First, we give a new proof of spectral concentration bounds for sparsifiers of graphs; it avoids linear-algebraic methods, replacing e.g. the usual application of the matrix Bernstein inequality and therefore applies to the (nonlinear) hypergraph setting. To achieve the result, we design a new sequence of hypergraphdependent epsilon-nets on the unit sphere in R-n. Second, we extend the weight-assignment technique of Chen, Khanna and Nagda [FOCS'20] to the spectral sparsification setting. Surprisingly, the number of spanning trees after the weight assignment can serve as a potential function guiding the reweighting process in the spectral setting.

Mikhail Kapralov, Jakab Tardos

Cut and spectral sparsification of graphs have numerous applications, including e.g. speeding up algorithms for cuts and Laplacian solvers. These powerful notions have recently been extended to hypergraphs, which are much richer and may offer new applications. However, the current bounds on the size of hypergraph sparsifiers are not as tight as the corresponding bounds for graphs. Our first result is a polynomial-time algorithm that, given a hypergraph on n vertices with maximum hyperedge size r, outputs an epsilon-spectral sparsifier with O* (nr) hyperedges, where O* suppresses (epsilon(-1) log n)(O(1) )factors. This size bound improves the two previous bounds: O*(n(3)) [Soma and Yoshida, SODA'19] and O* (nr(3)) [Bansal, Svensson and Trevisan, FOCS'19]. Our main technical tool is a new method for proving concentration of the nonlinear analogue of the quadratic form of the Laplacians for hypergraph expanders. We complement this with lower bounds on the bit complexity of any compression scheme that (1 + epsilon)-approximates all the cuts in a given hypergraph, and hence also on the bit complexity of every epsilon-cut/spectral sparsifier. These lower bounds are based on Ruzsa-Szemeredi graphs, and a particular instantiation yields an Omega(nr) lower bound on the bit complexity even for fixed constant epsilon. In the case of hypergraph cut sparsifiers, this is tight up to polylogarithmic factors in n, due to recent result of [Chen, Khanna and Nagda, FOCS'20]. For spectral sparsifiers it narrows the gap to O*(r). Finally, for directed hypergraphs, we present an algorithm that computes an c-spectral sparsifier with O*(n(2)r(3)) hyperarcs, where r is the maximum size of a hyperarc. For small r, this improves over O*(n(3)) known from [Soma and Yoshida, SODA'19], and is getting close to the trivial lower bound of Omega(n(2)) hyperarcs.

Mikhail Kapralov, Mikhail Makarov, Jakab Tardos

A motif is a frequently occurring subgraph of a given directed or undirected graph G (Milo et al.). Motifs capture higher order organizational structure of G beyond edge relationships, and, therefore, have found wide applications such as in graph clustering, community detection, and analysis of biological and physical networks to name a few (Benson at al., Tsourakakis at al.). In these applications, the cut structure of motifs plays a crucial role as vertices are partitioned into clusters by cuts whose conductance is based on the number of instances of a particular motif, as opposed to just the number of edges, crossing the cuts. In this paper, we introduce the concept of a motif cut sparsifier. We show that one can compute in polynomial time a sparse weighted subgraph G' with only (O) over tilde( n/is an element of(2)) edges such that for every cut, the weighted number of copies of M crossing the cut in G' is within a 1 + is an element of factor of the number of copies of M crossing the cut in G, for every constant size motif M. Our work carefully combines the viewpoints of both graph sparsification and hypergraph sparsification. We sample edges which requires us to extend and strengthen the concept of cut sparsifiers introduced in the seminal works of Karger and Benczur et al. to the motif setting. The task of adapting the importance sampling framework common to efficient graph sparsification algorithms to the motif setting turns out to be nontrivial due to the fact that cut sizes in a random subgraph of G depend non-linearly on the sampled edges. To overcome this, we adopt the viewpoint of hypergraph sparsification to define edge sampling probabilities which are derived from the strong connectivity values of a hypergraph whose hyperedges represent motif instances. Finally, an iterative sparsification primitive inspired by both viewpoints is used to reduce the number of edges in G to nearly linear. In addition, we present a strong lower bound ruling out a similar result for sparsification with respect to induced occurrences of motifs.