• Algorithmic Complexity of Clustering and Low-Rank Approximation Problems 

      Simonov, Kirill (Doctoral thesis, 2021-03-29)
      The two most popular unsupervised learning problems are k-Clustering and Low-Rank Approximation. Consider a set of n datapoints, in the k-Clustering problem, the objective is to partition these points into k clusters and ...
    • Approximating Long Cycle Above Dirac's Guarantee 

      Fomin, Fedor; Golovach, Petr; Simonov, Kirill; Sagunov, Danil (Journal article; Peer reviewed, 2023)
      Parameterization above (or below) a guarantee is a successful concept in parameterized algorithms. The idea is that many computational problems admit "natural" guarantees bringing to algorithmic questions whether a better ...
    • Building large k-cores from sparse graphs 

      Fomin, Fedor; Sagunov, Danil; Simonov, Kirill (Journal article; Peer reviewed, 2020)
      A popular model to measure network stability is the k-core, that is the maximal induced subgraph in which every vertex has degree at least k. For example, k-cores are commonly used to model the unraveling phenomena in ...
    • Building large k-cores from sparse graphs 

      Fomin, Fedor; Sagunov, Danil; Simonov, Kirill (Journal article; Peer reviewed, 2023)
      A k-core of a graph G is the maximal induced subgraph in which every vertex has degree at least k. In the Edge k-Core optimization problem, we are given a graph G and integers k, b and p. The task is to ensure that the ...
    • Detours in Directed Graphs 

      Fomin, Fedor; Golovach, Petr; Lochet, William Alexandre; Sagunov, Danil; Simonov, Kirill; Saurabh, Saket (Journal article; Peer reviewed, 2022)
      We study two "above guarantee" versions of the classical Longest Path problem on undirected and directed graphs and obtain the following results. In the first variant of Longest Path that we study, called Longest Detour, ...
    • Detours in directed graphs 

      Fomin, Fedor; Golovach, Petr; Lochet, William Alexandre; Sagunov, Danil; Saurabh, Saket; Simonov, Kirill (Journal article; Peer reviewed, 2023)
      We study two “above guarantee” versions of the classical Longest Path problem on undirected and directed graphs and obtain the following results. In the first variant of Longest Path that we study, called Longest Detour, ...
    • FPT Approximation for Fair Minimum-Load Clustering 

      Bandyapadhyay, Sayan; Fomin, Fedor; Golovach, Petr; Purohit, Nidhi; Simonov, Kirill (Journal article; Peer reviewed, 2022)
      In this paper, we consider the Minimum-Load k-Clustering/Facility Location (MLkC) problem where we are given a set P of n points in a metric space that we have to cluster and an integer k > 0 that denotes the number of ...
    • How to find a good explanation for clustering? 

      Bandyapadhyay, Sayan; Fomin, Fedor; Golovach, Petr; Lochet, William Alexandre; Purohit, Nidhi; Simonov, Kirill (Journal article; Peer reviewed, 2023)
      k-means and k-median clustering are powerful unsupervised machine learning techniques. However, due to complicated dependencies on all the features, it is challenging to interpret the resulting cluster assignments. Moshkovitz, ...
    • Long Cycles in Graphs: Extremal Combinatorics Meets Parameterized Algorithms 

      Fomin, Fedor; Golovach, Petr; Sagunov, Danil; Simonov, Kirill (Journal article; Peer reviewed, 2022)
      We discuss recent algorithmic extensions of two classic results of extremal combinatorics about long paths in graphs. First, the theorem of Dirac from 1952 asserts that a 2-connected graph G with the minimum vertex degree ...
    • Longest Cycle Above Erdös-Gallai Bound 

      Fomin, Fedor; Golovach, Petr; Sagunov, Danil; Simonov, Kirill (Journal article; Peer reviewed, 2022)
      In 1959, Erdős and Gallai proved that every graph G with average vertex degree ad(G) ≥ 2 contains a cycle of length at least ad(G). We provide an algorithm that for k ≥ 0 in time 2^𝒪(k)⋅n^𝒪(1) decides whether a 2-connected ...
    • Lossy Kernelization of Same-Size Clustering 

      Bandyapadhyay, Sayan; Fomin, Fedor; Golovach, Petr; Purohit, Nidhi; Simonov, Kirill (Journal article; Peer reviewed, 2023)
      In this work, we study the k-median clustering problem with an additional equal-size constraint on the clusters from the perspective of parameterized preprocessing. Our main result is the first lossy (2-approximate) ...
    • Low-Rank Binary Matrix Approximation in Column-Sum Norm 

      Fomin, Fedor; Golovach, Petr; Panolan, Fahad; Simonov, Kirill (Journal article; Peer reviewed, 2020)
      We consider 𝓁₁-Rank-r Approximation over {GF}(2), where for a binary m× n matrix 𝐀 and a positive integer constant r, one seeks a binary matrix 𝐁 of rank at most r, minimizing the column-sum norm ‖ 𝐀 -𝐁‖₁. We show ...
    • Parameterized Complexity of Feature Selection for Categorical Data Clustering 

      Bandyapadhyay, Sayan; Fomin, Fedor; Golovach, Petr; Simonov, Kirill (Journal article; Peer reviewed, 2021)
      We develop new algorithmic methods with provable guarantees for feature selection in regard to categorical data clustering. While feature selection is one of the most common approaches to reduce dimensionality in practice, ...
    • Parameterized complexity of PCA 

      Fomin, Fedor; Golovach, Petr; Simonov, Kirill (Journal article; Peer reviewed, 2020)
      We discuss some recent progress in the study of Principal Component Analysis (PCA) from the perspective of Parameterized Complexity.
    • Parameterized k-Clustering: Tractability Island 

      Fomin, Fedor; Golovach, Petr; Simonov, Kirill (Peer reviewed; Journal article, 2019)
      In k-Clustering we are given a multiset of n vectors X subset Z^d and a nonnegative number D, and we need to decide whether X can be partitioned into k clusters C_1, ..., C_k such that the cost sum_{i=1}^k min_{c_i in R^d} ...
    • Refined Complexity of PCA with Outliers 

      Fomin, Fedor; Golovach, Petr; Panolan, Fahad; Simonov, Kirill (Peer reviewed; Journal article, 2019)
      Principal component analysis (PCA) is one of the most fundamental procedures in exploratory data analysis and is the basic step in applications ranging from quantitative finance and bioinformatics to image analysis and ...
    • Turán's Theorem Through Algorithmic Lens 

      Fomin, Fedor; Golovach, Petr; Sagunov, Danil; Simonov, Kirill (Journal article; Peer reviewed, 2023)