arXiv subjects

Fedor Fomin

Publications and source records attributed to Fedor Fomin.

7 recordsLinked to original sources

arXiv2023

Polynomial-time Approximation of Independent Set Parameterized by Treewidth

We prove the following result about approximating the maximum independent set in a graph. Informally, we show that any approximation algorithm with a ``non-trivial'' approximation ratio (as a function of the number of vertices of the input graph $G$) can be turned into an approximation algorithm achieving almost the same ratio, albeit as a function of the treewidth of $G$. More formally, we prove that for any function $f$, the existence of a polynomial time $(n/f(n))$-approximation algorithm yields the existence of a polynomial time $O(tw \cdot\log{f(tw)}/f(tw))$-approximation algorithm, where $n$ and $tw$ denote the number of vertices and the width of a given tree decomposition of the input graph. By pipelining our result with the state-of-the-art $O(n \cdot (\log \log n)^2/\log^3 n)$-approximation algorithm by Feige (2004), this implies an $O(tw \cdot (\log \log tw)^3/\log^3 tw)$-approximation algorithm.

cs.DS↗

arXiv2022

Boolean and $\mathbb{F}_p$-Matrix Factorization: From Theory to Practice

Boolean Matrix Factorization (BMF) aims to find an approximation of a given binary matrix as the Boolean product of two low-rank binary matrices. Binary data is ubiquitous in many fields, and representing data by binary matrices is common in medicine, natural language processing, bioinformatics, computer graphics, among many others. Unfortunately, BMF is computationally hard and heuristic algorithms are used to compute Boolean factorizations. Very recently, the theoretical breakthrough was obtained independently by two research groups. Ban et al. (SODA 2019) and Fomin et al. (Trans. Algorithms 2020) show that BMF admits an efficient polynomial-time approximation scheme (EPTAS). However, despite the theoretical importance, the high double-exponential dependence of the running times from the rank makes these algorithms unimplementable in practice. The primary research question motivating our work is whether the theoretical advances on BMF could lead to practical algorithms. The main conceptional contribution of our work is the following. While EPTAS for BMF is a purely theoretical advance, the general approach behind these algorithms could serve as the basis in designing better heuristics. We also use this strategy to develop new algorithms for related $\mathbb{F}_p$-Matrix Factorization. Here, given a matrix $A$ over a finite field GF($p$) where $p$ is a prime, and an integer $r$, our objective is to find a matrix $B$ over the same field with GF($p$)-rank at most $r$ minimizing some norm of $A-B$. Our empirical research on synthetic and real-world data demonstrates the advantage of the new algorithms over previous works on BMF and $\mathbb{F}_p$-Matrix Factorization.

cs.LG↗

arXiv2020

Planar F-Deletion: Approximation, Kernelization and Optimal FPT Algorithms

Let F be a finite set of graphs. In the F-Deletion problem, we are given an n-vertex graph G and an integer k as input, and asked whether at most k vertices can be deleted from G such that the resulting graph does not contain a graph from F as a minor. F-Deletion is a generic problem and by selecting different sets of forbidden minors F, one can obtain various fundamental problems such as Vertex Cover, Feedback Vertex Set or Treewidth t-Deletion. In this paper we obtain a number of generic algorithmic results about Planar F-Deletion, when F contains at least one planar graph. The highlights of our work are - A constant factor approximation algorithm for the optimization version of Planar F-Deletion. - A randomized linear time and single exponential parameterized algorithm, that is, an algorithm running in time O(2^{O(k)} n), for the parameterized version of Planar F-deletion where all graphs in F are connected. The algorithm can be made deterministic at the cost of making the polynomial factor in the running time n*log^2 n rather than linear. - A polynomial kernel for parameterized Planar F-deletion These algorithms unify, generalize, and improve a multitude of results in the literature. Our main results have several direct applications, but also the methods we develop on the way have applicability beyond the scope of this paper. Our results -- constant factor approximation, polynomial kernelization and FPT algorithms -- are stringed together by a common theme of polynomial time preprocessing.

cs.DS↗

arXiv2019

A Fixed-Parameter Perspective on #BIS

The problem of (approximately) counting the independent sets of a bipartite graph (#BIS) is the canonical approximate counting problem that is complete in the intermediate complexity class #RHΠ_1. It is believed that #BIS does not have an efficient approximation algorithm but also that it is not NP-hard. We study the robustness of the intermediate complexity of #BIS by considering variants of the problem parameterised by the size of the independent set. We exhaustively map the complexity landscape for three problems, with respect to exact computation and approximation and with respect to conventional and parameterised complexity. The three problems are counting independent sets of a given size, counting independent sets with a given number of vertices in one vertex class and counting maximum independent sets amongst those with a given number of vertices in one vertex class. Among other things, we show that all of these problems are NP-hard to approximate within any polynomial ratio. (This is surprising because the corresponding problems without the size parameter are complete in #RHΠ_1, and hence are not believed to be NP-hard.) We also show that the first problem is #W[1]-hard to solve exactly but admits an FPTRAS, whereas the other two are W[1]-hard to approximate even within any polynomial ratio. Finally, we show that, when restricted to graphs of bounded degree, all three problems have efficient exact fixed-parameter algorithms.

cs.CC↗

arXiv2013

Large induced subgraphs via triangulations and CMSO

We obtain an algorithmic meta-theorem for the following optimization problem. Let ϕ be a Counting Monadic Second Order Logic (CMSO) formula and t be an integer. For a given graph G, the task is to maximize |X| subject to the following: there is a set of vertices F of G, containing X, such that the subgraph G[F] induced by F is of treewidth at most t, and structure (G[F],X) models ϕ. Some special cases of this optimization problem are the following generic examples. Each of these cases contains various problems as a special subcase: 1) "Maximum induced subgraph with at most l copies of cycles of length 0 modulo m", where for fixed nonnegative integers m and l, the task is to find a maximum induced subgraph of a given graph with at most l vertex-disjoint cycles of length 0 modulo m. 2) "Minimum Γ-deletion", where for a fixed finite set of graphs Γ containing a planar graph, the task is to find a maximum induced subgraph of a given graph containing no graph from Γ as a minor. 3) "Independent Π-packing", where for a fixed finite set of connected graphs Π, the task is to find an induced subgraph G[F] of a given graph G with the maximum number of connected components, such that each connected component of G[F] is isomorphic to some graph from Π. We give an algorithm solving the optimization problem on an n-vertex graph G in time O(#pmc n^{t+4} f(t,ϕ)), where #pmc is the number of all potential maximal cliques in G and f is a function depending of t and ϕ only. We also show how a similar running time can be obtained for the weighted version of the problem. Pipelined with known bounds on the number of potential maximal cliques, we deduce that our optimization problem can be solved in time O(1.7347^n) for arbitrary graphs, and in polynomial time for graph classes with polynomial number of minimal separators.

cs.DS↗

arXiv2008

Parameterized Low-distortion Embeddings - Graph metrics into lines and trees

We revisit the issue of low-distortion embedding of metric spaces into the line, and more generally, into the shortest path metric of trees, from the parameterized complexity perspective.Let $M=M(G)$ be the shortest path metric of an edge weighted graph $G=(V,E)$ on $n$ vertices. We describe algorithms for the problem of finding a low distortion non-contracting embedding of $M$ into line and tree metrics. We give an $O(nd^4(2d+1)^{2d})$ time algorithm that for an unweighted graph metric $M$ and integer $d$ either constructs an embedding of $M$ into the line with distortion at most $d$, or concludes that no such embedding exists. We find the result surprising, because the considered problem bears a strong resemblance to the notoriously hard Bandwidth Minimization problem which does not admit any FPT algorithm unless an unlikely collapse of parameterized complexity classes occurs. We show that our algorithm can also be applied to construct small distortion embeddings of weighted graph metrics. The running time of our algorithm is $O(n(dW)^4(2d+1)^{2dW})$ where $W$ is the largest edge weight of the input graph. We also show that deciding whether a weighted graph metric $M(G)$ with maximum weight $W < |V(G)|$ can be embedded into the line with distortion at most $d$ is NP-Complete for every fixed rational $d \geq 2$. This rules out any possibility of an algorithm with running time $O((nW)^{h(d)})$ where $h$ is a function of $d$ alone. We generalize the result on embedding into the line by proving that for any tree $T$ with maximum degree $Δ$, embedding of $M$ into a shortest path metric of $T$ is FPT, parameterized by $(Δ,d)$.

cs.DS↗

arXiv2007

Parameterized Algorithms for Directed Maximum Leaf Problems

We prove that finding a rooted subtree with at least $k$ leaves in a digraph is a fixed parameter tractable problem. A similar result holds for finding rooted spanning trees with many leaves in digraphs from a wide family $\cal L$ that includes all strong and acyclic digraphs. This settles completely an open question of Fellows and solves another one for digraphs in $\cal L$. Our algorithms are based on the following combinatorial result which can be viewed as a generalization of many results for a `spanning tree with many leaves' in the undirected case, and which is interesting on its own: If a digraph $D\in \cal L$ of order $n$ with minimum in-degree at least 3 contains a rooted spanning tree, then $D$ contains one with at least $(n/2)^{1/5}-1$ leaves.

cs.DS↗