arXiv subjects

Kirill Simonov

Publications and source records attributed to Kirill Simonov.

At least 37 records · Page 2Linked to original sources

arXiv2023

Turán's Theorem Through Algorithmic Lens

The fundamental theorem of Turán from Extremal Graph Theory determines the exact bound on the number of edges $t_r(n)$ in an $n$-vertex graph that does not contain a clique of size $r+1$. We establish an interesting link between Extremal Graph Theory and Algorithms by providing a simple compression algorithm that in linear time reduces the problem of finding a clique of size $\ell$ in an $n$-vertex graph $G$ with $m \ge t_r(n)-k$ edges, where $\ell\leq r+1$, to the problem of finding a maximum clique in a graph on at most $5k$ vertices. This also gives us an algorithm deciding in time $2.49^{k}\cdot(n + m)$ whether $G$ has a clique of size $\ell$. As a byproduct of the new compression algorithm, we give an algorithm that in time $2^{\mathcal{O}(td^2)} \cdot n^2$ decides whether a graph contains an independent set of size at least $n/(d+1) + t$. Here $d$ is the average vertex degree of the graph $G$. The multivariate complexity analysis based on ETH indicates that the asymptotical dependence on several parameters in the running times of our algorithms is tight.

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arXiv2023

Approximating Long Cycle Above Dirac's Guarantee

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 solution (above the guarantee) could be obtained efficiently. The above guarantee paradigm has led to several exciting discoveries in the areas of parameterized algorithms and kernelization. We argue that this paradigm could bring forth fresh perspectives on well-studied problems in approximation algorithms. Our example is the longest cycle problem. One of the oldest results in extremal combinatorics is the celebrated Dirac's theorem from 1952. Dirac's theorem provides the following guarantee on the length of the longest cycle: for every 2-connected n-vertex graph G with minimum degree δ(G)\leq n/2, the length of a longest cycle L is at least 2δ(G). Thus, the ``essential'' part in finding the longest cycle is in approximating the ``offset'' k = L - 2 δ(G). The main result of this paper is the above-guarantee approximation theorem for k. Informally, the theorem says that approximating the offset k is not harder than approximating the total length L of a cycle. In other words, for any (reasonably well-behaved) function f, a polynomial time algorithm constructing a cycle of length f(L) in an undirected graph with a cycle of length L, yields a polynomial time algorithm constructing a cycle of length 2δ(G)+Ω(f(k)).

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arXiv2023

A Parameterized Theory of PAC Learning

Probably Approximately Correct (i.e., PAC) learning is a core concept of sample complexity theory, and efficient PAC learnability is often seen as a natural counterpart to the class P in classical computational complexity. But while the nascent theory of parameterized complexity has allowed us to push beyond the P-NP ``dichotomy'' in classical computational complexity and identify the exact boundaries of tractability for numerous problems, there is no analogue in the domain of sample complexity that could push beyond efficient PAC learnability. As our core contribution, we fill this gap by developing a theory of parameterized PAC learning which allows us to shed new light on several recent PAC learning results that incorporated elements of parameterized complexity. Within the theory, we identify not one but two notions of fixed-parameter learnability that both form distinct counterparts to the class FPT -- the core concept at the center of the parameterized complexity paradigm -- and develop the machinery required to exclude fixed-parameter learnability. We then showcase the applications of this theory to identify refined boundaries of tractability for CNF and DNF learning as well as for a range of learning problems on graphs.

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arXiv2023

Proportionally Fair Matching with Multiple Groups

The study of fair algorithms has become mainstream in machine learning and artificial intelligence due to its increasing demand in dealing with biases and discrimination. Along this line, researchers have considered fair versions of traditional optimization problems including clustering, regression, ranking and voting. However, most of the efforts have been channeled into designing heuristic algorithms, which often do not provide any guarantees on the quality of the solution. In this work, we study matching problems with the notion of proportional fairness. Proportional fairness is one of the most popular notions of group fairness where every group is represented up to an extent proportional to the final selection size. Matching with proportional fairness or more commonly, proportionally fair matching, was introduced in [Chierichetti et al., AISTATS, 2019], where the problem was studied with only two groups. However, in many practical applications, the number of groups -- although often a small constant -- is larger than two. In this work, we make the first step towards understanding the computational complexity of proportionally fair matching with more than two groups. We design exact and approximation algorithms achieving reasonable guarantees on the quality of the matching as well as on the time complexity. Our algorithms are also supported by suitable hardness bounds.

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arXiv2022

The Fine-Grained Complexity of Graph Homomorphism Parameterized by Clique-Width

The generic homomorphism problem, which asks whether an input graph $G$ admits a homomorphism into a fixed target graph $H$, has been widely studied in the literature. In this article, we provide a fine-grained complexity classification of the running time of the homomorphism problem with respect to the clique-width of $G$ (denoted $\operatorname{cw}$) for virtually all choices of $H$ under the Strong Exponential Time Hypothesis. In particular, we identify a property of $H$ called the signature number $s(H)$ and show that for each $H$, the homomorphism problem can be solved in time $\mathcal{O}^*(s(H)^{\operatorname{cw}})$. Crucially, we then show that this algorithm can be used to obtain essentially tight upper bounds. Specifically, we provide a reduction that yields matching lower bounds for each $H$ that is either a projective core or a graph admitting a factorization with additional properties -- allowing us to cover all possible target graphs under long-standing conjectures.

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arXiv2022

Testing Upward Planarity of Partial $2$-Trees

We present an $O(n^2)$-time algorithm to test whether an $n$-vertex directed partial $2$-tree is upward planar. This result improves upon the previously best known algorithm, which runs in $O(n^4)$ time.

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arXiv2022

Fixed-Parameter Tractability of Maximum Colored Path and Beyond

We introduce a general method for obtaining fixed-parameter algorithms for problems about finding paths in undirected graphs, where the length of the path could be unbounded in the parameter. The first application of our method is as follows. We give a randomized algorithm, that given a colored $n$-vertex undirected graph, vertices $s$ and $t$, and an integer $k$, finds an $(s,t)$-path containing at least $k$ different colors in time $2^k n^{O(1)}$. This is the first FPT algorithm for this problem, and it generalizes the algorithm of Björklund, Husfeldt, and Taslaman [SODA 2012] on finding a path through $k$ specified vertices. It also implies the first $2^k n^{O(1)}$ time algorithm for finding an $(s,t)$-path of length at least $k$. Our method yields FPT algorithms for even more general problems. For example, we consider the problem where the input consists of an $n$-vertex undirected graph $G$, a matroid $M$ whose elements correspond to the vertices of $G$ and which is represented over a finite field of order $q$, a positive integer weight function on the vertices of $G$, two sets of vertices $S,T \subseteq V(G)$, and integers $p,k,w$, and the task is to find $p$ vertex-disjoint paths from $S$ to $T$ so that the union of the vertices of these paths contains an independent set of $M$ of cardinality $k$ and weight $w$, while minimizing the sum of the lengths of the paths. We give a $2^{p+O(k^2 \log (q+k))} n^{O(1)} w$ time randomized algorithm for this problem.

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arXiv2022

Weighted Model Counting with Twin-Width

Bonnet et al. (FOCS 2020) introduced the graph invariant twin-width and showed that many NP-hard problems are tractable for graphs of bounded twin-width, generalizing similar results for other width measures, including treewidth and clique-width. In this paper, we investigate the use of twin-width for solving the propositional satisfiability problem (SAT) and propositional model counting. We particularly focus on Bounded-ones Weighted Model Counting (BWMC), which takes as input a CNF formula $F$ along with a bound $k$ and asks for the weighted sum of all models with at most $k$ positive literals. BWMC generalizes not only SAT but also (weighted) model counting. We develop the notion of "signed" twin-width of CNF formulas and establish that BWMC is fixed-parameter tractable when parameterized by the certified signed twin-width of $F$ plus $k$. We show that this result is tight: it is neither possible to drop the bound $k$ nor use the vanilla twin-width instead if one wishes to retain fixed-parameter tractability, even for the easier problem SAT. Our theoretical results are complemented with an empirical evaluation and comparison of signed twin-width on various classes of CNF formulas.

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arXiv2022

Parameterized Algorithms for Upward Planarity

We obtain new parameterized algorithms for the classical problem of determining whether a directed acyclic graph admits an upward planar drawing. Our results include a new fixed-parameter algorithm parameterized by the number of sources, an XP-algorithm parameterized by treewidth, and a fixed-parameter algorithm parameterized by treedepth. All three algorithms are obtained using a novel framework for the problem that combines SPQR tree-decompositions with parameterized techniques. Our approach unifies and pushes beyond previous tractability results for the problem on series-parallel digraphs, single-source digraphs and outerplanar digraphs.

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arXiv2022

Longest Cycle above Erdős-Gallai Bound

In 1959, Erdős and Gallai proved that every graph G with average vertex degree ad(G)\geq 2 contains a cycle of length at least ad(G). We provide an algorithm that for k\geq 0 in time 2^{O(k)} n^{O(1)} decides whether a 2-connected n-vertex graph G contains a cycle of length at least ad(G)+k. This resolves an open problem explicitly mentioned in several papers. The main ingredients of our algorithm are new graph-theoretical results interesting on their own.

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arXiv2022

Detours in Directed Graphs

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, the task is to decide whether a graph has an (s,t)-path of length at least dist_G(s,t)+k (where dist_G(s,t) denotes the length of a shortest path from s to t). Bezáková et al. proved that on undirected graphs the problem is fixed-parameter tractable (FPT) by providing an algorithm of running time 2^{O (k)} n. Further, they left the parameterized complexity of the problem on directed graphs open. Our first main result establishes a connection between Longest Detour on directed graphs and 3-Disjoint Paths on directed graphs. Using these new insights, we design a 2^{O(k)} n^{O(1)} time algorithm for the problem on directed planar graphs. Further, the new approach yields a significantly faster FPT algorithm on undirected graphs. In the second variant of Longest Path, namely Longest Path Above Diameter, the task is to decide whether the graph has a path of length at least diam(G)+k (diam(G) denotes the length of a longest shortest path in a graph G). We obtain dichotomy results about Longest Path Above Diameter on undirected and directed graphs. For (un)directed graphs, Longest Path Above Diameter is NP-complete even for k=1. However, if the input undirected graph is 2-connected, then the problem is FPT. On the other hand, for 2-connected directed graphs, we show that Longest Path Above Diameter is solvable in polynomial time for each k\in{1,\dots, 4} and is NP-complete for every k\geq 5. The parameterized complexity of Longest Path Above Diameter on general directed graphs remains an interesting open problem.

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arXiv2021

How to Find a Good Explanation for Clustering?

$k$-means and $k$-median clustering are powerful unsupervised machine learning techniques. However, due to complicated dependences on all the features, it is challenging to interpret the resulting cluster assignments. Moshkovitz, Dasgupta, Rashtchian, and Frost [ICML 2020] proposed an elegant model of explainable $k$-means and $k$-median clustering. In this model, a decision tree with $k$ leaves provides a straightforward characterization of the data set into clusters. We study two natural algorithmic questions about explainable clustering. (1) For a given clustering, how to find the "best explanation" by using a decision tree with $k$ leaves? (2) For a given set of points, how to find a decision tree with $k$ leaves minimizing the $k$-means/median objective of the resulting explainable clustering? To address the first question, we introduce a new model of explainable clustering. Our model, inspired by the notion of outliers in robust statistics, is the following. We are seeking a small number of points (outliers) whose removal makes the existing clustering well-explainable. For addressing the second question, we initiate the study of the model of Moshkovitz et al. from the perspective of multivariate complexity. Our rigorous algorithmic analysis sheds some light on the influence of parameters like the input size, dimension of the data, the number of outliers, the number of clusters, and the approximation ratio, on the computational complexity of explainable clustering.

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arXiv2021

Parameterized Complexity of Feature Selection for Categorical Data Clustering

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, most of the known feature selection methods are heuristics. We study the following mathematical model. We assume that there are some inadvertent (or undesirable) features of the input data that unnecessarily increase the cost of clustering. Consequently, we want to select a subset of the original features from the data such that there is a small-cost clustering on the selected features. More precisely, for given integers $\ell$ (the number of irrelevant features) and $k$ (the number of clusters), budget $B$, and a set of $n$ categorical data points (represented by $m$-dimensional vectors whose elements belong to a finite set of values $Σ$), we want to select $m-\ell$ relevant features such that the cost of any optimal $k$-clustering on these features does not exceed $B$. Here the cost of a cluster is the sum of Hamming distances ($\ell_0$-distances) between the selected features of the elements of the cluster and its center. The clustering cost is the total sum of the costs of the clusters. We use the framework of parameterized complexity to identify how the complexity of the problem depends on parameters $k$, $B$, and $|Σ|$. Our main result is an algorithm that solves the Feature Selection problem in time $f(k,B,|Σ|)\cdot m^{g(k,|Σ|)}\cdot n^2$ for some functions $f$ and $g$. In other words, the problem is fixed-parameter tractable parameterized by $B$ when $|Σ|$ and $k$ are constants. Our algorithm is based on a solution to a more general problem, Constrained Clustering with Outliers. We also complement our algorithmic findings with complexity lower bounds.

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arXiv2021

FPT Approximation for Fair Minimum-Load Clustering

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 $k$ that denotes the number of clusters. Additionally, we are given a set $F$ of cluster centers in the same metric space. The goal is to select a set $C\subseteq F$ of $k$ centers and assign each point in $P$ to a center in $C$, such that the maximum load over all centers is minimized. Here the load of a center is the sum of the distances between it and the points assigned to it. Although clustering/facility location problems have a rich literature, the minimum-load objective is not studied substantially, and hence MLkC has remained a poorly understood problem. More interestingly, the problem is notoriously hard even in some special cases including the one in line metrics as shown by Ahmadian et al. [ACM Trans. Algo. 2018]. They also show APX-hardness of the problem in the plane. On the other hand, the best-known approximation factor for MLkC is $O(k)$, even in the plane. In this work, we study a fair version of MLkC inspired by the work of Chierichetti et al. [NeurIPS, 2017], which generalizes MLkC. Here the input points are colored by one of the $\ell$ colors denoting the group they belong to. MLkC is the special case with $\ell=1$. Considering this problem, we are able to obtain a $3$-approximation in $f(k,\ell)\cdot n^{O(1)}$ time. Also, our scheme leads to an improved $(1 + ε)$-approximation in case of Euclidean norm, and in this case, the running time depends only polynomially on the dimension $d$. Our results imply the same approximations for MLkC with running time $f(k)\cdot n^{O(1)}$, achieving the first constant approximations for this problem in general and Euclidean metric spaces.

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arXiv2021

Lossy Kernelization of Same-Size Clustering

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) polynomial kernel for this problem, parameterized by the cost of clustering. We complement this result by establishing lower bounds for the problem that eliminate the existences of an (exact) kernel of polynomial size and a PTAS.

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arXiv2020

EPTAS for $k$-means Clustering of Affine Subspaces

We consider a generalization of the fundamental $k$-means clustering for data with incomplete or corrupted entries. When data objects are represented by points in $\mathbb{R}^d$, a data point is said to be incomplete when some of its entries are missing or unspecified. An incomplete data point with at most $Δ$ unspecified entries corresponds to an axis-parallel affine subspace of dimension at most $Δ$, called a $Δ$-point. Thus we seek a partition of $n$ input $Δ$-points into $k$ clusters minimizing the $k$-means objective. For $Δ=0$, when all coordinates of each point are specified, this is the usual $k$-means clustering. We give an algorithm that finds an $(1+ ε)$-approximate solution in time $f(k,ε, Δ) \cdot n^2 \cdot d$ for some function $f$ of $k,ε$, and $Δ$ only.

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arXiv2020

On Coresets for Fair Clustering in Metric and Euclidean Spaces and Their Applications

Fair clustering is a constrained variant of clustering where the goal is to partition a set of colored points, such that the fraction of points of any color in every cluster is more or less equal to the fraction of points of this color in the dataset. This variant was recently introduced by Chierichetti et al. [NeurIPS, 2017] in a seminal work and became widely popular in the clustering literature. In this paper, we propose a new construction of coresets for fair clustering based on random sampling. The new construction allows us to obtain the first coreset for fair clustering in general metric spaces. For Euclidean spaces, we obtain the first coreset whose size does not depend exponentially on the dimension. Our coreset results solve open questions proposed by Schmidt et al. [WAOA, 2019] and Huang et al. [NeurIPS, 2019]. The new coreset construction helps to design several new approximation and streaming algorithms. In particular, we obtain the first true constant-approximation algorithm for metric fair clustering, whose running time is fixed-parameter tractable (FPT). In the Euclidean case, we derive the first $(1+ε)$-approximation algorithm for fair clustering whose time complexity is near-linear and does not depend exponentially on the dimension of the space. Besides, our coreset construction scheme is fairly general and gives rise to coresets for a wide range of constrained clustering problems. This leads to improved constant-approximations for these problems in general metrics and near-linear time $(1+ε)$-approximations in the Euclidean metric.

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arXiv2020

Building large k-cores from sparse graphs

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 social networks. In this model, users having less than $k$ connections within the network leave it, so the remaining users form exactly the $k$-core. In this paper we study the question whether it is possible to make the network more robust by spending only a limited amount of resources on new connections. A mathematical model for the $k$-core construction problem is the following Edge $k$-Core optimization problem. We are given a graph $G$ and integers $k$, $b$ and $p$. The task is to ensure that the $k$-core of $G$ has at least $p$ vertices by adding at most $b$ edges. The previous studies on Edge $k$-Core demonstrate that the problem is computationally challenging. In particular, it is NP-hard when $k=3$, W[1]-hard being parameterized by $k+b+p$ (Chitnis and Talmon, 2018), and APX-hard (Zhou et al, 2019). Nevertheless, we show that there are efficient algorithms with provable guarantee when the $k$-core has to be constructed from a sparse graph with some additional structural properties. Our results are 1) When the input graph is a forest, Edge $k$-Core is solvable in polynomial time; 2) Edge $k$-Core is fixed-parameter tractable (FPT) being parameterized by the minimum size of a vertex cover in the input graph. On the other hand, with such parameterization, the problem does not admit a polynomial kernel subject to a widely-believed assumption from complexity theory; 3) Edge $k$-Core is FPT parameterized by $\mathrm{tw}+k$. This improves upon a result of Chitnis and Talmon by not requiring $b$ to be small. Each of our algorithms is built upon a new graph-theoretical result interesting in its own.

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