arXiv subjects

Petr A. Golovach

Publications and source records attributed to Petr A. Golovach.

At least 55 records · Page 3Linked 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

Parameterized Complexity of Broadcasting in Graphs

The task of the broadcast problem is, given a graph G and a source vertex s, to compute the minimum number of rounds required to disseminate a piece of information from s to all vertices in the graph. It is assumed that, at each round, an informed vertex can transmit the information to at most one of its neighbors. The broadcast problem is known to NP-hard. We show that the problem is FPT when parametrized by the size k of a feedback edge-set, or by the size k of a vertex-cover, or by k=n-t where t is the input deadline for the broadcast protocol to complete.

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arXiv2023

Computing paths of large rank in planar frameworks deterministically

A framework consists of an undirected graph $G$ and a matroid $M$ whose elements correspond to the vertices of $G$. Recently, Fomin et al. [SODA 2023] and Eiben et al. [ArXiV 2023] developed parameterized algorithms for computing paths of rank $k$ in frameworks. More precisely, for vertices $s$ and $t$ of $G$, and an integer $k$, they gave FPT algorithms parameterized by $k$ deciding whether there is an $(s,t)$-path in $G$ whose vertex set contains a subset of elements of $M$ of rank $k$. These algorithms are based on Schwartz-Zippel lemma for polynomial identity testing and thus are randomized, and therefore the existence of a deterministic FPT algorithm for this problem remains open. We present the first deterministic FPT algorithm that solves the problem in frameworks whose underlying graph $G$ is planar. While the running time of our algorithm is worse than the running times of the recent randomized algorithms, our algorithm works on more general classes of matroids. In particular, this is the first FPT algorithm for the case when matroid $M$ is represented over rationals. Our main technical contribution is the nontrivial adaptation of the classic irrelevant vertex technique to frameworks to reduce the given instance to one of bounded treewidth. This allows us to employ the toolbox of representative sets to design a dynamic programming procedure solving the problem efficiently on instances of bounded treewidth.

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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

Two-sets cut-uncut on planar graphs

We study the following Two-Sets Cut-Uncut problem on planar graphs. Therein, one is given an undirected planar graph $G$ and two sets of vertices $S$ and $T$. The question is, what is the minimum number of edges to remove from $G$, such that we separate all of $S$ from all of $T$, while maintaining that every vertex in $S$, and respectively in $T$, stays in the same connected component. We show that this problem can be solved in time $2^{|S|+|T|} n^{O(1)}$ with a one-sided error randomized algorithm. Our algorithm implies a polynomial-time algorithm for the network diversion problem on planar graphs, which resolves an open question from the literature. More generally, we show that Two-Sets Cut-Uncut remains fixed-parameter tractable even when parameterized by the number $r$ of faces in the plane graph covering the terminals $S \cup T$, by providing an algorithm of running time $4^{r + O(\sqrt r)} n^{O(1)}$.

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arXiv2022

(Re)packing Equal Disks into Rectangle

The problem of packing of equal disks (or circles) into a rectangle is a fundamental geometric problem. (By a packing here we mean an arrangement of disks in a rectangle without overlapping.) We consider the following algorithmic generalization of the equal disk packing problem. In this problem, for a given packing of equal disks into a rectangle, the question is whether by changing positions of a small number of disks, we can allocate space for packing more disks. More formally, in the repacking problem, for a given set of $n$ equal disks packed into a rectangle and integers $k$ and $h$, we ask whether it is possible by changing positions of at most $h$ disks to pack $n+k$ disks. Thus the problem of packing equal disks is the special case of our problem with $n=h=0$. While the computational complexity of packing equal disks into a rectangle remains open, we prove that the repacking problem is NP-hard already for $h=0$. Our main algorithmic contribution is an algorithm that solves the repacking problem in time $(h+k)^{O(h+k)}\cdot |I|^{O(1)}$, where $I$ is the input size. That is, the problem is fixed-parameter tractable parameterized by $k$ and $h$.

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arXiv2022

Shortest Cycles With Monotone Submodular Costs

We introduce the following submodular generalization of the Shortest Cycle problem. For a nonnegative monotone submodular cost function $f$ defined on the edges (or the vertices) of an undirected graph $G$, we seek for a cycle $C$ in $G$ of minimum cost $\textsf{OPT}=f(C)$. We give an algorithm that given an $n$-vertex graph $G$, parameter $\varepsilon > 0$, and the function $f$ represented by an oracle, in time $n^{\mathcal{O}(\log 1/\varepsilon)}$ finds a cycle $C$ in $G$ with $f(C)\leq (1+\varepsilon)\cdot \textsf{OPT}$. This is in sharp contrast with the non-approximability of the closely related Monotone Submodular Shortest $(s,t)$-Path problem, which requires exponentially many queries to the oracle for finding an $n^{2/3-\varepsilon}$-approximation [Goel et al., FOCS 2009]. We complement our algorithm with a matching lower bound. We show that for every $\varepsilon > 0$, obtaining a $(1+\varepsilon)$-approximation requires at least $n^{Ω(\log 1/ \varepsilon)}$ queries to the oracle. When the function $f$ is integer-valued, our algorithm yields that a cycle of cost $\textsf{OPT}$ can be found in time $n^{\mathcal{O}(\log \textsf{OPT})}$. In particular, for $\textsf{OPT}=n^{\mathcal{O}(1)}$ this gives a quasipolynomial-time algorithm computing a cycle of minimum submodular cost. Interestingly, while a quasipolynomial-time algorithm often serves as a good indication that a polynomial time complexity could be achieved, we show a lower bound that $n^{\mathcal{O}(\log n)}$ queries are required even when $\textsf{OPT} = \mathcal{O}(n)$.

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arXiv2022

An Algorithmic Meta-Theorem for Graph Modification to Planarity and FOL

In general, a graph modification problem is defined by a graph modification operation $\boxtimes$ and a target graph property ${\cal P}$. Typically, the modification operation $\boxtimes$ may be vertex removal}, edge removal}, edge contraction}, or edge addition and the question is, given a graph $G$ and an integer $k$, whether it is possible to transform $G$ to a graph in ${\cal P}$ after applying $k$ times the operation $\boxtimes$ on $G$. This problem has been extensively studied for particilar instantiations of $\boxtimes$ and ${\cal P}$. In this paper we consider the general property ${\cal P}_{ϕ}$ of being planar and, moreover, being a model of some First-Order Logic sentence $ϕ$ (an FOL-sentence). We call the corresponding meta-problem Graph $\boxtimes$-Modification to Planarity and $ϕ$ and prove the following algorithmic meta-theorem: there exists a function $f:\Bbb{N}^{2}\to\Bbb{N}$ such that, for every $\boxtimes$ and every FOL sentence $ϕ$, the Graph $\boxtimes$-Modification to Planarity and $ϕ$ is solvable in $f(k,|ϕ|)\cdot n^2$ time. The proof constitutes a hybrid of two different classic techniques in graph algorithms. The first is the irrelevant vertex technique that is typically used in the context of Graph Minors and deals with properties such as planarity or surface-embeddability (that are not FOL-expressible) and the second is the use of Gaifman's Locality Theorem that is the theoretical base for the meta-algorithmic study of FOL-expressible problems.

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arXiv2022

Compound Logics for Modification Problems

We introduce a novel model-theoretic framework inspired from graph modification and based on the interplay between model theory and algorithmic graph minors. The core of our framework is a new compound logic operating with two types of sentences, expressing graph modification: the modulator sentence, defining some property of the modified part of the graph, and the target sentence, defining some property of the resulting graph. In our framework, modulator sentences are in counting monadic second-order logic (CMSOL) and have models of bounded treewidth, while target sentences express first-order logic (FOL) properties along with minor-exclusion. Our logic captures problems that are not definable in first-order logic and, moreover, may have instances of unbounded treewidth. Also, it permits the modeling of wide families of problems involving vertex/edge removals, alternative modulator measures (such as elimination distance or $\mathcal{G}$-treewidth), multistage modifications, and various cut problems. Our main result is that, for this compound logic, model-checking can be done in quadratic time. All derived algorithms are constructive and this, as a byproduct, extends the constructibility horizon of the algorithmic applications of the Graph Minors theorem of Robertson and Seymour. The proposed logic can be seen as a general framework to capitalize on the potential of the irrelevant vertex technique. It gives a way to deal with problem instances of unbounded treewidth, for which Courcelle's theorem does not apply. The proof of our meta-theorem combines novel combinatorial results related to the Flat Wall theorem along with elements of the proof of Courcelle's theorem and Gaifman's theorem. We finally prove extensions where the target property is expressible in FOL+DP, i.e., the enhancement of FOL with disjoint-paths predicates.

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arXiv2022

Hitting Topological Minor Models in Planar Graphs is Fixed Parameter Tractable

For a finite collection of graphs ${\cal F}$, the \textsc{${\cal F}$-TM-Deletion} problem has as input an $n$-vertex graph $G$ and an integer $k$ and asks whether there exists a set $S \subseteq V(G)$ with $|S| \leq k$ such that $G \setminus S$ does not contain any of the graphs in ${\cal F}$ as a topological minor. We prove that for every such ${\cal F}$, \textsc{${\cal F}$-TM-Deletion} is fixed parameter tractable on planar graphs. Our algorithm runs in a $2^{\mathcal{O}(k^2)}\cdot n^{2}$ time or, alternatively in $2^{\mathcal{O}(k)}\cdot n^{4}$ time. Our techniques can easily be extended to graphs that are embeddable on any fixed surface.

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arXiv2022

Exact Exponential Algorithms for Clustering Problems

In this paper we initiate a systematic study of exact algorithms for well-known clustering problems, namely $k$-Median and $k$-Means. In $k$-Median, the input consists of a set $X$ of $n$ points belonging to a metric space, and the task is to select a subset $C \subseteq X$ of $k$ points as centers, such that the sum of the distances of every point to its nearest center is minimized. In $k$-Means, the objective is to minimize the sum of squares of the distances instead. It is easy to design an algorithm running in time $\max_{k\leq n} {n \choose k} n^{O(1)} = O^*(2^n)$ ($O^*(\cdot)$ notation hides polynomial factors in $n$). We design first non-trivial exact algorithms for these problems. In particular, we obtain an $O^*((1.89)^n)$ time exact algorithm for $k$-Median that works for any value of $k$. Our algorithm is quite general in that it does not use any properties of the underlying (metric) space -- it does not even require the distances to satisfy the triangle inequality. In particular, the same algorithm also works for $k$-Means. We complement this result by showing that the running time of our algorithm is asymptotically optimal, up to the base of the exponent. That is, unless ETH fails, there is no algorithm for these problems running in time $2^{o(n)} \cdot n^{O(1)}$. Finally, we consider the "supplier" versions of these clustering problems, where, in addition to the set $X$ we are additionally given a set of $m$ candidate centers $F$, and objective is to find a subset of $k$ centers from $F$. The goal is still to minimize the $k$-Median/$k$-Means/$k$-Center objective. For these versions we give a $O(2^n (mn)^{O(1)})$ time algorithms using subset convolution. We complement this result by showing that, under the Set Cover Conjecture, the supplier versions of these problems do not admit an exact algorithm running in time $2^{(1-ε) n} (mn)^{O(1)}$.

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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

Combing a Linkage in an Annulus

A linkage in a graph $G$ of size $k$ is a subgraph $L$ of $G$ whose connected components are $k$ paths. The pattern of a linkage of size $k$ is the set of $k$ pairs formed by the endpoints of these paths. A consequence of the Unique Linkage Theorem is the following: there exists a function $f:\mathbb{N}\to\mathbb{N}$ such that if a plane graph $G$ contains a sequence $\mathcal{C}$ of at least $f(k)$ nested cycles and a linkage of size at most $k$ whose pattern vertices lay outside the outer cycle of $\mathcal{C},$ then $G$ contains a linkage with the same pattern avoiding the inner cycle of $\mathcal{C}$. In this paper we prove the following variant of this result: Assume that all the cycles in $\mathcal{C}$ are "orthogonally" traversed by a linkage $P$ and $L$ is a linkage whose pattern vertices may lay either outside the outer cycle or inside the inner cycle of $\mathcal{C}:=[C_{1},\ldots,C_{p},\ldots,C_{2p-1}]$. We prove that there are two functions $g,f:\mathbb{N}\to\mathbb{N}$, such that if $L$ has size at most $k$, $P$ has size at least $f(k),$ and $|\mathcal{C}|\geq g(k)$, then there is a linkage with the same pattern as $L$ that is "internally combed" by $P$, in the sense that $L\cap C_{p}\subseteq P\cap C_{p}$. In fact, we prove this result in the most general version where the linkage $L$ is $s$-scattered: no two vertices of distinct paths of $L$ are within distance at most $s$. We deduce several variants of this result in the cases where $s=0$ and $s>0$. These variants permit the application of the unique linkage theorem on several path routing problems on embedded graphs.

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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

Induced Disjoint Paths in AT-free Graphs

Paths $P_1,\ldots,P_k$ in a graph $G=(V,E)$ are mutually induced if any two distinct $P_i$ and $P_j$ have neither common vertices nor adjacent vertices (except perhaps their end-vertices). The Induced Disjoint Paths problem is to decide if a graph $G$ with $k$ pairs of specified vertices $(s_i,t_i)$ contains $k$ mutually induced paths $P_i$ such that each $P_i$ connects $s_i$ and $t_i$. This is a classical graph problem that is NP-complete even for $k=2$. We study it for AT-free graphs. Unlike its subclasses of permutation graphs and cocomparability graphs, the class of AT-free graphs has no geometric intersection model. However, by a new, structural analysis of the behaviour of Induced Disjoint Paths for AT-free graphs, we prove that it can be solved in polynomial time for AT-free graphs even when $k$ is part of the input. This is in contrast to the situation for other well-known graph classes, such as planar graphs, claw-free graphs, or more recently, (theta,wheel)-free graphs, for which such a result only holds if $k$ is fixed. As a consequence of our main result, the problem of deciding if a given AT-free graph contains a fixed graph $H$ as an induced topological minor admits a polynomial-time algorithm. In addition, we show that such an algorithm is essentially optimal by proving that the problem is W[1]-hard with parameter $|V_H|$, even on a subclass of AT-free graph, namely cobipartite graphs. We also show that the problems $k$-in-a-Path and $k$-in-a-Tree are polynomial-time solvable on AT-free graphs even if $k$ is part of the input. These problems are to test if a graph has an induced path or induced tree, respectively, spanning $k$ given vertices.

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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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