arXiv subjects

Stefan Kratsch

Publications and source records attributed to Stefan Kratsch.

At least 37 records · Page 2Linked to original sources

arXiv2022

Tight Bounds for Connectivity Problems Parameterized by Cutwidth

In this work we start the investigation of tight complexity bounds for connectivity problems parameterized by cutwidth assuming the Strong Exponential-Time Hypothesis (SETH). Van Geffen et al. posed this question for odd cycle transversal and feedback vertex set. We answer it for these two and four further problems, namely connected vertex cover, connected domintaing set, steiner tree, and connected odd cycle transversal. For the latter two problems it sufficed to prove lower bounds that match the running time inherited from parameterization by treewidth; for the others we provide faster algorithms than relative to treewidth and prove matching lower bounds. For upper bounds we first extend the idea of Groenland et al.~[STACS~2022] to solve what we call coloring-like problem. Such problems are defined by a symmetric matrix $M$ over $\mathbb{F}_2$ indexed by a set of colors. The goal is to count the number (modulo some prime $p$) of colorings of a graph such that $M$ has a $1$-entry if indexed by the colors of the end-points of any edge. We show that this problem can be solved faster if $M$ has small rank over $\mathbb{F}_p$. We apply this result to get our upper bounds for connected vertex cover and connected dominating set. The upper bounds for odd cycle transversal and feedback vertex set use a subdivision trick to get below the bounds that matrix rank would yield.

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arXiv2022

Towards exact structural thresholds for parameterized complexity

Parameterized complexity seeks to use input structure to obtain faster algorithms for NP-hard problems. This has been most successful for graphs of low treewidth: Many problems admit fast algorithms relative to treewidth and many of them are optimal under SETH. Fewer such results are known for more general structure such as low clique-width and more restrictive structure such as low deletion distance to a sparse graph class. Despite these successes, such results remain "islands'' within the realm of possible structure. Rather than adding more islands, we seek to determine the transitions between them, that is, we aim for structural thresholds where the complexity increases as input structure becomes more general. Going from deletion distance to treewidth, is a single deletion set to a graph with simple components enough to yield the same lower bound as for treewidth or does it take many disjoint separators? Going from treewidth to clique-width, how much more density entails the same complexity as clique-width? Conversely, what is the most restrictive structure that yields the same lower bound? For treewidth, we obtain both refined and new lower bounds that apply already to graphs with a single separator $X$ such that $G-X$ has treewidth $r=O(1)$, while $G$ has treewidth $|X|+O(1)$. We rule out algorithms running in time $O^*((r+1-ε)^{k})$ for Deletion to $r$-Colorable parameterized by $k=|X|$. For clique-width, we rule out time $O^*((2^r-ε)^k)$ for Deletion to $r$-Colorable, where $X$ is now allowed to consist of $k$ twinclasses. There are further results on Vertex Cover, Dominating Set and Maximum Cut. All lower bounds are matched by existing and newly designed algorithms.

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arXiv2022

Optimal Discretization is Fixed-parameter Tractable

Given two disjoint sets $W_1$ and $W_2$ of points in the plane, the Optimal Discretization problem asks for the minimum size of a family of horizontal and vertical lines that separate $W_1$ from $W_2$, that is, in every region into which the lines partition the plane there are either only points of $W_1$, or only points of $W_2$, or the region is empty. Equivalently, Optimal Discretization can be phrased as a task of discretizing continuous variables: we would like to discretize the range of $x$-coordinates and the range of $y$-coordinates into as few segments as possible, maintaining that no pair of points from $W_1 \times W_2$ are projected onto the same pair of segments under this discretization. We provide a fixed-parameter algorithm for the problem, parameterized by the number of lines in the solution. Our algorithm works in time $2^{O(k^2 \log k)} n^{O(1)}$, where $k$ is the bound on the number of lines to find and $n$ is the number of points in the input. Our result answers in positive a question of Bonnet, Giannopolous, and Lampis [IPEC 2017] and of Froese (PhD thesis, 2018) and is in contrast with the known intractability of two closely related generalizations: the Rectangle Stabbing problem and the generalization in which the selected lines are not required to be axis-parallel.

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arXiv2022

Efficient parameterized algorithms on graphs with heterogeneous structure: Combining tree-depth and modular-width

Many computational problems admit fast algorithms on special inputs, however, the required properties might be quite restrictive. E.g., many graph problems can be solved much faster on interval or cographs, or on graphs of small modular-width or small tree-width, than on general graphs. One challenge is to attain the greatest generality of such results, i.e., being applicable to less restrictive input classes, without losing much in terms of running time. Building on the use of algebraic expressions we present a clean and robust way of combining such homogeneous structure into more complex heterogeneous structure, and we show-case this for the combination of modular-width, tree-depth, and a natural notion of modular tree-depth. We give a generic framework for designing efficient parameterized algorithms on the created graph classes, aimed at getting competitive running times that match the homogeneous cases. To show the applicability we give efficient parameterized algorithms for Negative Cycle Detection, Vertex-Weighted All-Pairs Shortest Paths, and Triangle Counting.

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arXiv2022

On Triangle Counting Parameterized by Twin-Width

In this report we present an algorithm solving Triangle Counting in time $O(d^2n+m)$, where n and m, respectively, denote the number of vertices and edges of a graph G and d denotes its twin-width, a recently introduced graph parameter. We assume that a compact representation of a d-contraction sequence of G is given.

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arXiv2021

Solving connectivity problems parameterized by treedepth in single-exponential time and polynomial space

A breakthrough result of Cygan et al. (FOCS 2011) showed that connectivity problems parameterized by treewidth can be solved much faster than the previously best known time $\mathcal{O}^*(2^{\mathcal{O}(tw \log(tw))})$. Using their inspired Cut\&Count technique, they obtained $\mathcal{O}^*(α^{tw})$ time algorithms for many such problems. Moreover, they proved these running times to be optimal assuming the Strong Exponential-Time Hypothesis. Unfortunately, like other dynamic programming algorithms on tree decompositions, these algorithms also require exponential space, and this is widely believed to be unavoidable. In contrast, for the slightly larger parameter called treedepth, there are already several examples of algorithms matching the time bounds obtained for treewidth, but using only polynomial space. Nevertheless, this has remained open for connectivity problems. In the present work, we close this knowledge gap by applying the Cut\&Count technique to graphs of small treedepth. While the general idea is unchanged, we have to design novel procedures for counting consistently cut solution candidates using only polynomial space. Concretely, we obtain time $\mathcal{O}^*(3^d)$ and polynomial space for Connected Vertex Cover, Feedback Vertex Set, and Steiner Tree on graphs of treedepth $d$. Similarly, we obtain time $\mathcal{O}^*(4^d)$ and polynomial space for Connected Dominating Set and Connected Odd Cycle Transversal.

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arXiv2020

Approximate Turing Kernelization for Problems Parameterized by Treewidth

We extend the notion of lossy kernelization, introduced by Lokshtanov et al. [STOC 2017], to approximate Turing kernelization. An $α$-approximate Turing kernel for a parameterized optimization problem is a polynomial-time algorithm that, when given access to an oracle that outputs $c$-approximate solutions in $O(1)$ time, obtains an $(α\cdot c)$-approximate solution to the considered problem, using calls to the oracle of size at most $f(k)$ for some function $f$ that only depends on the parameter. Using this definition, we show that Independent Set parameterized by treewidth $\ell$ has a $(1+\varepsilon)$-approximate Turing kernel with $O(\frac{\ell^2}{\varepsilon})$ vertices, answering an open question posed by Lokshtanov et al. [STOC 2017]. Furthermore, we give $(1+\varepsilon)$-approximate Turing kernels for the following graph problems parameterized by treewidth: Vertex Cover, Edge Clique Cover, Edge-Disjoint Triangle Packing and Connected Vertex Cover. We generalize the result for Independent Set and Vertex Cover, by showing that all graph problems that we will call "friendly" admit $(1+\varepsilon)$-approximate Turing kernels of polynomial size when parameterized by treewidth. We use this to obtain approximate Turing kernels for Vertex-Disjoint $H$-packing for connected graphs $H$, Clique Cover, Feedback Vertex Set and Edge Dominating Set.

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arXiv2020

Efficient parameterized algorithms for computing all-pairs shortest paths

Computing all-pairs shortest paths is a fundamental and much-studied problem with many applications. Unfortunately, despite intense study, there are still no significantly faster algorithms for it than the $\mathcal{O}(n^3)$ time algorithm due to Floyd and Warshall (1962). Somewhat faster algorithms exist for the vertex-weighted version if fast matrix multiplication may be used. Yuster (SODA 2009) gave an algorithm running in time $\mathcal{O}(n^{2.842})$, but no combinatorial, truly subcubic algorithm is known. Motivated by the recent framework of efficient parameterized algorithms (or "FPT in P"), we investigate the influence of the graph parameters clique-width ($cw$) and modular-width ($mw$) on the running times of algorithms for solving All-Pairs Shortest Paths. We obtain efficient (and combinatorial) parameterized algorithms on non-negative vertex-weighted graphs of times $\mathcal{O}(cw^2n^2)$, resp. $\mathcal{O}(mw^2n + n^2)$. If fast matrix multiplication is allowed then the latter can be improved to $\mathcal{O}(mw^{1.842}n + n^2)$ using the algorithm of Yuster as a black box. The algorithm relative to modular-width is adaptive, meaning that the running time matches the best unparameterized algorithm for parameter value $mw$ equal to $n$, and they outperform them already for $mw \in \mathcal{O}(n^{1 - \varepsilon})$ for any $\varepsilon > 0$.

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arXiv2019

Assessing the Computational Complexity of Multi-Layer Subgraph Detection

Multi-layer graphs consist of several graphs (layers) over the same vertex set. They are motivated by real-world problems where entities (vertices) are associated via multiple types of relationships (edges in different layers). We chart the border of computational (in)tractability for the class of subgraph detection problems on multi-layer graphs, including fundamental problems such as maximum matching, finding certain clique relaxations (motivated by community detection), or path problems. Mostly encountering hardness results, sometimes even for two or three layers, we can also spot some islands of tractability.

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arXiv2019

Multi-budgeted directed cuts

We study multi-budgeted variants of the classic minimum cut problem and graph separation problems that turned out to be important in parameterized complexity: Skew Multicut and Directed Feedback Arc Set. In our generalization, we assign colors $1,2,...,\ell$ to some edges and give separate budgets $k_{1},k_{2},...,k_{\ell}$. Let $E_{i}$ be the set of edges of color $i$. The solution $C$ for the multi-budgeted variant of a graph separation problem not only needs to satisfy the usual separation requirements, but also needs to satisfy that $|C\cap E_{i}|\leq k_{i}$ for every $i\in \{1,...,\ell\}$. Contrary to the classic minimum cut problem, the multi-budgeted variant turns out to be NP-hard even for $\ell = 2$. We propose FPT algorithms parameterized by $k=k_{1}+...+k_{\ell}$ for all three problems. To this end, we develop a branching procedure for the multi-budgeted minimum cut problem that measures the progress of the algorithm not by reducing $k$ as usual, by but elevating the capacity of some edges and thus increasing the size of maximum source-to-sink flow. Using the fact that a similar strategy is used to enumerate all important separators of a given size, we merge this process with the flow-guided branching and show an FPT bound on the number of (appropriately defined) important multi-budgeted separators. This allows us to extend our algorithm to the Skew Multicut and Directed Feedback Arc Set problems. Furthermore, we show connections of the multi-budgeted variants with weighted variants of the directed cut problems and the Chain $\ell$-SAT problem, whose parameterized complexity remains an open problem. We show that these problems admit a bounded-in-parameter number of "maximally pushed" solutions (in a similar spirit as important separators are maximally pushed), giving somewhat weak evidence towards their tractability.

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arXiv2019

Parameterized Approximation Schemes for Independent Set of Rectangles and Geometric Knapsack

The area of parameterized approximation seeks to combine approximation and parameterized algorithms to obtain, e.g., (1+eps)-approximations in f(k,eps)n^{O(1)} time where k is some parameter of the input. We obtain the following results on parameterized approximability: 1) In the maximum independent set of rectangles problem (MISR) we are given a collection of n axis parallel rectangles in the plane. Our goal is to select a maximum-cardinality subset of pairwise non-overlapping rectangles. This problem is NP-hard and also W[1]-hard [Marx, ESA'05]. The best-known polynomial-time approximation factor is O(loglog n) [Chalermsook and Chuzhoy, SODA'09] and it admits a QPTAS [Adamaszek and Wiese, FOCS'13; Chuzhoy and Ene, FOCS'16]. Here we present a parameterized approximation scheme (PAS) for MISR, i.e. an algorithm that, for any given constant eps>0 and integer k>0, in time f(k,eps)n^{g(eps)}, either outputs a solution of size at least k/(1+eps), or declares that the optimum solution has size less than k. 2) In the (2-dimensional) geometric knapsack problem (TDK) we are given an axis-aligned square knapsack and a collection of axis-aligned rectangles in the plane (items). Our goal is to translate a maximum cardinality subset of items into the knapsack so that the selected items do not overlap. In the version of TDK with rotations (TDKR), we are allowed to rotate items by 90 degrees. Both variants are NP-hard, and the best-known polynomial-time approximation factors are 558/325+eps and 4/3+eps, resp. [Galvez et al., FOCS'17]. These problems admit a QPTAS for polynomially bounded item sizes [Adamaszek and Wiese, SODA'15]. We show that both variants are W[1]-hard. Furthermore, we present a PAS for TDKR. For all considered problems, getting time f(k,eps)n^{O(1)}, rather than f(k,eps)n^{g(eps)}, would give FPT time f'(k)n^{O(1)} exact algorithms using eps=1/(k+1), contradicting W[1]-hardness.

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arXiv2019

Elimination Distances, Blocking Sets, and Kernels for Vertex Cover

The Vertex Cover problem plays an essential role in the study of polynomial kernelization in parameterized complexity, i.e., the study of provable and efficient preprocessing for NP-hard problems. Motivated by the great variety of positive and negative results for kernelization for Vertex Cover subject to different parameters and graph classes, we seek to unify and generalize them using so-called blocking sets, which have played implicit and explicit roles in many results. We show that in the most-studied setting, parameterized by the size of a deletion set to a specified graph class $\mathcal{C}$, bounded minimal blocking set size is necessary but not sufficient to get a polynomial kernelization. Under mild technical assumptions, bounded minimal blocking set size is showed to allow an essentially tight efficient reduction in the number of connected components. We then determine the exact maximum size of minimal blocking sets for graphs of bounded elimination distance to any hereditary class $\mathcal{C}$, including the case of graphs of bounded treedepth. We get similar but not tight bounds for certain non-hereditary classes $\mathcal{C}$, including the class $\mathcal{C}_{LP}$ of graphs where integral and fractional vertex cover size coincide. These bounds allow us to derive polynomial kernels for Vertex Cover parameterized by the size of a deletion set to graphs of bounded elimination distance to, e.g., forest, bipartite, or $\mathcal{C}_{LP}$ graphs.

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arXiv2019

On adaptive algorithms for maximum matching

In the fundamental Maximum Matching problem the task is to find a maximum cardinality set of pairwise disjoint edges in a given undirected graph. The fastest algorithm for this problem, due to Micali and Vazirani, runs in time $\mathcal{O}(\sqrt{n}m)$ and stands unbeaten since 1980. It is complemented by faster, often linear-time, algorithms for various special graph classes. Moreover, there are fast parameterized algorithms, e.g., time $\mathcal{O}(km\log n)$ relative to tree-width $k$, which outperform $\mathcal{O}(\sqrt{n}m)$ when the parameter is sufficiently small. We show that the Micali-Vazirani algorithm, and in fact any algorithm following the phase framework of Hopcroft and Karp, is adaptive to beneficial input structure. We exhibit several graph classes for which such algorithms run in linear time $\mathcal{O}(n+m)$. More strongly, we show that they run in time $\mathcal{O}(\sqrt{k}m)$ for graphs that are $k$ vertex deletions away from any of several such classes, without explicitly computing an optimal or approximate deletion set; before, most such bounds were at least $Ω(km)$. Thus, any phase-based matching algorithm with linear-time phases obliviously interpolates between linear time for $k=\mathcal{O}(1)$ and the worst case of $\mathcal{O}(\sqrt{n}m)$ when $k=Θ(n)$. We complement our findings by proving that the phase framework by itself still allows $Ω(\sqrt{n})$ phases, and hence time $Ω(\sqrt{n}m)$, even on paths, cographs, and bipartite chain graphs.

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arXiv2019

On Kernelization for Edge Dominating Set under Structural Parameters

In the NP-hard Edge Dominating Set problem (EDS) we are given a graph $G=(V,E)$ and an integer $k$, and need to determine whether there is a set $F\subseteq E$ of at most $k$ edges that are incident with all (other) edges of $G$. It is known that this problem is fixed-parameter tractable and admits a polynomial kernel when parameterized by $k$. A caveat for this parameter is that it needs to be large, i.e., at least equal to half the size of a maximum matching of $G$, for instances not to be trivially negative. Motivated by this, we study the existence of polynomial kernels for EDS when parameterized by structural parameters that may be much smaller than $k$. Unfortunately, at first glance this looks rather hopeless: Even when parameterized by the deletion distance to a disjoint union of paths $P_3$ of length two there is no polynomial kernelization (under standard assumptions), ruling out polynomial kernels for many smaller parameters like the feedback vertex set size. In contrast, somewhat surprisingly, there is a polynomial kernelization for deletion distance to a disjoint union of paths $P_5$ of length four. As our main result, we fully classify for all finite sets $\mathcal{H}$ of graphs, whether a kernel size polynomial in $|X|$ is possible when given $X$ such that each connected component of $G-X$ is isomorphic to a graph in $\mathcal{H}$.

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arXiv2018

Bipartite Graphs of Small Readability

We study a parameter of bipartite graphs called readability, introduced by Chikhi et al. (Discrete Applied Mathematics, 2016) and motivated by applications of overlap graphs in bioinformatics. The behavior of the parameter is poorly understood. The complexity of computing it is open and it is not known whether the decision version of the problem is in NP. The only known upper bound on the readability of a bipartite graph (following from a work of Braga and Meidanis, LATIN 2002) is exponential in the maximum degree of the graph. Graphs that arise in bioinformatic applications have low readability. In this paper, we focus on graph families with readability $o(n)$, where $n$ is the number of vertices. We show that the readability of $n$-vertex bipartite chain graphs is between $Ω(\log n)$ and $O(\sqrt{n})$. We give an efficiently testable characterization of bipartite graphs of readability at most $2$ and completely determine the readability of grids, showing in particular that their readability never exceeds $3$. As a consequence, we obtain a polynomial time algorithm to determine the readability of induced subgraphs of grids. One of the highlights of our techniques is the appearance of Euler's totient function in the analysis of the readability of bipartite chain graphs. We also develop a new technique for proving lower bounds on readability, which is applicable to dense graphs with a large number of distinct degrees.

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arXiv2018

Efficient and adaptive parameterized algorithms on modular decompositions

We study the influence of a graph parameter called modular-width on the time complexity for optimally solving well-known polynomial problems such as Maximum Matching, Triangle Counting, and Maximum $s$-$t$ Vertex-Capacitated Flow. The modular-width of a graph depends on its (unique) modular decomposition tree, and can be computed in linear time $O(n+m)$ for graphs with $n$ vertices and $m$ edges. Modular decompositions are an important tool for graph algorithms, e.g., for linear-time recognition of certain graph classes. Throughout, we obtain efficient parameterized algorithms of running times $O(f(mw)n+m)$, $O(n+f(mw)m)$ , or $O(f(mw)+n+m)$ for graphs of modular-width $mw$. Our algorithm for Maximum Matching, running in time $O(mw^2\log mw \cdot n+m)$, is both faster and simpler than the recent $O(mw^4n+m)$ time algorithm of Coudert et al. (SODA 2018). For several other problems, e.g., Triangle Counting and Maximum $b$-Matching, we give adaptive algorithms, meaning that their running times match the best unparameterized algorithms for worst-case modular-width of $mw=Θ(n)$ and they outperform them already for $mw=o(n)$, until reaching linear time for $mw=O(1)$.

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arXiv2018

The parameterized complexity of finding a 2-sphere in a simplicial complex

We consider the problem of finding a subcomplex K' of a simplicial complex K such that K' is homeomorphic to the 2-dimensional sphere, S^2. We study two variants of this problem. The first asks if there exists such a K' with at most k triangles, and we show that this variant is W[1]-hard and, assuming ETH, admits no O(n^{o(sqrt(k))}) time algorithm. We also give an algorithm that is tight with regards to this lower bound. The second problem is the dual of the first, and asks if K' can be found by removing at most k triangles from K. This variant has an immediate O(3^k poly(|K|)) time algorithm, and we show that it admits a polynomial kernelization to O(k^2) triangles, as well as a polynomial compression to a weighted version with bit-size O(k log k).

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arXiv2017

Smaller parameters for vertex cover kernelization

We revisit the topic of polynomial kernels for Vertex Cover relative to structural parameters. Our starting point is a recent paper due to Fomin and Strømme [WG 2016] who gave a kernel with $\mathcal{O}(|X|^{12})$ vertices when $X$ is a vertex set such that each connected component of $G-X$ contains at most one cycle, i.e., $X$ is a modulator to a pseudoforest. We strongly generalize this result by using modulators to $d$-quasi-forests, i.e., graphs where each connected component has a feedback vertex set of size at most $d$, and obtain kernels with $\mathcal{O}(|X|^{3d+9})$ vertices. Our result relies on proving that minimal blocking sets in a $d$-quasi-forest have size at most $d+2$. This bound is tight and there is a related lower bound of $\mathcal{O}(|X|^{d+2-ε})$ on the bit size of kernels. In fact, we also get bounds for minimal blocking sets of more general graph classes: For $d$-quasi-bipartite graphs, where each connected component can be made bipartite by deleting at most $d$ vertices, we get the same tight bound of $d+2$ vertices. For graphs whose connected components each have a vertex cover of cost at most $d$ more than the best fractional vertex cover, which we call $d$-quasi-integral, we show that minimal blocking sets have size at most $2d+2$, which is also tight. Combined with existing randomized polynomial kernelizations this leads to randomized polynomial kernelizations for modulators to $d$-quasi-bipartite and $d$-quasi-integral graphs. There are lower bounds of $\mathcal{O}(|X|^{d+2-ε})$ and $\mathcal{O}(|X|^{2d+2-ε})$ for the bit size of such kernels.

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