arXiv subjects

Petr Hliněný

Publications and source records attributed to Petr Hliněný.

At least 55 records · Page 3Linked to original sources

arXiv2019

Shrub-depth: Capturing Height of Dense Graphs

The recent increase of interest in the graph invariant called tree-depth and in its applications in algorithms and logic on graphs led to a natural question: is there an analogously useful "depth" notion also for dense graphs (say; one which is stable under graph complementation)? To this end, in a 2012 conference paper, a new notion of shrub-depth has been introduced, such that it is related to the established notion of clique-width in a similar way as tree-depth is related to tree-width. Since then shrub-depth has been successfully used in several research papers. Here we provide an in-depth review of the definition and basic properties of shrub-depth, and we focus on its logical aspects which turned out to be most useful. In particular, we use shrub-depth to give a characterization of the lower $ω$ levels of the MSO1 transduction hierarchy of simple graphs.

cs.LO↗

arXiv2018

A New Perspective on FO Model Checking of Dense Graph Classes

We study the first-order (FO) model checking problem of dense graphs, namely those which have FO interpretations in (or are FO transductions of) some sparse graph classes. We give a structural characterization of the graph classes which are FO interpretable in graphs of bounded degree. This characterization allows us to efficiently compute such an FO interpretation for an input graph. As a consequence, we obtain an FPT algorithm for successor-invariant FO model checking of any graph class which is FO interpretable in (or an FO transduction of) a graph class of bounded degree. The approach we use to obtain these results may also be of independent interest.

cs.LO↗

arXiv2018

A Simpler Self-reduction Algorithm for Matroid Path-width

Path-width of matroids naturally generalizes the better known parameter of path-width for graphs, and is NP-hard by a reduction from the graph case. While the term matroid path-width was formally introduced by Geelen-Gerards-Whittle [JCTB 2006] in pure matroid theory, it was soon recognized by Kashyap [SIDMA 2008] that it is the same concept as long-studied so called trellis complexity in coding theory, later named trellis-width, and hence it is an interesting notion also from the algorithmic perspective. It follows from a result of Hlineny [JCTB 2006] that the decision problem, whether a given matroid over a finite field has path-width at most t, is fixed-parameter tractable (FPT) in t, but this result does not give any clue about constructing a path-decomposition. The first constructive and rather complicated FPT algorithm for path-width of matroids over a finite field was given by Jeong-Kim-Oum [SODA 2016]. Here we propose a simpler "self-reduction" FPT algorithm for a path-decomposition. Precisely, we design an efficient routine that constructs an optimal path-decomposition of a matroid by calling any subroutine for testing whether the path-width of a matroid is at most t (such as the aforementioned decision algorithm for matroid path-width).

cs.DS↗

arXiv2018

FO model checking of geometric graphs

Over the past two decades the main focus of research into first-order (FO) model checking algorithms has been on sparse relational structures - culminating in the FPT algorithm by Grohe, Kreutzer and Siebertz for FO model checking of nowhere dense classes of graphs. On contrary to that, except the case of locally bounded clique-width only little is currently known about FO model checking of dense classes of graphs or other structures. We study the FO model checking problem for dense graph classes definable by geometric means (intersection and visibility graphs). We obtain new nontrivial FPT results, e.g., for restricted subclasses of circular-arc, circle, box, disk, and polygon-visibility graphs. These results use the FPT algorithm by Gajarský et al. for FO model checking of posets of bounded width. We also complement the tractability results by related hardness reductions.

cs.LO↗

arXiv2017

Parameterized Extension Complexity of Independent Set and Related Problems

Let $G$ be a graph on $n$ vertices and $\mathrm{STAB}_k(G)$ be the convex hull of characteristic vectors of its independent sets of size at most $k$. We study extension complexity of $\mathrm{STAB}_k(G)$ with respect to a fixed parameter $k$ (analogously to, e.g., parameterized computational complexity of problems). We show that for graphs $G$ from a class of bounded expansion it holds that $\mathrm{xc}(\mathrm{STAB}_k(G))\leqslant \mathcal{O}(f(k)\cdot n)$ where the function $f$ depends only on the class. This result can be extended in a simple way to a wide range of similarly defined graph polytopes. In case of general graphs we show that there is {\em no function $f$} such that, for all values of the parameter $k$ and for all graphs on $n$ vertices, the extension complexity of $\mathrm{STAB}_k(G)$ is at most $f(k)\cdot n^{\mathcal{O}(1)}.$ While such results are not surprising since it is known that optimizing over $\mathrm{STAB}_k(G)$ is $FPT$ for graphs of bounded expansion and $W[1]$-hard in general, they are also not trivial and in both cases stronger than the corresponding computational complexity results.

cs.CC↗

arXiv2017

Parameterized Shifted Combinatorial Optimization

Shifted combinatorial optimization is a new nonlinear optimization framework which is a broad extension of standard combinatorial optimization, involving the choice of several feasible solutions at a time. This framework captures well studied and diverse problems ranging from so-called vulnerability problems to sharing and partitioning problems. In particular, every standard combinatorial optimization problem has its shifted counterpart, which is typically much harder. Already with explicitly given input set the shifted problem may be NP-hard. In this article we initiate a study of the parameterized complexity of this framework. First we show that shifting over an explicitly given set with its cardinality as the parameter may be in XP, FPT or P, depending on the objective function. Second, we study the shifted problem over sets definable in MSO logic (which includes, e.g., the well known MSO partitioning problems). Our main results here are that shifted combinatorial optimization over MSO definable sets is in XP with respect to the MSO formula and the treewidth (or more generally clique-width) of the input graph, and is W[1]-hard even under further severe restrictions.

cs.CC↗

arXiv2016

Crossing Number is Hard for Kernelization

The graph crossing number problem, cr(G)<=k, asks for a drawing of a graph G in the plane with at most k edge crossings. Although this problem is in general notoriously difficult, it is fixed- parameter tractable for the parameter k [Grohe]. This suggests a closely related question of whether this problem has a polynomial kernel, meaning whether every instance of cr(G)<=k can be in polynomial time reduced to an equivalent instance of size polynomial in k (and independent of |G|). We answer this question in the negative. Along the proof we show that the tile crossing number problem of twisted planar tiles is NP-hard, which has been an open problem for some time, too, and then employ the complexity technique of cross-composition. Our result holds already for the special case of graphs obtained from planar graphs by adding one edge.

cs.CC↗

arXiv2015

Faster Existential FO Model Checking on Posets

We prove that the model checking problem for the existential fragment of first-order (FO) logic on partially ordered sets is fixed-parameter tractable (FPT) with respect to the formula and the width of a poset (the maximum size of an antichain). While there is a long line of research into FO model checking on graphs, the study of this problem on posets has been initiated just recently by Bova, Ganian and Szeider (CSL-LICS 2014), who proved that the existential fragment of FO has an FPT algorithm for a poset of fixed width. We improve upon their result in two ways: (1) the runtime of our algorithm is O(f(|ϕ|,w).n^2) on n-element posets of width w, compared to O(g(|ϕ|). n^{h(w)}) of Bova et al., and (2) our proofs are simpler and easier to follow. We complement this result by showing that, under a certain complexity-theoretical assumption, the existential FO model checking problem does not have a polynomial kernel.

cs.LO↗

arXiv2015

Inserting Multiple Edges into a Planar Graph

Let $G$ be a connected planar (but not yet embedded) graph and $F$ a set of additional edges not yet in $G$. The {multiple edge insertion} problem (MEI) asks for a drawing of $G+F$ with the minimum number of pairwise edge crossings, such that the subdrawing of $G$ is plane. An optimal solution to this problem approximates the crossing number of the graph $G+F$. Finding an exact solution to MEI is NP-hard for general $F$, but linear time solvable for the special case of $|F|=1$ (SODA01, Algorithmica) or when all of $F$ are incident to a new vertex (SODA09). The complexity for general $F$ but with constant $k=|F|$ was open, but algorithms both with relative and absolute approximation guarantees have been presented (SODA11, ICALP11). We show that the problem is fixed parameter tractable (FPT) in $k$ for biconnected $G$, or if the cut vertices of $G$ have degrees bounded by a constant. We give the first exact algorithm for this problem; it requires only $O(|V(G)|)$ time for any constant $k$.

cs.DS↗

arXiv2015

On Hardness of the Joint Crossing Number

The Joint Crossing Number problem asks for a simultaneous embedding of two disjoint graphs into one surface such that the number of edge crossings (between the two graphs) is minimized. It was introduced by Negami in 2001 in connection with diagonal flips in triangulations of surfaces, and subsequently investigated in a general form for small-genus surfaces. We prove that all of the commonly considered variants of this problem are NP-hard already in the orientable surface of genus 6, by a reduction from a special variant of the anchored crossing number problem of Cabello and Mohar.

cs.DM↗

arXiv2015

Strategy Synthesis in Adversarial Patrolling Games

Patrolling is one of the central problems in operational security. Formally, a patrolling problem is specified by a set $U$ of nodes (admissible defender's positions), a set $T \subseteq U$ of vulnerable targets, an admissible defender's moves over $U$, and a function which to every target assigns the time needed to complete an intrusion at it. The goal is to design an optimal strategy for a defender who is moving from node to node and aims at detecting possible intrusions at the targets. The goal of the attacker is to maximize the probability of a successful attack. We assume that the attacker is adversarial, i.e., he knows the strategy of the defender and can observe her moves. We prove that the defender has an optimal strategy for every patrolling problem. Further, we show that for every $\varepsilon$ > 0, there exists a finite-memory $\varepsilon$-optimal strategy for the defender constructible in exponential time, and we observe that such a strategy cannot be computed in polynomial time unless P=NP. Then we focus ourselves to unrestricted defender's moves. Here, a patrolling problem is fully determined by its signature, the number of targets of each attack length. We bound the maximal probability of successfully defended attacks. Then, we introduce a decomposition method which allows to split a given patrolling problem $G$ into smaller subproblems and construct a defender's strategy for $G$ by "composing" the strategies constructed for these subproblems. Finally, for patrolling problems with $T = U$ and a well-formed signature, we give an exact classification of all sufficiently connected environments where the defender can achieve the same value as in the fully connected uniform environment. This result is useful for designing "good" environments where the defender can act optimally.

cs.GT↗

arXiv2015

FO Model Checking on Posets of Bounded Width

Over the past two decades the main focus of research into first-order (FO) model checking algorithms have been sparse relational structures-culminating in the FPT-algorithm by Grohe, Kreutzer and Siebertz for FO model checking of nowhere dense classes of graphs [STOC'14], with dense structures starting to attract attention only recently. Bova, Ganian and Szeider [LICS'14] initiated the study of the complexity of FO model checking on partially ordered sets (posets). Bova, Ganian and Szeider showed that model checking existential FO logic is fixed-parameter tractable (FPT) on posets of bounded width, where the width of a poset is the size of the largest antichain in the poset. The existence of an FPT algorithm for general FO model checking on posets of bounded width, however, remained open. We resolve this question in the positive by giving an algorithm that takes as its input an $n$-element poset $\cal P$ of width $w$ and an FO logic formula $ϕ$, and determines whether $ϕ$ holds on $\cal P$ in time $f(ϕ,w)\cdot n^2$.

cs.LO↗

arXiv2015

Kernelization Using Structural Parameters on Sparse Graph Classes

Meta-theorems for polynomial (linear) kernels have been the subject of intensive research in parameterized complexity. Heretofore, meta-theorems for linear kernels exist on graphs of bounded genus, $H$-minor-free graphs, and $H$-topological-minor-free graphs. To the best of our knowledge, no meta-theorems for polynomial kernels are known for any larger sparse graph classes; e.g., for classes of bounded expansion or for nowhere dense ones. In this paper we prove such meta-theorems for the two latter cases. More specifically, we show that graph problems that have finite integer index (FII) have linear kernels on graphs of bounded expansion when parameterized by the size of a modulator to constant-treedepth graphs. For nowhere dense graph classes, our result yields almost-linear kernels. While our parameter may seem rather strong, we argue that a linear kernelization result on graphs of bounded expansion with a weaker parameter (than treedepth modulator) would fail to include some of the problems covered by our framework. Moreover, we only require the problems to have FII on graphs of constant treedepth. This allows us to prove linear kernels for problems such as Longest Path/Cycle, Exact $s,t$-Path, Treewidth, and Pathwidth, which do not have FII on general graphs (and the first two not even on bounded treewidth graphs).

cs.DS↗

arXiv2014

Tree-depth and Vertex-minors

In a recent paper, Kwon and Oum claim that every graph of bounded rank-width is a pivot-minor of a graph of bounded tree-width (while the converse has been known true already before). We study the analogous questions for "depth" parameters of graphs, namely for the tree-depth and related new shrub-depth. We show that shrub-depth is monotone under taking vertex-minors, and that every graph class of bounded shrub-depth can be obtained via vertex-minors of graphs of bounded tree-depth. We also consider the same questions for bipartite graphs and pivot-minors.

cs.DM↗

arXiv2012

Lower Bounds on the Complexity of MSO1 Model-Checking

One of the most important algorithmic meta-theorems is a famous result by Courcelle, which states that any graph problem definable in monadic second-order logic with edge-set quantifications (i.e., MSO2 model-checking) is decidable in linear time on any class of graphs of bounded tree-width. Recently, Kreutzer and Tazari proved a corresponding complexity lower-bound - that MSO2 model-checking is not even in XP wrt. the formula size as parameter for graph classes that are subgraph-closed and whose tree-width is poly-logarithmically unbounded. Of course, this is not an unconditional result but holds modulo a certain complexity-theoretic assumption, namely, the Exponential Time Hypothesis (ETH). In this paper we present a closely related result. We show that even MSO1 model-checking with a fixed set of vertex labels, but without edge-set quantifications, is not in XP wrt. the formula size as parameter for graph classes which are subgraph-closed and whose tree-width is poly-logarithmically unbounded unless the non-uniform ETH fails. In comparison to Kreutzer and Tazari; $(1)$ we use a stronger prerequisite, namely non-uniform instead of uniform ETH, to avoid the effectiveness assumption and the construction of certain obstructions used in their proofs; and $(2)$ we assume a different set of problems to be efficiently decidable, namely MSO1-definable properties on vertex labeled graphs instead of MSO2-definable properties on unlabeled graphs. Our result has an interesting consequence in the realm of digraph width measures: Strengthening the recent result, we show that no subdigraph-monotone measure can be "algorithmically useful", unless it is within a poly-logarithmic factor of undirected tree-width.

cs.LO↗

arXiv2012

Detours in Scope-Based Route Planning

We study a dynamic scenario of the static route planning problem in road networks. Particularly, we put accent on the most practical dynamic case - increased edge weights (up to infinity). We show how to enhance the scope-based route planning approach presented at ESA'11 to intuitively by-pass closures by detours. Three variants of a detour "admissibility" are presented - from a simple one with straightforward implementation through its enhanced version to a full and very complex variant variant which always returns an optimal detour.

cs.DS↗

arXiv2011

How Not to Characterize Planar-emulable Graphs

We investigate the question of which graphs have planar emulators (a locally-surjective homomorphism from some finite planar graph) -- a problem raised already in Fellows' thesis (1985) and conceptually related to the better known planar cover conjecture by Negami (1986). For over two decades, the planar emulator problem lived poorly in a shadow of Negami's conjecture--which is still open--as the two were considered equivalent. But, in the end of 2008, a surprising construction by Rieck and Yamashita falsified the natural "planar emulator conjecture", and thus opened a whole new research field. We present further results and constructions which show how far the planar-emulability concept is from planar-coverability, and that the traditional idea of likening it to projective embeddability is actually very out-of-place. We also present several positive partial characterizations of planar-emulable graphs.

cs.DM↗

arXiv2010

Better algorithms for satisfiability problems for formulas of bounded rank-width

We provide a parameterized polynomial algorithm for the propositional model counting problem #SAT, the runtime of which is single-exponential in the rank-width of a formula. Previously, analogous algorithms have been known -- e.g.~[Fischer, Makowsky, and Ravve] -- with a single-exponential dependency on the clique-width of a formula. Our algorithm thus presents an exponential runtime improvement (since clique-width reaches up to exponentially higher values than rank-width), and can be of practical interest for small values of rank-width. We also provide an algorithm for the MAX-SAT problem along the same lines.

cs.DM↗