arXiv subjects

Jakub Gajarský

Publications and source records attributed to Jakub Gajarský.

22 records · Page 2Linked to original sources

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

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↗

arXiv2013

Parameterized Algorithms for Modular-Width

It is known that a number of natural graph problems which are FPT parameterized by treewidth become W-hard when parameterized by clique-width. It is therefore desirable to find a different structural graph parameter which is as general as possible, covers dense graphs but does not incur such a heavy algorithmic penalty. The main contribution of this paper is to consider a parameter called modular-width, defined using the well-known notion of modular decompositions. Using a combination of ILPs and dynamic programming we manage to design FPT algorithms for Coloring and Partitioning into paths (and hence Hamiltonian path and Hamiltonian cycle), which are W-hard for both clique-width and its recently introduced restriction, shrub-depth. We thus argue that modular-width occupies a sweet spot as a graph parameter, generalizing several simpler notions on dense graphs but still evading the "price of generality" paid by clique-width.

cs.DS↗