arXiv subjects

Roman Rabinovich

Publications and source records attributed to Roman Rabinovich.

24 records · Page 2Linked to original sources

arXiv2016

The Generalised Colouring Numbers on Classes of Bounded Expansion

The generalised colouring numbers $\mathrm{adm}_r(G)$, $\mathrm{col}_r(G)$, and $\mathrm{wcol}_r(G)$ were introduced by Kierstead and Yang as generalisations of the usual colouring number, also known as the degeneracy of a graph, and have since then found important applications in the theory of bounded expansion and nowhere dense classes of graphs, introduced by Nešetřil and Ossona de Mendez. In this paper, we study the relation of the colouring numbers with two other measures that characterise nowhere dense classes of graphs, namely with uniform quasi-wideness, studied first by Dawar et al. in the context of preservation theorems for first-order logic, and with the splitter game, introduced by Grohe et al. We show that every graph excluding a fixed topological minor admits a universal order, that is, one order witnessing that the colouring numbers are small for every value of $r$. Finally, we use our construction of such orders to give a new proof of a result of Eickmeyer and Kawarabayashi, showing that the model-checking problem for successor-invariant first-order formulas is fixed-parameter tractable on classes of graphs with excluded topological minors.

cs.DM↗

arXiv2016

Routing with Congestion in Acyclic Digraphs

We study the version of the $k$-disjoint paths problem where $k$ demand pairs $(s_1,t_1)$, $\dots$, $(s_k,t_k)$ are specified in the input and the paths in the solution are allowed to intersect, but such that no vertex is on more than $c$ paths. We show that on directed acyclic graphs the problem is solvable in time $n^{O(d)}$ if we allow congestion $k-d$ for $k$ paths. Furthermore, we show that, under a suitable complexity theoretic assumption, the problem cannot be solved in time $f(k)n^{o(d/\log d)}$ for any computable function $f$.

cs.DS↗

arXiv2014

Directed Width Measures and Monotonicity of Directed Graph Searching

We consider generalisations of tree width to directed graphs, that attracted much attention in the last fifteen years. About their relative strength with respect to "bounded width in one measure implies bounded width in the other" many problems remain unsolved. Only some results separating directed width measures are known. We give an almost complete picture of this relation. For this, we consider the cops and robber games characterising DAG-width and directed tree width (up to a constant factor). For DAG-width games, it is an open question whether the robber-monotonicity cost (the difference between the minimal numbers of cops capturing the robber in the general and in the monotone case) can be bounded by any function. Examples show that this function (if it exists) is at least $f(k) > 4k/3$ (Kreutzer, Ordyniak 2008). We approach a solution by defining weak monotonicity and showing that if $k$ cops win weakly monotonically, then $O(k^2)$ cops win monotonically. It follows that bounded Kelly-width implies bounded DAG-width, which has been open since the definition of Kelly-width by Hunter and Kreutzer in 2008. For directed tree width games we show that, unexpectedly, the cop-monotonicity cost (no cop revisits any vertex) is not bounded by any function. This separates directed tree width from D-width defined by Safari in 2005, refuting his conjecture.

cs.DM↗

arXiv2012

Down the Borel Hierarchy: Solving Muller Games via Safety Games

We transform a Muller game with n vertices into a safety game with (n!)^3 vertices whose solution allows to determine the winning regions of the Muller game and to compute a finite-state winning strategy for one player. This yields a novel antichain-based memory structure and a natural notion of permissive strategies for Muller games. Moreover, we generalize our construction by presenting a new type of game reduction from infinite games to safety games and show its applicability to several other winning conditions.

cs.LO↗

arXiv2012

The discrete strategy improvement algorithm for parity games and complexity measures for directed graphs

For some time the discrete strategy improvement algorithm due to Jurdzinski and Voge had been considered as a candidate for solving parity games in polynomial time. However, it has recently been proved by Oliver Friedmann that the strategy improvement algorithm requires super-polynomially many iteration steps, for all popular local improvements rules, including switch-all (also with Fearnley's snare memorisation), switch-best, random-facet, random-edge, switch-half, least-recently-considered, and Zadeh's Pivoting rule. We analyse the examples provided by Friedmann in terms of complexity measures for directed graphs such as treewidth, DAG-width, Kelly-width, entanglement, directed pathwidth, and cliquewidth. It is known that for every class of parity games on which one of these parameters is bounded, the winning regions can be efficiently computed. It turns out that with respect to almost all of these measures, the complexity of Friedmann's counterexamples is bounded, and indeed in most cases by very small numbers. This analysis strengthens in some sense Friedmann's results and shows that the discrete strategy improvement algorithm is even more limited than one might have thought. Not only does it require super-polynomial running time in the general case, where the problem of polynomial-time solvability is open, it even has super-polynomial lower time bounds on natural classes of parity games on which efficient algorithms are known.

cs.CC↗

arXiv2011

Graph Searching, Parity Games and Imperfect Information

We investigate the interrelation between graph searching games and games with imperfect information. As key consequence we obtain that parity games with bounded imperfect information can be solved in PTIME on graphs of bounded DAG-width which generalizes several results for parity games on graphs of bounded complexity. We use a new concept of graph searching where several cops try to catch multiple robbers instead of just a single robber. The main technical result is that the number of cops needed to catch r robbers monotonously is at most r times the DAG-width of the graph. We also explore aspects of this new concept as a refinement of directed path-width which accentuates its connection to the concept of imperfect information.

cs.GT↗