39 research outputs found
The PACE 2022 Parameterized Algorithms and Computational Experiments Challenge: Directed Feedback Vertex Set
The Parameterized Algorithms and Computational Experiments challenge (PACE) 2022 was devoted to engineer algorithms solving the NP-hard Directed Feedback Vertex Set (DFVS) problem. The DFVS problem is to find a minimum subset in a given directed graph such that, when all vertices of and their adjacent edges are deleted from , the remainder is acyclic.
Overall, the challenge had 90 participants from 26 teams, 12 countries, and 3 continents that submitted their implementations to this year’s competition. In this report, we briefly describe the setup of the challenge, the selection of benchmark instances, as well as the ranking of the participating teams. We also briefly outline the approaches used in the submitted solvers
Grundy Distinguishes Treewidth from Pathwidth
Structural graph parameters, such as treewidth, pathwidth, and clique-width,
are a central topic of study in parameterized complexity. A main aim of
research in this area is to understand the "price of generality" of these
widths: as we transition from more restrictive to more general notions, which
are the problems that see their complexity status deteriorate from
fixed-parameter tractable to intractable? This type of question is by now very
well-studied, but, somewhat strikingly, the algorithmic frontier between the
two (arguably) most central width notions, treewidth and pathwidth, is still
not understood: currently, no natural graph problem is known to be W-hard for
one but FPT for the other. Indeed, a surprising development of the last few
years has been the observation that for many of the most paradigmatic problems,
their complexities for the two parameters actually coincide exactly, despite
the fact that treewidth is a much more general parameter. It would thus appear
that the extra generality of treewidth over pathwidth often comes "for free".
Our main contribution in this paper is to uncover the first natural example
where this generality comes with a high price. We consider Grundy Coloring, a
variation of coloring where one seeks to calculate the worst possible coloring
that could be assigned to a graph by a greedy First-Fit algorithm. We show that
this well-studied problem is FPT parameterized by pathwidth; however, it
becomes significantly harder (W[1]-hard) when parameterized by treewidth.
Furthermore, we show that Grundy Coloring makes a second complexity jump for
more general widths, as it becomes para-NP-hard for clique-width. Hence, Grundy
Coloring nicely captures the complexity trade-offs between the three most
well-studied parameters. Completing the picture, we show that Grundy Coloring
is FPT parameterized by modular-width.Comment: To be published in proceedings of ESA 202
Modification to Planarity is Fixed Parameter Tractable
A replacement action is a function L that maps each k-vertex labeled graph to another k-vertex graph. We consider a general family of graph modification problems, called L-Replacement to C, where the input is a graph G and the question is whether it is possible to replace in G some k-vertex subgraph H of it by L(H) so that the new graph belongs to the graph class C. L-Replacement to C can simulate several modification operations such as edge addition, edge removal, edge editing, and diverse completion and superposition operations. In this paper, we prove that for any action L, if C is the class of planar graphs, there is an algorithm that solves L-Replacement to C in O(|G|^{2}) steps. We also present several applications of our approach to related problems
More applications of the d-neighbor equivalence: acyclicity and connectivity constraints
In this paper, we design a framework to obtain efficient algorithms for
several problems with a global constraint (acyclicity or connectivity) such as
Connected Dominating Set, Node Weighted Steiner Tree, Maximum Induced Tree,
Longest Induced Path, and Feedback Vertex Set. We design a meta-algorithm that
solves all these problems and whose running time is upper bounded by
, , and where is respectively the clique-width,
-rank-width, rank-width and maximum induced matching width of a
given decomposition. Our meta-algorithm simplifies and unifies the known
algorithms for each of the parameters and its running time matches
asymptotically also the running times of the best known algorithms for basic
NP-hard problems such as Vertex Cover and Dominating Set. Our framework is
based on the -neighbor equivalence defined in [Bui-Xuan, Telle and
Vatshelle, TCS 2013]. The results we obtain highlight the importance of this
equivalence relation on the algorithmic applications of width measures.
We also prove that our framework could be useful for -hard problems
parameterized by clique-width such as Max Cut and Maximum Minimal Cut. For
these latter problems, we obtain , and time
algorithms where is respectively the clique-width, the
-rank-width and the rank-width of the input graph