45 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
Linear MIM-Width of Trees
We provide an algorithm computing the linear maximum induced
matching width of a tree and an optimal layout.Comment: 19 pages, 7 figures, full version of WG19 paper of same nam
Tight bounds for planar strongly connected Steiner subgraph with fixed number of terminals (and extensions)
(see paper for full abstract)
Given a vertex-weighted directed graph and a set of terminals, the objective of the SCSS problem is to find a
vertex set of minimum weight such that contains a
path for each . The problem is NP-hard, but
Feldman and Ruhl [FOCS '99; SICOMP '06] gave a novel algorithm for
the SCSS problem, where is the number of vertices in the graph and is
the number of terminals. We explore how much easier the problem becomes on
planar directed graphs:
- Our main algorithmic result is a algorithm
for planar SCSS, which is an improvement of a factor of in the
exponent over the algorithm of Feldman and Ruhl.
- Our main hardness result is a matching lower bound for our algorithm: we
show that planar SCSS does not have an algorithm
for any computable function , unless the Exponential Time Hypothesis (ETH)
fails.
The following additional results put our upper and lower bounds in context:
- In general graphs, we cannot hope for such a dramatic improvement over the
algorithm of Feldman and Ruhl: assuming ETH, SCSS in general graphs
does not have an algorithm for any computable
function .
- Feldman and Ruhl generalized their algorithm to the more general
Directed Steiner Network (DSN) problem; here the task is to find a subgraph of
minimum weight such that for every source there is a path to the
corresponding terminal . We show that, assuming ETH, there is no
time algorithm for DSN on acyclic planar graphs.Comment: To appear in SICOMP. Extended abstract in SODA 2014. This version has
a new author (Andreas Emil Feldmann), and the algorithm is faster and
considerably simplified as compared to conference versio
Parameterized Inapproximability of Independent Set in -Free Graphs
We study the Independent Set (IS) problem in -free graphs, i.e., graphs
excluding some fixed graph as an induced subgraph. We prove several
inapproximability results both for polynomial-time and parameterized
algorithms.
Halld\'orsson [SODA 1995] showed that for every IS has a
polynomial-time -approximation in -free
graphs. We extend this result by showing that -free graphs admit a
polynomial-time -approximation, where is the
size of a maximum independent set in . Furthermore, we complement the result
of Halld\'orsson by showing that for some there is
no polynomial-time -approximation for these graphs, unless NP = ZPP.
Bonnet et al. [IPEC 2018] showed that IS parameterized by the size of the
independent set is W[1]-hard on graphs which do not contain (1) a cycle of
constant length at least , (2) the star , and (3) any tree with two
vertices of degree at least at constant distance.
We strengthen this result by proving three inapproximability results under
different complexity assumptions for almost the same class of graphs (we weaken
condition (2) that does not contain ). First, under the ETH, there
is no algorithm for any computable function .
Then, under the deterministic Gap-ETH, there is a constant such that
no -approximation can be computed in time. Also,
under the stronger randomized Gap-ETH there is no such approximation algorithm
with runtime .
Finally, we consider the parameterization by the excluded graph , and show
that under the ETH, IS has no algorithm in -free graphs
and under Gap-ETH there is no -approximation for -free
graphs with runtime .Comment: Preliminary version of the paper in WG 2020 proceeding
Maximum Independent Set when excluding an induced minor: and
Dallard, Milani\v{c}, and \v{S}torgel [arXiv '22] ask if for every class
excluding a fixed planar graph as an induced minor, Maximum Independent Set
can be solved in polynomial time, and show that this is indeed the case when
is any planar complete bipartite graph, or the 5-vertex clique minus one
edge, or minus two disjoint edges. A positive answer would constitute a
far-reaching generalization of the state-of-the-art, when we currently do not
know if a polynomial-time algorithm exists when is the 7-vertex path.
Relaxing tractability to the existence of a quasipolynomial-time algorithm, we
know substantially more. Indeed, quasipolynomial-time algorithms were recently
obtained for the -vertex cycle, [Gartland et al., STOC '21] and the
disjoint union of triangles, [Bonamy et al., SODA '23].
We give, for every integer , a polynomial-time algorithm running in
when is the friendship graph ( disjoint edges
plus a vertex fully adjacent to them), and a quasipolynomial-time algorithm
running in when is (the
disjoint union of triangles and a 4-vertex cycle). The former extends a
classical result on graphs excluding as an induced subgraph [Alekseev,
DAM '07], while the latter extends Bonamy et al.'s result.Comment: 15 pages, 2 figure