16 research outputs found
Reduction Rules and ILP Are All You Need: Minimal Directed Feedback Vertex Set
This note describes the development of an exact solver for Minimal Directed
Feedback Vertex Set as part of the PACE 2022 competition. The solver is powered
largely by aggressively trying to reduce the DFVS problem to a Minimal Cover
problem, and applying reduction rules adapted from Vertex Cover literature. The
resulting problem is solved as an Integer Linear Program (ILP) using SCIP. The
resulting solver performed the second-best in the competition, although a bug
at submission time disqualified it. As an additional note, we describe a new
vertex cover reduction generalizing the Desk reduction rule.Comment: 11 page
Graph editing to a fixed target
For a fixed graph H, the H-Minor Edit problem takes as input a graph G and an integer k and asks whether G can be modified into H by a total of at most k edge contractions, edge deletions and vertex deletions. Replacing edge contractions by vertex dissolutions yields the H-Topological Minor Edit problem. For each problem we show polynomial-time solvable and NP-complete cases depending on the choice of H. Moreover, when G is AT-free, chordal or planar, we show that H-Minor Edit is polynomial-time solvable for all graphs H
Graph editing to a fixed target
For a fixed graph H, the H-Minor Edit problem takes as input a graph G and an integer k and asks whether G can be modified into H by a total of at most k edge contractions, edge deletions and vertex deletions. Replacing edge contractions by vertex dissolutions yields the H-Topological Minor Edit problem. For each problem we show polynomial-time solvable and NP-complete cases depending on the choice of H. Moreover, when G is AT-free, chordal or planar, we show that H-Minor Edit is polynomial-time solvable for all graphs H
Blazing a Trail via Matrix Multiplications: A Faster Algorithm for Non-Shortest Induced Paths
For vertices and of an -vertex graph , a -trail of is
an induced -path of that is not a shortest -path of . Berger,
Seymour, and Spirkl [Discrete Mathematics 2021] gave the previously only known
polynomial-time algorithm, running in time, to either output a
-trail of or ensure that admits no -trail. We reduce the
complexity to the time required to perform a poly-logarithmic number of
multiplications of Boolean matrices, leading to a largely
improved -time algorithm.Comment: 18 pages, 6 figures, a preliminary version appeared in STACS 202
Improved Algorithms for Recognizing Perfect Graphs and Finding Shortest Odd and Even Holes
Various classes of induced subgraphs are involved in the deepest results of
graph theory and graph algorithms. A prominent example concerns the {\em
perfection} of that the chromatic number of each induced subgraph of
equals the clique number of . The seminal Strong Perfect Graph Theorem
confirms that the perfection of can be determined by detecting odd holes in
and its complement. Chudnovsky et al. show in 2005 an algorithm
for recognizing perfect graphs, which can be implemented to run in
time for the exponent of square-matrix
multiplication. We show the following improved algorithms.
1. The tractability of detecting odd holes was open for decades until the
major breakthrough of Chudnovsky et al. in 2020. Their algorithm is
later implemented by Lai et al. to run in time, leading to the best
formerly known algorithm for recognizing perfect graphs. Our first result is an
algorithm for detecting odd holes, implying an algorithm for
recognizing perfect graphs.
2. Chudnovsky et al. extend in 2021 the algorithms for detecting odd
holes (2020) and recognizing perfect graphs (2005) into the first polynomial
algorithm for obtaining a shortest odd hole, which runs in time. We
reduce the time for finding a shortest odd hole to .
3. Conforti et al. show in 1997 the first polynomial algorithm for detecting
even holes, running in about time. It then takes a line of
intensive efforts in the literature to bring down the complexity to
, , , and finally . On the other hand,
the tractability of finding a shortest even hole has been open for 16 years
until the very recent algorithm of Cheong and Lu in 2022. We
improve the time of finding a shortest even hole to .Comment: 29 pages, 5 figure