13 research outputs found

### Degree-doubling graph families

Let G be a family of n-vertex graphs of uniform degree 2 with the property
that the union of any two member graphs has degree four. We determine the
leading term in the asymptotics of the largest cardinality of such a family.
Several analogous problems are discussed.Comment: 9 page

### Well-quasi-ordering digraphs with no long alternating paths by the strong immersion relation

Nash-Williams' Strong Immersion Conjecture states that graphs are
well-quasi-ordered by the strong immersion relation. That is, given infinitely
many graphs, one graph contains another graph as a strong immersion. In this
paper we study the analogous problem for directed graphs. It is known that
digraphs are not well-quasi-ordered by the strong immersion relation, but for
all known such infinite antichains, paths that change direction arbitrarily
many times can be found. This paper proves that the converse statement is true:
for every positive integer $k$, the digraphs that do not contain a path that
changes direction $k$ times are well-quasi-ordered by the strong immersion
relation, even when vertices are labelled by a well-quasi-order. This result is
optimal since paths that change direction arbitrarily many times with
vertex-labels form an infinite antichain with respect to the strong immersion
relation

### Half-Integral Linkages in Highly Connected Directed Graphs

We study the half-integral k-Directed Disjoint Paths Problem (1/2 kDDPP) in highly strongly connected digraphs. The integral kDDPP is NP-complete even when restricted to instances where k=2, and the input graph is L-strongly connected, for any L >= 1. We show that when the integrality condition is relaxed to allow each vertex to be used in two paths, the problem becomes efficiently solvable in highly connected digraphs (even with k as part of the input).
Specifically, we show that there is an absolute constant c such that for each k >= 2 there exists L(k) such that 1/2 kDDPP is solvable in time O(|V(G)|^c) for a L(k)-strongly connected directed graph G. As the function L(k) grows rather quickly, we also show that 1/2 kDDPP is solvable in time O(|V(G)|^{f(k)}) in (36k^3+2k)-strongly connected directed graphs. We show that for each epsilon<1, deciding half-integral feasibility of kDDPP instances is NP-complete when k is given as part of the input, even when restricted to graphs with strong connectivity epsilon k

### Computing complexity measures of degenerate graphs

We show that the VC-dimension of a graph can be computed in time $n^{\log
d+1} d^{O(d)}$, where $d$ is the degeneracy of the input graph. The core idea
of our algorithm is a data structure to efficiently query the number of
vertices that see a specific subset of vertices inside of a (small) query set.
The construction of this data structure takes time $O(d2^dn)$, afterwards
queries can be computed efficiently using fast M\"obius inversion.
This data structure turns out to be useful for a range of tasks, especially
for finding bipartite patterns in degenerate graphs, and we outline an
efficient algorithms for counting the number of times specific patterns occur
in a graph. The largest factor in the running time of this algorithm is
$O(n^c)$, where $c$ is a parameter of the pattern we call its left covering
number.
Concrete applications of this algorithm include counting the number of
(non-induced) bicliques in linear time, the number of co-matchings in quadratic
time, as well as a constant-factor approximation of the ladder index in linear
time.
Finally, we supplement our theoretical results with several implementations
and run experiments on more than 200 real-world datasets -- the largest of
which has 8 million edges -- where we obtain interesting insights into the
VC-dimension of real-world networks.Comment: Accepted for publication in the 18th International Symposium on
Parameterized and Exact Computation (IPEC 2023

### Packing Directed Cycles Quarter- and Half-Integrally

The celebrated Erd\H{o}s-P\'osa theorem states that every undirected graph
that does not admit a family of $k$ vertex-disjoint cycles contains a feedback
vertex set (a set of vertices hitting all cycles in the graph) of size $O(k
\log k)$. After being known for long as Younger's conjecture, a similar
statement for directed graphs has been proven in 1996 by Reed, Robertson,
Seymour, and Thomas. However, in their proof, the dependency of the size of the
feedback vertex set on the size of vertex-disjoint cycle packing is not
elementary.
We show that if we compare the size of a minimum feedback vertex set in a
directed graph with the quarter-integral cycle packing number, we obtain a
polynomial bound. More precisely, we show that if in a directed graph $G$ there
is no family of $k$ cycles such that every vertex of $G$ is in at most four of
the cycles, then there exists a feedback vertex set in $G$ of size $O(k^4)$.
Furthermore, a variant of our proof shows that if in a directed graph $G$ there
is no family of $k$ cycles such that every vertex of $G$ is in at most two of
the cycles, then there exists a feedback vertex set in $G$ of size $O(k^6)$.
On the way there we prove a more general result about quarter-integral
packing of subgraphs of high directed treewidth: for every pair of positive
integers $a$ and $b$, if a directed graph $G$ has directed treewidth
$\Omega(a^6 b^8 \log^2(ab))$, then one can find in $G$ a family of $a$
subgraphs, each of directed treewidth at least $b$, such that every vertex of
$G$ is in at most four subgraphs.Comment: Accepted to European Symposium on Algorithms (ESA '19

### Optimal Discretization is Fixed-parameter Tractable

Given two disjoint sets $W_1$ and $W_2$ of points in the plane, the Optimal
Discretization problem asks for the minimum size of a family of horizontal and
vertical lines that separate $W_1$ from $W_2$, that is, in every region into
which the lines partition the plane there are either only points of $W_1$, or
only points of $W_2$, or the region is empty. Equivalently, Optimal
Discretization can be phrased as a task of discretizing continuous variables:
we would like to discretize the range of $x$-coordinates and the range of
$y$-coordinates into as few segments as possible, maintaining that no pair of
points from $W_1 \times W_2$ are projected onto the same pair of segments under
this discretization.
We provide a fixed-parameter algorithm for the problem, parameterized by the
number of lines in the solution. Our algorithm works in time $2^{O(k^2 \log k)}
n^{O(1)}$, where $k$ is the bound on the number of lines to find and $n$ is the
number of points in the input.
Our result answers in positive a question of Bonnet, Giannopolous, and Lampis
[IPEC 2017] and of Froese (PhD thesis, 2018) and is in contrast with the known
intractability of two closely related generalizations: the Rectangle Stabbing
problem and the generalization in which the selected lines are not required to
be axis-parallel.Comment: Accepted to ACM-SIAM Symposium on Discrete Algorithms (SODA 2021

### Being Even Slightly Shallow Makes Life Hard

We study the computational complexity of identifying dense substructures, namely r/2-shallow topological minors and r-subdivisions. Of particular interest is the case r = 1, when these substructures correspond to very localized relaxations of subgraphs. Since Densest Subgraph can be solved in polynomial time, we ask whether these slight relaxations also admit efficient algorithms.
In the following, we provide a negative answer: Dense r/2-Shallow Topological Minor and Dense r-Subdivsion are already NP-hard for r = 1 in very sparse graphs. Further, they do not admit algorithms with running time 2^(o(tw^2)) n^O(1) when parameterized by the treewidth of the input graph for r > 2 unless ETH fails

### Algorithmic Properties of Sparse Digraphs

The notions of bounded expansion [Nesetril and Ossona de Mendez, 2008] and nowhere denseness [Nesetril and Ossona de Mendez, 2011], introduced by Nesetril and Ossona de Mendez as structural measures for undirected graphs, have been applied very successfully in algorithmic graph theory. We study the corresponding notions of directed bounded expansion and nowhere crownfulness on directed graphs, introduced by Kreutzer and Tazari [Kreutzer and Tazari, 2012]. The classes of directed graphs having those properties are very general classes of sparse directed graphs, as they include, on one hand, all classes of directed graphs whose underlying undirected class has bounded expansion, such as planar, bounded-genus, and H-minor-free graphs, and on the other hand, they also contain classes whose underlying undirected class is not even nowhere dense. We show that many of the algorithmic tools that were developed for undirected bounded expansion classes can, with some care, also be applied in their directed counterparts, and thereby we highlight a rich algorithmic structure theory of directed bounded expansion and nowhere crownful classes