6 research outputs found
Improved Bounds for the Excluded-Minor Approximation of Treedepth
Treedepth, a more restrictive graph width parameter than treewidth and pathwidth, plays a major role in the theory of sparse graph classes. We show that there exists a constant C such that for every integers a,b >= 2 and a graph G, if the treedepth of G is at least Cab log a, then the treewidth of G is at least a or G contains a subcubic (i.e., of maximum degree at most 3) tree of treedepth at least b as a subgraph.
As a direct corollary, we obtain that every graph of treedepth Omega(k^3 log k) is either of treewidth at least k, contains a subdivision of full binary tree of depth k, or contains a path of length 2^k. This improves the bound of Omega(k^5 log^2 k) of Kawarabayashi and Rossman [SODA 2018].
We also show an application for approximation algorithms of treedepth: given a graph G of treedepth k and treewidth t, one can in polynomial time compute a treedepth decomposition of G of width O(kt log^{3/2} t). This improves upon a bound of O(kt^2 log t) stemming from a tradeoff between known results.
The main technical ingredient in our result is a proof that every tree of treedepth d contains a subcubic subtree of treedepth at least d * log_3 ((1+sqrt{5})/2)
Approximating pathwidth for graphs of small treewidth
We describe a polynomial-time algorithm which, given a graph with
treewidth , approximates the pathwidth of to within a ratio of
. This is the first algorithm to achieve an
-approximation for some function .
Our approach builds on the following key insight: every graph with large
pathwidth has large treewidth or contains a subdivision of a large complete
binary tree. Specifically, we show that every graph with pathwidth at least
has treewidth at least or contains a subdivision of a complete
binary tree of height . The bound is best possible up to a
multiplicative constant. This result was motivated by, and implies (with
), the following conjecture of Kawarabayashi and Rossman (SODA'18): there
exists a universal constant such that every graph with pathwidth
has treewidth at least or contains a subdivision of a
complete binary tree of height .
Our main technical algorithm takes a graph and some (not necessarily
optimal) tree decomposition of of width in the input, and it computes
in polynomial time an integer , a certificate that has pathwidth at
least , and a path decomposition of of width at most . The
certificate is closely related to (and implies) the existence of a subdivision
of a complete binary tree of height . The approximation algorithm for
pathwidth is then obtained by combining this algorithm with the approximation
algorithm of Feige, Hajiaghayi, and Lee (STOC'05) for treewidth
Efficient fully dynamic elimination forests with applications to detecting long paths and cycles
We present a data structure that in a dynamic graph of treedepth at most ,
which is modified over time by edge insertions and deletions, maintains an
optimum-height elimination forest. The data structure achieves worst-case
update time , which matches the best known parameter
dependency in the running time of a static fpt algorithm for computing the
treedepth of a graph. This improves a result of Dvo\v{r}\'ak et al. [ESA 2014],
who for the same problem achieved update time for some non-elementary
(i.e. tower-exponential) function . As a by-product, we improve known upper
bounds on the sizes of minimal obstructions for having treedepth from
doubly-exponential in to .
As applications, we design new fully dynamic parameterized data structures
for detecting long paths and cycles in general graphs. More precisely, for a
fixed parameter and a dynamic graph , modified over time by edge
insertions and deletions, our data structures maintain answers to the following
queries:
- Does contain a simple path on vertices?
- Does contain a simple cycle on at least vertices?
In the first case, the data structure achieves amortized update time
. In the second case, the amortized update time is . In both cases we assume access to a dictionary
on the edges of .Comment: 74 pages, 5 figure