174 research outputs found
PACE Solver Description: Tweed-Plus: A Subtree-Improving Heuristic Solver for Treedepth
This paper introduces Tweed-Plus, a heuristic solver for the treedepth problem. The solver uses two well-known algorithms to create an initial elimination tree: nested dissection (making use of the Metis library) and the minimum-degree heuristic. After creating an elimination tree of the entire input graph, the solver continues to apply nested dissection and the minimum-degree heuristic to parts of the graph with the aim of replacing subtrees of the elimination tree with alternatives of lower depth
Solving Connectivity Problems Parameterized by Treedepth in Single-Exponential Time and Polynomial Space
A breakthrough result of Cygan et al. (FOCS 2011) showed that connectivity problems parameterized by treewidth can be solved much faster than the previously best known time ?^*(2^{?(twlog tw)}). Using their inspired Cut&Count technique, they obtained ?^*(?^tw) time algorithms for many such problems. Moreover, they proved these running times to be optimal assuming the Strong Exponential-Time Hypothesis. Unfortunately, like other dynamic programming algorithms on tree decompositions, these algorithms also require exponential space, and this is widely believed to be unavoidable. In contrast, for the slightly larger parameter called treedepth, there are already several examples of matching the time bounds obtained for treewidth, but using only polynomial space. Nevertheless, this has remained open for connectivity problems.
In the present work, we close this knowledge gap by applying the Cut&Count technique to graphs of small treedepth. While the general idea is unchanged, we have to design novel procedures for counting consistently cut solution candidates using only polynomial space. Concretely, we obtain time ?^*(3^d) and polynomial space for Connected Vertex Cover, Feedback Vertex Set, and Steiner Tree on graphs of treedepth d. Similarly, we obtain time ?^*(4^d) and polynomial space for Connected Dominating Set and Connected Odd Cycle Transversal
A Faster Parameterized Algorithm for Treedepth
The width measure \emph{treedepth}, also known as vertex ranking, centered
coloring and elimination tree height, is a well-established notion which has
recently seen a resurgence of interest. We present an algorithm which---given
as input an -vertex graph, a tree decomposition of the graph of width ,
and an integer ---decides Treedepth, i.e. whether the treedepth of the graph
is at most , in time . If necessary, a witness structure
for the treedepth can be constructed in the same running time. In conjunction
with previous results we provide a simple algorithm and a fast algorithm which
decide treedepth in time and ,
respectively, which do not require a tree decomposition as part of their input.
The former answers an open question posed by Ossona de Mendez and Nesetril as
to whether deciding Treedepth admits an algorithm with a linear running time
(for every fixed ) that does not rely on Courcelle's Theorem or other heavy
machinery. For chordal graphs we can prove a running time of for the same algorithm.Comment: An extended abstract was published in ICALP 2014, Track
An Algorithm for the Exact Treedepth Problem
We present a novel algorithm for the minimum-depth elimination tree problem, which is equivalent to the optimal treedepth decomposition problem. Our algorithm makes use of two cheaply-computed lower bound functions to prune the search tree, along with symmetry-breaking and domination rules. We present an empirical study showing that the algorithm outperforms the current state-of-the-art solver (which is based on a SAT encoding) by orders of magnitude on a range of graph classes
Treedepth Parameterized by Vertex Cover Number
To solve hard graph problems from the parameterized perspective, structural parameters have commonly been used. In particular, vertex cover number is frequently used in this context. In this paper, we study the problem of computing the treedepth of a given graph G. We show that there are an O(tau(G)^3) vertex kernel and an O(4^{tau(G)}*tau(G)*n) time fixed-parameter algorithm for this problem, where tau(G) is the size of a minimum vertex cover of G and n is the number of vertices of G
Circumference and Pathwidth of Highly Connected Graphs
Birmele [J. Graph Theory, 2003] proved that every graph with circumference t
has treewidth at most t-1. Under the additional assumption of 2-connectivity,
such graphs have bounded pathwidth, which is a qualitatively stronger result.
Birmele's theorem was extended by Birmele, Bondy and Reed [Combinatorica, 2007]
who showed that every graph without k disjoint cycles of length at least t has
bounded treewidth (as a function of k and t). Our main result states that,
under the additional assumption of (k + 1)- connectivity, such graphs have
bounded pathwidth. In fact, they have pathwidth O(t^3 + tk^2). Moreover,
examples show that (k + 1)-connectivity is required for bounded pathwidth to
hold. These results suggest the following general question: for which values of
k and graphs H does every k-connected H-minor-free graph have bounded
pathwidth? We discuss this question and provide a few observations.Comment: 11 pages, 4 figure
PACE Solver Description: Bute-Plus: A Bottom-Up Exact Solver for Treedepth
This note introduces Bute-Plus, an exact solver for the treedepth problem.
The core of the solver is a positive-instance driven dynamic program that
constructs an elimination tree of minimum depth in a bottom-up fashion. Three
features greatly improve the algorithm's run time. The first of these is a
specialised trie data structure. The second is a domination rule. The third is
a heuristic presolve step can quickly find a treedepth decomposition of optimal
depth for many instances.Comment: 4 pages, 1 appendix pages, 0 figures. A version of this tool
description paper without the appendix is published in the proceedings of
IPEC 2020:
https://drops.dagstuhl.de/opus/volltexte/2020/13337/pdf/LIPIcs-IPEC-2020-34.pdf
. Changes: this version expands the paper from a preliminary versio
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