Dagstuhl Seminar Proceedings, Volume 8431
Dagstuhl Seminar Proceedings
DagSemProc
https://www.dagstuhl.de/dagpub/1862-4405
https://dblp.org/db/series/dagstuhl
1862-4405
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
8431
2008
https://drops.dagstuhl.de/entities/volume/DagSemProc-volume-8431
08431 Abstracts Collection – Moderately Exponential Time Algorithms
From $19/10/2008$ to $24/10/2008$, the Dagstuhl Seminar 08431 ``Moderately Exponential Time Algorithms '' was held in Schloss Dagstuhl~--~Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.
Algorithms
Exponential time algorithms
Graphs
SAT
1-22
Regular Paper
Fedor V.
Fomin
Fedor V. Fomin
Kazuo
Iwama
Kazuo Iwama
Dieter
Kratsch
Dieter Kratsch
10.4230/DagSemProc.08431.1
Creative Commons Attribution 4.0 International license
https://creativecommons.org/licenses/by/4.0/legalcode
08431 Executive Summary – Moderately Exponential Time Algorithms
The Dagstuhl seminar on Moderately Exponential Time Algorithms took place
from 19.10.08 to 24.10.08. The 54 participants came from 18 countries.
There were 27 talks and 2 open problem sessions. Talks were complemented
by intensive informal discussions, and many new research directions
and open problems will result from these discussions. The warm and encouraging Dagstuhl atmosphere stimulated new research projects.
Algorithms
NP-hard problems
Exact algorithms
Moderately Exponential Time Algorithms
1-2
Regular Paper
Fedor V.
Fomin
Fedor V. Fomin
Kazuo
Iwama
Kazuo Iwama
Dieter
Kratsch
Dieter Kratsch
10.4230/DagSemProc.08431.2
Creative Commons Attribution 4.0 International license
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08431 Open Problems – Moderately Exponential Time Algorithms
Two problem sessions were part of the seminar on Moderately Exponential Time Algorithms. Some of the open problems presented at those sessions have been collected.
Algorithms
NP-hard problems
Moderately Exponential Time Algorithms
1-8
Regular Paper
Fedor V.
Fomin
Fedor V. Fomin
Kazuo
Iwama
Kazuo Iwama
Dieter
Kratsch
Dieter Kratsch
Petteri
Kaski
Petteri Kaski
Mikko
Koivisto
Mikko Koivisto
Lukasz
Kowalik
Lukasz Kowalik
Yoshio
Okamoto
Yoshio Okamoto
Johan
van Rooij
Johan van Rooij
Ryan
Williams
Ryan Williams
10.4230/DagSemProc.08431.3
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Spanning Trees of Bounded Degree Graphs
We consider lower bounds on the number of spanning trees of connected graphs
with degree bounded by $d$.
The question is of interest because such bounds may improve the analysis of the
improvement produced by memorisation in the runtime of exponential algorithms.
The value of interest is the constant $beta_d$ such that all connected graphs with degree bounded by $d$ have at least $beta_d^mu$ spanning trees where $mu$ is the cyclomatic number or
excess of the graph, namely $m-n+1$.
We conjecture that $beta_d$ is achieved by the complete graph $K_{d+1}$ but we have not proved this for any $d$ greater than $3$. We give weaker lower bounds on $beta_d$ for $dle 11$.
First we establish lower bounds on the factor by which the number of spanning trees is multiplied when one new vertex is added to an existing graph so that the new vertex has degree $c$ and the maximum degree of the resulting graph is at most $d$. In all the cases analysed, this lower bound $f_{c,d}$ is attained when the graph before the addition was a complete graph of order $d$ but we have not proved this in general.
Next we show that, for any cut of size $c$ cutting a graph $G$ of degree bounded by $d$
into two connected components $G_1$ and $G_2$, the number of spanning trees of $G$ is
at least the product of this number for $G_1$ and $G_2$ multiplied by the same
factor $f_{c,d}$.
Finally we examine the process of repeatedly cutting a graph until no edges remain. The number of spanning trees is at least the product of the multipliers associated with all the cuts. Some obvious constraints on the number of cuts of each size give linear constraints on the normalised numbers of cuts of each size which are then used to lower bound $beta_d$ by the solution of a linear program.
The lower bound obtained is significantly improved by imposing a rule that, at each stage, a cut of the minimum available size is chosen and adding some new constraints implied by this rule.
Spanning trees
memorisation
cyclomatic number
bounded degree graphs
cut
linear program.
1-8
Regular Paper
John Michael
Robson
John Michael Robson
10.4230/DagSemProc.08431.4
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