eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2008-12-23
8431
1
22
10.4230/DagSemProc.08431.1
article
08431 Abstracts Collection – Moderately Exponential Time Algorithms
Fomin, Fedor V.
Iwama, Kazuo
Kratsch, Dieter
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol08431/DagSemProc.08431.1/DagSemProc.08431.1.pdf
Algorithms
Exponential time algorithms
Graphs
SAT
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2008-12-23
8431
1
2
10.4230/DagSemProc.08431.2
article
08431 Executive Summary – Moderately Exponential Time Algorithms
Fomin, Fedor V.
Iwama, Kazuo
Kratsch, Dieter
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol08431/DagSemProc.08431.2/DagSemProc.08431.2.pdf
Algorithms
NP-hard problems
Exact algorithms
Moderately Exponential Time Algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2008-12-23
8431
1
8
10.4230/DagSemProc.08431.3
article
08431 Open Problems – Moderately Exponential Time Algorithms
Fomin, Fedor V.
Iwama, Kazuo
Kratsch, Dieter
Kaski, Petteri
Koivisto, Mikko
Kowalik, Lukasz
Okamoto, Yoshio
van Rooij, Johan
Williams, Ryan
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol08431/DagSemProc.08431.3/DagSemProc.08431.3.pdf
Algorithms
NP-hard problems
Moderately Exponential Time Algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2008-12-23
8431
1
8
10.4230/DagSemProc.08431.4
article
Spanning Trees of Bounded Degree Graphs
Robson, John Michael
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol08431/DagSemProc.08431.4/DagSemProc.08431.4.pdf
Spanning trees
memorisation
cyclomatic number
bounded degree graphs
cut
linear program.