Paper V
Exact algorithms for treewidth and minimum
fill-in
∗†
Fedor V. Fomin
‡
Department of Informatics
University of Bergen
5020 Bergen, Norway
fomin@ii.uib.no
Dieter Kratsch
LITA, Universit´e de Metz
57045 Metz Cedex 01, France
kratsch@univ-metz.fr
Ioan Todinca
LIFO, Universit´e d’Orl´eans
45067 Orl´eans Cedex 2, France
Ioan.Todinca@lifo.univ-orleans.fr
Yngve Villanger
Department of Informatics
University of Bergen
5020 Bergen, Norway
yngvev@ii.uib.no
Abstract
We show that there are O(1.8899
n
) time algorithms to compute the
treewidth and the minimum fill-in of each graph G on n vertices. Our
result is based on a combinatorial proof that e ach graph on n vertices
has at most n ·1.7087
n
minimal separators and that all potential maximal
cliques can be listed in O(1.8899
n
) time. For the class of AT-free graphs
we obtain O(1.4142
n
) time algorithms to compute treewidth and minimum
fill-in.
Keywords: Exact exponential algorithm, treewidth, fill-in, minimal separators,
potential maximal clique, minimal triangulation
∗
A preliminary version [26] of this paper has been presented at the 31st International Col-
loquium on Automata, Languages and Programming, Turku, Finland, July 2004.
†
Supp orted by The Aurora Programme Collaboration Research Project between Norway
and France.
‡
Fedor Fomin acknowledges support of Norges forskningsr˚ad, project 160778/V30.
2 Exact algorithms for treewidth and minimum fill-in
1 Introduction
Exact exponential algorithms. The interest in exact (fast) exponential al-
gorithms dates back to Held and Karp’s paper [29] on the travelling salesman
problem in the early sixties. Mention just a few examples: time O
∗
(1.4422
n
) algo-
rithm for Knapsack (Horowitz and Sahni [30]); time O
∗
(1.2600
n
) and O
∗
(1.2109
n
)
algorithms for Independent Set (Tarjan and Trojanowski [44], Robson [40]); 3-
Coloring in time O
∗
(1.4422
n
) (Lawler [34]); 3-SAT in time O
∗
(1.6181
n
) (Monien
and Speckenmeyer [35]).
In this paper we use a modified big-Oh notation that suppresses all other
(polynomially bounded) terms. For functions f and g we write f(n) = O
∗
(g(n))
if f(n) = O(g(n)poly(n)), where poly(n) is a polynomial. This modification may
be justified by the exponential growth of f (n).
Nowadays, it is common believe that NP-hard problems can not be solved
in polynomial time. For a number of NP-hard problems, we even have strong
evidence that they cannot be solved in sub-exponential time. In order to obtain
exact solutions to these problems, the only hope is to design exact algorithms
with good exponential running times. How good can these exponential running
times be? Can we reach 2
n
2
for instances of size n? Can we reach 10
n
? Or even
2
n
? Or can we reach c
n
for some constant c that is very close to 1? The last
years have seen an emerging interest in attacking these questions for concrete
combinatorial problems: There is an O
∗
(2.4150
n
) time algorithm for Coloring
(Byskov [15]); an O
∗
(1.3289
n
) time algorithm for 3-Coloring (Beigel and Eppstein
[3]); an O
∗
(1.7325
n
) time algorithm for Max-Cut (Williams [47]); an algorithms
for 3-SAT in time O
∗
(1.4726
n
) (Brueggemann and Kern [14]); an O
∗
(1.5129
n
)
time algorithm for Dominating Set (Fomin et al. [25]).
There can be several explanations why now the algorithmic community wit-
nesses the revival of the interest in fast exponential algorithms:
• The design and analysis of exact algorithms leads to a better understand-
ing of NP-hard problems and initiates interesting new combinatorial and
algorithmic challenges.
• For certain applications it is important to find exact solutions. With the
increased speed of modern computers, fast algorithms, even though they
have e xponential running times in the worst case, may actually lead to
practical algorithms for certain NP-hard problems, at least for moderate
instance sizes.
• Approximation, randomized algorithms and different heuristics are not al-
ways satisfactory. Each of these approaches has weak points like necessity
of exact solutions, difficulty of approximation, limited power of the method
itself and many others.
Exact algorithms for treewidth and minimum fill-in 3
• A reduction of the base of the exponential running time, say from O(2
n
) to
O(1.8
n
), increases the size of the instances solvable within a given amount
of time by a constant multiplicative factor. However running a given expo-
nential algorithm on a faster computer can enlarge the mentioned size only
by a constant additive factor.
For overviews and introductions to the field see the recent surveys by Iwama
[31], Sch¨oning [42], and Woeginger [48, 49].
Treewidth and minimum fill-in. Treewidth is one of the most basic parame-
ters in Graph Algorithms [6] and it plays an important role in structural Graph
Theory. It serves as one of the main tools in Robertson and Seymour’s Graph
Minors project [39]. Treewidth also plays a crucial role in parameterized complex-
ity theory [21]. The minimum fill-in problem (also known as minimum chordal
graph completion) has important applications in sparse matrix computations and
computational biology.
The problems to compute the treewidth and minimum fill-in of a graph are
known to be NP-hard even when the input is restricted to complements of bi-
partite graphs (so called cobipartite graphs) [2, 50]. Despite of the importance
of treewidth almost nothing is known on how to cope with its intractability. For
a long time the best known approximation algorithm for treewidth had a fac-
tor log OP T [1, 11] (see also [7]). Recently, Feige et al. [23] obtained factor
√
log OP T approximation algorithm for treewidth. Furthermore it is an old open
question whether the treewidth can be approximated within a constant factor.
Treewidth is known to be fixed parameter tractable. Moreover, for any fixed
k, there is a linear time algorithm to compute the treewidth of graphs of treewidth
at most k (unfortunately there is a huge hidden constant in the running time) [5].
There is a number of algorithms that for a given graph G and integer k, either
report that the treewidth of G is at least k, or produce a tree decomp osition of
width at most c
1
·k in time c
2
k
·n
O(1)
, where c
1
, c
2
are some constants (see e.g.
[1]). Fixed parameter algorithms are known for the minimum fill-in problem as
well [17, 32].
We are not aware about any previous work on exact algorithms for the treewidth
or minimum fill-in problem. There are three relatively simple approaches result-
ing in time O
∗
(2
n
) algorithms:
• One can reformulate the problems as problems of finding special vertex elim-
ination orderings and then find an optimal permutation by using the dy-
namic programming based technique like in the article of Held & Karp [29]
for the travelling salesman problem;
• With some modifications, the algorithm of Arnborg et al. [2] for a given k
deciding in time O(n
k+1
) if the treewidth of a graph is at most k, can be
used to compute the treewidth (and similarly fill-in) in time O
∗
(2
n
);
