arXiv:1110.4765v1 [cs.DS] 21 Oct 2011
Finding small separators in linear time via treewidth reduction
∗
D´aniel Marx
†
Barry O’Sullivan
‡
Igor Razgon
§
Abstract
We present a method for reducing the treewidth of a graph while preserving all of its minimal s−t
separators up to a certain fixed size k. This technique allows us to solve s−t Cut and Multicut problems
with various additional restrictions (e.g., the vertices being removed from the graph form an independent
set or induce a connected graph) in linear time for every fixed number k of removed vertices.
Our results have applications for problems that are not directly defined by separators, but the known
solution methods depend on some variant of separation. For example, we can solve similarly restricted
generalizations of Bipartization (delete at most k vertices from G to make it bipartite) in almost linear
time for every fixed number k of removed vertices. These results answer a number of open questions in
the area of parameterized complexity. Furthermore, our technique turns out to be relevant for (H,C,K)-
and (H,C, ≤K)-coloring problems as well, which are cardinality constrained variants of the classical
H-coloring problem. We make progress in the classification of the parameterized complexity of these
problems by identifying new cases that can be solved in almost linear time for every fixed cardinality
bound.
1 Introduction
Finding cuts and separators is a classical topic in combinatorial optimization and in recent years there has
been an increase of interest in the fixed-parameter tractability of such problems [7, 11, 24, 28, 30, 32, 50,
53, 54, 65]. Recall that a problem is fixed-parameter tractable (or FPT) with respect to a parameter k if
instances of size n can be solved in time f(k) ·n
O(1)
for some computable function f(k) depending only on
the parameter k of the instance [20,25,56]. In typical parameterized separation problems, the parameter k is
the size of the separator we are looking for, thus fixed-parameter tractability with respect to this parameter
means that the combinatorial explosion is restricted to the size of the separator, but otherwise the running
time depends polynomially on the size of the graph.
The main message of our paper is the following: the small s −t separators live in a part of the graph
that has bounded treewidth. Therefore, if a separation problem is FPT in bounded treewidth graphs, then it
is FPT in general graphs as well. As there are general techniques for obtaining linear-time algorithms for
problems on bounded-treewidth graphs (e.g., dynamic programming and Courcelle’s Theorem), it follows
that a surprisingly large number of generalized separation problems can be shown to be linear-time FPT with
this approach (we say that a problem is linear-time FPT with parameter k if it can be solved in time f(k)·n for
some function f). For example, one of the consequences our general argument is a theorem stating that given
a graph G, two terminal vertices s,t, and a parameter k, we can compute in a FPT-time a graph G
∗
having the
treewidth bounded by a function of k while (roughly speaking) preserving all the inclusionwise minimal s−t
separators of size at most k. Therefore, combining this theorem with the well-known Courcelle’s Theorem,
∗
A subset of the results was presented at STACS 2010 [52].
†
Institut f¨ur Informatik, Humboldt-Universit¨at zu Berlin, dmarx@cs.bme.hu
‡
Cork Constraint Computation Centre, University College Cork, b.osullivan@cs.ucc.ie
§
Department of Computer Science, University of Leicester, ir45@mcs.le.ac.uk
1
we obtain a powerful tool for finding s −t separators obeying additional constraints expressible in monadic
second order logic.
Algorithms for separation problems are often based on interesting mathematical properties of the prob-
lem. For example, the classical s−t cut algorithm of Ford and Fulkerson is essentially based on the tight
connection between maximum flows and minimum cuts. However, algorithms based on nice mathematical
properties and connections are inherently fragile, and any slight generalization of the problem can break
these connection and make the problem NP-hard (unless the generalization involves very special conditions,
e.g., submodular functions). On the other hand, the main thesis of the paper is that the fixed-parameter
tractability of separation problems has a highly robust theory: the technique of treewidth reduction pre-
sented in the paper allows us to show the fixed-parameter tractability of several generalizations with very
little additional effort. As separation problems are crucial ingredients for solving other type of problems
(e.g., bipartization), this robustness propagates into other problem areas as well.
1.1 Results
We demonstrate the power of the methodology with the following results.
• We prove that the MINIMUM STABLE s −t CUT problem (Is there an independent set S of size at
most k whose removal separates s and t?) is fixed-parameter tractable and in fact can be solved in
linear time for every fixed k. Our techniques allow us to prove various generalizations of this result
very easily. First, instead of requiring that S is independent, we can require that it induces a graph that
belongs to a hereditary class G (hereditary means that if G ∈ G, then every induced subgraph of G is
in G as well); the problem remains linear-time solvable for every fixed k. Second, in the MULTICUT
problem a list of pairs of terminals are given (s
1
,t
1
), ..., (s
ℓ
,t
ℓ
) and S is a set of at most k vertices that
induces a graph from G and separates s
i
from t
i
for every i. We show that this problem can be solved
in linear time for every fixed k and ℓ (i.e., linear-time FPT parameterized by k and ℓ), which is a very
strong generalization of previous results [30,50,65]. Third, the results generalize to the MULTICUT-
UNCUT problem, where two sets T
1
, T
2
of pairs of terminals are given, and S has to separate every
pair of T
1
and should not separate any pair of T
2
.
• We show that CONNECTED s-t CUT (Is there a set of of at most k vertices that induces a connected
graph and whose removal separates terminals s and t?) is linear-time FPT. The significance of the
result is that at first sight this problem does not seem to be amenable to our techniques: connectivity
is not a hereditary property, and therefore the solution is not necessarily a minimal s-t separator.
However, with some problem-specific ideas related to connectivity, we can extend our approach to
handle such a requirement. This suggests that our technique might be applicable to a much wider
range of cut problems than the hereditary problems described above.
• As a demonstration, we show that the EDGE-INDUCED VERTEX CUT (Is there a set of at most k
edges such that removal of their endpoints separates two given terminals s and t?) is linear-time FPT,
answering an open problem posed in 2007 by Samer [14]. The motivation behind this problem is
described in [61]. While the reader might not be particularly interested in this exotic variant of s −t
cut, we believe that it nicely demonstrates the message of the paper. Slightly changing the definition of
a well-understood cut problem usually makes the problem NP-hard and determining the parameterized
complexity of such variants directly is by no means obvious and seems to require problem-specific
ideas in each case. On the other hand, using our techniques, the fixed-parameter tractability of many
such problems can be shown in a uniform way with very little effort. Let us mention (without proofs)
three more variants that can be treated in a similar way: (1) separate s and t by deleting at most k
edges and at most k vertices, (2) in a 2-colored graph, separate s and t by deleting at most k black and
2
at most k white vertices, (3) in a k-colored graph, separate s and t by deleting one vertex from each
color class.
• The BIPARTIZATION problem asks if a given graph G can be made bipartite by deleting at most k
vertices. Reed et al. [60] showed that the problem is FPT and Kawarabayashi and Reed [44] proved
that the problem is almost linear-time FPT, i.e., can be solved in time f(k) ·n ·
α
(n,n), where
α
is the inverse Ackermann function. We prove that the variant STABLE BIPARTIZATION (Is there
an independent set of size at most k whose removal makes the graph bipartite?) is almost linear-
time FPT, answering an open question posed by Fernau [14]. Furthermore, we prove that EXACT
STABLE BIPARTIZATION (Is there an independent set of size exactly k whose removal makes the
graph bipartite?) is also almost linear-time FPT, answering an open question posed in 2001 by D´ıaz
et al. [16]. This latter result might be somewhat surprising, as finding an independent set of size
exactly k is W[1]-hard, and hence unlikely to be FPT. As in the case of s −t cuts, we introduce the
generalization G-BIPARTIZATION, where the at most k vertices of the solution have to induce a graph
belonging to the class G; we show that this problem is almost linear-time FPT whenever G is decidable
and hereditary. We also study the analogous edge-deletion version G-EDGEBIPARTIZATION and show
it to be FPT if G is decidable and closed under taking subgraphs.
• The BIPARTITE CONTRACTION problem asks if a given graph G can be made bipartite by the con-
traction of at most k edges. Very recently, Heggernes et al. [38] showed that this problem is FPT
by presenting a nontrivial problem-specific algorithm. We observe that a simple corollary of our
results on G-EDGEBIPARTIZATION immediately shows that BIPARTITE CONTRACTION is almost-
linear time FPT.
• Finally, we analyze the constrained bipartization problems in a more general environment of (H,C,≤K)-
coloring [16], where the parameter is the maximum number of vertices mapped to C in the ho-
momorphism and prove that the problem is almost linear-time FPT if the graph H \C consists of
two adjacent vertices without loops. There have been significant efforts in the literature to fully
characterize the complexity (i.e., to prove dichotomy theorems) of various versions of H-coloring
[8, 22, 23, 26, 33–35,39, 40]. The version studied here was introduced in [16–19], where it was ob-
served that this problem family contains several classical concrete problems as special case, including
some significant open problems. Thus obtaining a full dichotomy would require breakthroughs in
parameterized complexity. Our result removes one of the roadblocks towards this goal.
As the results listed above demonstrate, our method leads to the solution of several independent prob-
lems; it seems that the same combinatorial difficulty lies at the heart of these problems. Our technique
manages to overcome this difficulty and it is expected to be of use for further problems of similar flavor.
We would like to emphasize that while designing FPT-time algorithms for bounded-treewidth graphs and in
particular the use of Courcelle’s Theorem is a fairly standard method, we use this technique for problems
where there is no bound on the treewidth in the input.
Various versions of (multiterminal) cut problems [11, 28, 32, 50] play a mysterious, not yet fully un-
derstood role in the fixed-parameter tractability of certain problems. Proving that BIPARTIZATION [60],
DIRECTED FEEDBACK VERTEX SET [12], and ALMOST 2-SAT [58] are FPT answered longstanding open
questions, and in each case the algorithm relies on a nonobvious use of separators. Furthermore, EDGE
MULTICUT has been observed to be equivalent to FUZZY CLUSTER EDITING, a correlation clustering prob-
lem [1,6,15]. Thus aiming for a better understanding of separators in a parameterized setting seems to be a
fruitful direction of research. The results of this paper extend our understanding of separators by showing
that various additional constraints can be easily accommodated. It is important to point out that our algorithm
is very different from previous parameterized algorithms for separation problems [7,11,28,30,32,50, 54].
3
Those algorithms in the literature exploited certain nice properties of separators, and hence it seems very
difficult to generalize them for the problems we consider here. On the other hand, our approach is very
robust and, as demonstrated by our examples, it is able to handle many variants.
2 Treewidth reduction
The main combinatorial result of the paper is presented in this section. We start by introducing the main
tools required to prove the result: the notions of treewidth and torso.
2.1 Treewidth, brambles, and monadic second order logic
A tree decomposition of a graph G(V,E) is a pair (T,B) in which T(I, F) is a tree and B = {B
i
| i ∈ I} is a
family of subsets of V(G) such that
1.
S
i∈I
B
i
= V ;
2. for each edge e = (u,v) ∈ E, there exists an i ∈ I such that both u and v belong to B
i
; and
3. for every v ∈V , the set of nodes {i ∈ I |v ∈B
i
} forms a connected subtree of T.
The width of the tree decomposition is the maximum size of a bag in B minus 1. The treewidth of a graph G,
denoted by tw(G), is the minimum width over all possible tree decompositions of G. For more background
on the combinatorial and algorithmic consequences, the reader is referred to e.g., [5,29]. A useful fact that
we will use later on is that for every clique K of G, there is a bag B
i
with K ⊆ B
i
.
Treewidth has a dual characterization in terms of brambles [59,62]. A bramble in a graph G is a family
of connected subgraphs of G such that any two of these subgraphs either have a nonempty intersection or
are joined by an edge. The order of a bramble is the least number of vertices required to cover all subgraphs
in the bramble. The bramble number bn(G) of a graph G is the largest order of a bramble of G. Seymour
and Thomas [62] proved that bramble number tightly characterizes treewidth:
Theorem 2.1 (Seymour and Thomas [62]). For every graph G, bn(G) = tw(G) + 1.
Typically, the definition of treewidth is useful when we are trying to prove upper bounds, and brambles
are useful when we are trying to prove lower bounds. Interestingly, in the current paper we use brambles to
prove upper bounds on the treewidth. The reason for this is that we are relating the treewidth of different
graphs appearing in our construction and want to show that if the resulting graph has large treewidth, then
the earlier graphs have large treewidth as well.
The algorithmic importance of treewidth comes from the fact that a large number of NP-hard problems
can be solved in linear time if we have a bound on the treewidth of the input graph. Most of these algo-
rithms use a bottom-up dynamic programming approach, which generalizes dynamic programming on trees.
Courcelle’s Theorem [13] (see also [20, Section 6.5], [29]) gives a powerful way of quickly showing that a
problem is linear-time solvable on bounded treewidth graphs. Sentences in Monadic Second Order Logic of
Graphs (MSO) contain quantifiers, logical connectives (¬, ∨, and ∧), vertex variables, vertex set variables,
binary relations ∈ and =, and the atomic formula E(u,v) expressing that u and v are adjacent. If a graph
property can be described in this language, then this description can be turned into an algorithm:
Theorem 2.2 (Courcelle [13]). If a graph property can be described as a formula
φ
in the Monadic Second
Order Logic of Graphs, then it can be recognized in time f
φ
(tw(G)) ·(|E(G)|+ |V(G)|) if a given graph G
has this property.
4
Theorem 2.2 can be extended to labeled graphs, where the sentence contains additional atomic formulas
P
i
(x) meaning that vertex x has label i. We can implement labels on the edges by additional atomic formulas
E
i
(x,y) with the meaning that there is an edge of label i connecting vertices x and y. We informally call these
labels as “colors” and talk e.g., about colored graphs with black and white vertices and red and blue edges.
Most of the results in the paper go through for graphs colored with fixed constant number of colors: the
colors do not play a role in graph-theoretic properties (such as separation, treewidth, etc.) and requirements
on colors can be easily accommodated in MSO formulas.
Constructing an MSO formula for a given graph problem is usually a straightforward, but somewhat
lengthy exercise. Thus when we use Theorem 2.2, the construction of the formula is relegated to the ap-
pendix.
2.2 Separators
Two slightly different notions of separation will be used in the paper:
Definition 2.3. We say that a set S of vertices separates sets of vertices A and B if no component of G \S
contains vertices from both A\S and B\S. If s and t are two distinct vertices of G, then an s−t separator is
a set S of vertices disjoint from {s,t} such that s and t are in different components of G \S.
Thus if we say that S separates A and B, then we do not require that S is disjoint from A and B. In
particular, if S separates A and B, then A ∩B ⊆ S.
We say that an s −t separator S is minimum if there is no s−t separator S
′
with |S
′
| < |S|. We say that
an s−t separator S is (inclusionwise) minimal if there is no s −t separator S
′
with S
′
⊂ S.
If X is a set of vertices, we denote by N
G
(X) the set of those vertices in V(G) \X that are adjacent to at
least one vertex of X. (We omit the subscript G if it clear from the context.) We use the folklore result that
all the minimum cuts can be covered by a sequence of noncrossing minimum cuts: there exists a sequence
X
1
⊂···⊂X
q
such that every N(X
i
) is a minimum s−t separator and every vertex that appears in a minimum
separator is covered by one of the N(X
i
)’s. The existence of these sets can be proved by a simple application
of the uncrossing technique. We present a different proof here (related to [57]) that allows us to find such a
sequence in linear time. Strictly speaking, it is not possible to construct the sets X
1
, ..., X
q
in linear time,
as their total size could be quadratic. However, it is sufficient to produce the differences X
i+1
\X
i
, as they
contain all the information in the sequence.
Lemma 2.4. Let s,t be two vertices in graph G such that the minimum size of an s −t separator is ℓ > 0.
Then there is a collection X = {X
1
,... X
q
} of sets where {s} ⊆ X
i
⊆V(G) \({t}∪N({t})) (1 ≤ i ≤ q), such
that
1. X
1
⊂ X
2
⊂ ··· ⊂ X
q
,
2. |N(X
i
)| = ℓ for every 1 ≤ i ≤q, and
3. every s−t separator of size ℓ is fully contained in
S
q
i=1
N(X
i
).
Furthermore, there is an O(ℓ(|V(G)|+ |E(G)|)) time algorithm that produces the sets X
1
, X
1
\X
2
, .. . ,
X
q
\X
q−1
corresponding to such a collection X.
Proof. Let us construct a directed network D the following way. There are 2|V(G)| nodes in D: for every
v ∈V (G), there are two nodes v
1
, v
2
, there is an arc
−−→
v
1
v
2
with capacity 1, and there is an arc
−−→
v
2
v
1
with infinite
capacity. For every edge xy ∈ E(G), we add two arcs
−−→
x
2
y
1
and
−−→
y
2
x
1
with infinite capacity.
For Y ⊆V(D), let ∆
+
D
(Y) be the set of edges leaving Y in D. We say that F ⊆ E(D) is an s
2
→ t
1
cut,
if there is no path from s
2
to t
1
in D \F. It is clear that a set S ⊆V(G) \{s,t} is an s−t separator if and
only if the corresponding set {
−−→
v
1
v
2
| v ∈S} of |S| arcs of D form an s
2
→t
1
cut. Therefore, if we can find a
5
