Partitioning into Colorful Components by
Minimum Edge Deletions
Sharon Bruckner
1?
, Falk H¨uffner
2??
, Christian Komusiewicz
2
, Rolf Niedermeier
2
,
Sven Thiel
3
, and Johannes Uhlmann
2? ? ?
1
Institut f¨ur Mathematik, Freie Universit¨at Berlin, sharonb@mi.fu-berlin.de
2
Institut f¨ur Softwaretechnik und Theoretische Informatik, TU Berlin,
{falk.hueffner,christian.komusiewicz,rolf.niedermeier}@tu-berlin.de
johannes.uhlmann@campus.tu-berlin.de
3
Institut f¨ur Informatik, Friedrich-Schiller-Universit¨at Jena,
sven.thiel@uni-jena.de
Abstract.
The NP-hard Colorful Components problem is, given a
vertex-colored graph, to delete a minimum number of edges such that
no connected component contains two vertices of the same color. It has
applications in multiple sequence alignment and in multiple network align-
ment where the colors correspond to species. We initiate a systematic
complexity-theoretic study of Colorful Components by presenting
NP-hardness as well as fixed-parameter tractability results for differ-
ent variants of Colorful Components. We also perform experiments
with our algorithms and additionally develop an efficient and very accu-
rate heuristic algorithm clearly outperforming a previous min-cut-based
heuristic on multiple sequence alignment data.
1 Introduction
We study a maximum parsimony approach to the discovery of heterogeneous
components in vertex-colored graphs:
Colorful Components
Instance:
An undirected graph
G
= (
V, E
) and a coloring of the ver-
tices χ : V → {1, . . . , c}.
Task:
Find a minimum subset of edges
E
0
⊆ E
such that in
G
0
=
(
V, E \ E
0
), all connected components are colorful, that is, they do not
contain two vertices of the same color.
Such an edge set
E
0
is called a solution, and we denote its size by
k
. Color-
ful Components is an edge modification problem originating from biological
applications in sequence and network alignment as described next.
The first application of Colorful Components stems from Multiple Se-
quence Alignment. This is the process of aligning at least three protein, DNA, or
?
Supported by project NANOPOLY (PITN-GA-2009–238700).
??
Supported by DFG project PABI (NI 369/7-2).
? ? ?
Supported by DFG project PABI (NI 369/7-2).
2 Proc. CPM 2012, Vol. 7354 of LNCS
RNA sequences such that positions believed to be homologous, that is, resulting
from inheritance from a common ancestor, are written in a common column.
This serves to illustrate the similarity or dissimilarity between the sequences and
makes it possible to investigate their evolutionary relationship. Corel et al.
[6]
present an algorithm for this problem where a central step is to find connected
subgraphs in graphs whose vertices are positions of the sequences, edges indicate
that a pair of positions should be aligned, and the colors one-to-one correspond
to sequences. These subgraphs correspond to partial alignment columns and
thus may contain at most one vertex from each input sequence. This yields the
Colorful Components problem. The solution of Colorful Components
is then used by the DIALIGN software to compute a multiple alignment. Corel
et al.
[6]
solve Colorful Components using a greedy algorithm, subsequently
called “min-cut heuristic”: Find two vertices of the same color in some connected
component, find a minimum edge cut between them, and remove it; repeat this
until all connected components are colorful.
A second biological motivation for Colorful Components arises in Network
Alignment for multiple protein–protein interaction (PPI) networks. We propose a
method for network alignment that is based on solving Colorful Components.
Given networks
G
i
= (
V
i
, E
i
) and a similarity relation
S
between the proteins
of different species, first create a network whose vertex set is
S
V
i
and in which
vertex
v ∈ V
i
receives color
i
. Then, add an edge
{u, v}
if
uSv
. The detected
colorful components are then sets of matched proteins. Every protein appears in
exactly one component, and every component has at most one protein from each
species, which is a very strict model. The results can then be viewed as functional
orthologs [
14
], or they can form the basis for further analysis. Deni´elou et al.
[7]
suggest a three-step framework for network alignment where the first step is to
aggregate the proteins from the different species into subsets, and Colorful
Components offers a way of performing this task that results in consistent,
disjoint aggregated groups.
Related combinatorial problems. Colorful Components can be seen as the
problem of destroying by edge deletions all bad paths, that is, simple paths between
two vertices of the same color. Thus, it is a special case of the well-known NP-hard
Multicut problem, which has as input an undirected graph and a set of vertex
pairs and asks for a minimum number of edges to delete to disconnect each given
vertex pair. Multicut is fixed-parameter tractable with respect to the number
k
of edge deletions, with a running time of 2
O(k
3
)
· |V |
O(1)
[4, 13].
Colorful Components is also a special case of Multi-Multiway Cut [
1
].
This problem asks to disconnect by edge deletions all paths between vertices from
the same vertex set of some given vertex sets. Thus, Colorful Components is
the special case where the vertex sets form a partition. Finally, there is related
work on “clustering with diversity” [
11
] which extends a traditional clustering
problem by asking that in each resulting cluster all points of the underlying
colored metric space must have different colors.
Proc. CPM 2012, Vol. 7354 of LNCS 3
Contributions. On the theoretical side, we present a first systematic study on
the computational complexity of Colorful Components, exhibiting both
tractable and intractable cases. First, we observe that Colorful Components
is NP-hard even in trees. Then, we present a complexity dichotomy concerning
the number
c
of colors showing that Colorful Components is polynomial-
time solvable for two or less colors and NP-hard otherwise. For three or more
colors, we also obtain super-polynomial running time lower bounds (based on
the Exponential Time Hypothesis) even in the case that the input graph has
bounded degree. On the positive side, we present fixed-parameter algorithms with
running time 2
c
·|V |
O(1)
for Colorful Components in trees and with running
time
O
((
c −
1)
k
· |E|
) in general graphs. In experimental work we demonstrate
that, somewhat surprisingly, we can get better results by solving the more general
Weighted Multi-Multiway Cut problem, since this allows us to merge
vertices. We take advantage of this in data reduction rules, a simplified branching,
and a new heuristic. With the branching algorithms, we can solve to optimality
more than half of the instances generated from the BAliBASE 3.0 benchmark [
15
]
each time within five minutes on a standard PC, with up to 5 000 vertices and
13 000 edges. Our heuristic has an average error of 0.6 %, a large improvement
over the 29.2 % of the previously suggested min-cut heuristic [
6
]. We also show
the strength of the developed data reduction rules.
Preliminaries. We consider only undirected and simple graphs
G
= (
V, E
)
where
n
:=
|V |
and
m
:=
|E|
. We assume that
n
=
O
(
m
) since isolated vertices
can be removed from the input in linear time. A bad path is a simple (that is,
cycle-free) path between two vertices of the same color. The length of a path is
the number of its edges. An edge cut is a set of edges whose deletion increases
the number of connected components. For a nonnegative number
t
, a graph is
t-edge connected if it does not have an edge cut of size less than t.
The Exponential Time Hypothesis (ETH) states that, for all
x ≥
3,
x
-SAT,
which asks whether a boolean input formula in conjunctive normal form with
n
variables and
m
clauses and at most
x
variables per clause is satisfiable, cannot
be solved within a running time of 2
o(n)
or 2
o(m)
; see Lokshtanov et al.
[12]
for a recent survey. A problem is fixed-parameter tractable with respect to a
parameter
k
if it can be solved in
f
(
k
)
· n
O(1)
time for an arbitrary (typically
exponential) function f in k.
2 Computational Hardness
In this section, we present hardness results for two restricted variants of Color-
ful Components.
First, we consider the special case where the input graph is a tree. For
obtaining our hardness result, we exploit the connection between Colorful
Components and Multicut. Note that Multicut is NP-hard and MaxSNP-
hard even if the input is a star, that is, a tree consisting of a central vertex with
attached degree-1 vertices [
8
]. Multicut in stars can be reduced to Colorful
4 Proc. CPM 2012, Vol. 7354 of LNCS
Components as follows: for every pair
{s, t}
to be disconnected, create degree-1
vertices
s
0
and
t
0
attached to
s
and
t
, respectively, and color
s
0
and
t
0
with the
same unique color. Each original vertex gets a further unique color. Since this
reduction produces trees whose diameter is four, we arrive at the following.
Proposition 1.
Colorful Components is NP-hard even in trees with diam-
eter four.
In stars, however, Colorful Components turns out to be polynomial-time
solvable: If the central vertex
v
has two neighbors with the same color, one can
delete the edge between
v
and one of the two identically colored degree-one
vertices. If
v
has no two neighbors of the same color, then every connected
component is colorful.
Second, we study the computational complexity of Colorful Components
if the number of colors is fixed. This is of interest since the number of colors may
be small in practical cases.
Theorem 1.
Colorful Components with three colors in graphs with max-
imum degree six is NP-hard; it cannot be solved in 2
o(k)
· n
O(1)
, 2
o(n)
· n
O(1)
,
or 2
o(m)
· n
O(1)
time unless the ETH is false.
Proof.
We present a polynomial-time many-to-one reduction from the NP-hard
3-SAT problem which has as input a Boolean formula
φ
in 3-CNF.
4
For simplicity,
we assume that every clause contains exactly three literals.
The basic idea of the reduction is as follows. For each variable
x
i
of a given
3-CNF formula
φ
, we construct a variable cycle of length 4
m
i
, where
m
i
denotes
the number of clauses that contain
x
i
. These cycles are colored alternatingly
with two colors
c
e
and
c
o
such that deleting every second edge yields a minimum-
cardinality edge deletion set for obtaining colorful components for this cycle. The
corresponding two possibilities are used to represent the two choices for the value
of
x
i
. Then, for each clause
C
j
of
φ
containing the variables
x
p
,
x
q
, and
x
r
, we
connect the three corresponding variable cycles by a clause gadget. This gadget
has the property that if the solutions for the variable gadgets correspond to an
assignment that satisfies
C
j
, then one needs only four edge deletions for the
clause gadget. Conversely, if four edge deletions are sufficient, then the assignment
corresponding to the deletions in the variable cycle satisfies
C
j
. Let
m
be the
number of clauses in
φ
and observe that, since
φ
is a 3-CNF formula, the overall
number of vertices in the variable cycles is 12
m
. Our construction guarantees
that there is a satisfying assignment for
φ
if and only if the constructed graph can
be transformed into one with colorful components by exactly 6
m
+ 4
m
= 10
m
edge deletions, where 6
m
edge deletions are used for the variable cycles and
4m modifications are used for the clause gadgets. The details follow.
Given a 3-CNF formula
φ
consisting of the clauses
C
0
, . . . , C
m−1
over the
variables
{x
0
, . . . , x
n−1
}
, construct a Colorful Components-instance (
G
=
4
A similar reduction type was previously used to show analogous results for Transi-
tivity Editing [
16
] and Cluster Editing [
10
], which, in contrast, are defined on
uncolored graphs.
Proc. CPM 2012, Vol. 7354 of LNCS 5
r
4π(r,j)
r
4π(r,j)+1
p
4π(p,j)
p
4π(p,j)+1
q
4π(q,j)+1
q
4π(q,j)+2
a
j
Fig. 1.
The clause gadget for clause
C
j
= (
x
p
∨¯x
q
∨x
r
). White vertices have color
c
e
, gray
vertices have color
c
o
, and black vertices have color
c
g
. The vertex
a
j
is the reserved
vertex for
C
j
, the other vertices lie on the variable cycles for
x
p
,
x
q
, and
x
r
, respectively.
(
V, E
)
, k
) as follows. For each variable
x
i
, 0
≤ i < n
,
G
contains a variable cycle
consisting of the vertices
V
v
i
:=
{i
0
, . . . , i
4m
i
−1
}
and the edges
E
v
i
:=
{{i
k
, i
k+1
} |
0
≤ k <
4
m
i
}
(for ease of presentation let
i
4m
i
=
i
0
). An edge
{i
x
, i
x+1
}
is even
if
x
is even, and odd otherwise. A vertex
i
x
receives color
c
e
if
x
is even; otherwise,
it receives color
c
o
. So far, the constructed graph consists of a disjoint union of
cycles and has 12m vertices and edges.
Next, add a clause gadget to
G
for each clause of
φ
. In the construction of the
clause gadgets, we need for each clause
C
j
in the variable cycles of
C
j
’s variables
a fixed set of vertices that are “reserved” for
C
j
. To this end, suppose that for
each variable
x
i
an arbitrary but fixed ordering of the clauses containing
x
i
is
given, and let
π
(
i, j
)
∈ {
0
, . . . ,
4
m
i
−
1
}
denote the position of a clause
C
j
that
contains
x
i
in this ordering. We now give the details of the construction of the
clause gadgets. Let
C
j
be a clause containing the variables
x
p
,
x
q
, and
x
r
(either
negated or nonnegated). We construct a clause gadget connecting the variable
cycles of
x
p
,
x
q
, and
x
r
. First, let
a
j
be a new vertex that appears only in the
clause gadget for clause
C
j
and color
a
j
with a third color
c
g
. Let
E
g
j
denote
the edge set of the clause gadget and let
E
g
j
contain for each
i ∈ {p, q, r}
the
edges
{a
j
, i
4π(i,j)
}
and
{a
j
, i
4π(i,j)+1
}
if
x
i
occurs nonnegated in
C
j
or the edges
{a
j
, i
4π( i,j)+1
}
and
{a
j
, i
4π(i,j)+2
}
, otherwise. See Fig. 1 for an illustration. The
construction of
G
= (
V, E
) is completed by setting
V
:=
S
n−1
i=0
V
v
i
∪
S
m−1
j=0
{a
j
}
and E :=
S
n−1
i=0
E
v
i
∪
S
m−1
j=0
E
g
j
.
We show the correctness of the reduction by showing the following claim.
φ
is satisfiable
⇔ G
can be transformed into a graph with colorful
components by deleting at most k := 10m edges.
“
⇒
”: Given a satisfying assignment
β
for
φ
, we can transform
G
into a graph
with colorful connected components as follows. For each variable
x
i
delete the
odd edges of the variable cycle of
x
i
if
β
(
x
i
) =
true
and the even edges otherwise.
After these deletions, there are no bad paths that contain only vertices from the
variable cycles. Then, proceed as follows for each clause
C
j
. Assume without loss
of generality that
C
j
contains the variables
x
p
,
x
q
, and
x
r
, and that the literal
corresponding to
x
p
is true. Then, delete the four edges that are incident with
a
j