Peeters, M.J.P.

English

Research Papers in Economics

The maximum edge biclique problem is
NP-complete
Ren´ e Peeters
Department of Econometrics and Operations Research
Tilburg University
The Netherlands
April 5, 2005
File No. DA5734
Running head: Maximum edge biclique problem
Correspondence:
Ren´ e Peeters
Department of Econometrics and Operations Research
Tilburg University
P.O. Box 90153
5000 LE Tilburg
The Netherlands
E-mail: m.j.p.peeters@uvt.nl
Phone: +31-13-4663425
Fax: +31-13-4663280
1Abstract
We prove that the maximum edge biclique problem in bipartite
graphs is NP-complete.
Key words: Complexity, bipartite graphs, biclique
21 Introduction
Let G = (V,E) be a graph with vertex set V and edge set E. A pair of
two disjoint subsets A and B of V is called a biclique if {a,b} ∈ E for all
a ∈ A and b ∈ B. Thus the edges {a,b} form a complete bipartite subgraph
of G (which is not necessarily an induced subgraph if G is not bipartite). A
biclique {A,B} clearly has |A|+|B| vertices and |A|∗|B| edges. In this note
we restrict ourselves to case when G is bipartite. The two colour classes of
G will be denoted by V1 and V2, so V = V1 ∪ V2.
Already in the book of Garey and Johnson [2, GT24] the complexity of
deciding whether or not a bipartite graph contains a biclique of a certain
size is discussed. If the requirement is that |A| = |B| = K for some in-
teger K (this is called the balanced complete bipartite subgraph problem or
balanced biclique problem), then the problem is NP-complete. If however the
requirement is that |A| + |B| ≥ K (the maximum vertex biclique problem),
the problem can be solved in polynomial time via the matching algorithm.
The complexity of the maximum vertex biclique problem for general graphs
depends on the precise deﬁnition of a biclique in this case. With the above
deﬁnition the problem is solvable in polynomial time since there is a one to
one correspondence between bicliques in the bipartite double1 of the graph
and bicliques in the graph itself (see also [4]). If one deﬁnes a biclique as
an induced complete bipartite subgraph (so A and B are independent sets
1The bipartite double of a graph with adjacency matrix A is the bipartite graph with
adjacency matrix

 O A
A O

.
3in G), then the maximum vertex biclique problem for general graphs is NP-
complete (see [8]). A natural third variant is the so-called maximum edge
biclique problem (MBP) where the requirement is that |A| ∗ |B| ≥ K. Up to
now the complexity of this problem was still undecided.
In various papers the complexity of MBP is mentioned and guessed to be
NP-complete ([1, 4, 3, 7]. In [1] some applications of MBP are discussed and
it is shown that the weighted version of MBP is NP-complete. Furthermore
the authors show that four variants of MBP are NP-complete. Using diﬀerent
techniques Hochbaum [4], Haemers [3] and Pasechnik [7] derive upper bounds
for the maximum number of edges in a biclique. Hochbaum [4] presents a
2-approximation algorithm for the minimum number of edges needed to be re-
moved so that the remainder is a biclique based on an LP-relaxation. Inspired
by the work of Lov´ asz on the Shannon capacity of a graph ([6]), Haemers [3]
and Pasechnik [7] derive similar inequalities for the maximum biclique prob-
lem. Pasechnik uses semideﬁnite programming techniques wereas Haemers
uses eigenvalue techniques.
In the next section we prove that indeed MBP is NP-complete. The
reduction used is inspired by the reduction that is used to prove the NP-
completeness of the balanced biclique problem (see [5]). As a consequence
MBP is also NP-complete for general graphs.
2 The reduction
We deﬁne MBP as follows:
4Maximum edge biclique problem (MBP): Given a bipartite graph
G = (V1 ∪V2,E) and a positive integer K, does G contain a biclique with at
least K edges?
Theorem 1 MBP is NP-complete.
Proof: We shall reduce CLIQUE to MBP. This reduction is a modiﬁcation
of the reduction from CLIQUE to BALANCED COMPLETE BIPARTITE
SUBGRAPH referred to in [2, GT24] and published in [5].
Let G = (V,E) and K provide an instance of CLIQUE. Without loss of
generality we may assume that K = 1
2|V |.
Now construct an instance G0 = (V1 ∪V2,E0), K0 of MBP as follows: Let
V1 = V,
V2 = E ∪ W,
where W is a set of 1
2K2 − K new elements.
E
0 = {{v,e} : v ∈ V ;e ∈ E;v 6∈ e} ∪ {{v,w} : v ∈ V ;w ∈ W}
K
0 = K
3 −
3
2
K
2
This construction can clearly be performed in polynomial time. Suppose
G has a clique C of size K. Take A := V − C and B := W ∪ {{c,d} : c,d ∈
C;c 6= d}. Then {A,B} is a biclique in G0 with |A| ∗ |B| = K ∗ (1
2K2 − K +
1
2K(K − 1)) = K3 − 3
2K2 edges. So if G has a clique of size K then G0 has
a biclique with K0 edges.
5Now suppose G has no clique of size K. Let {A,B} be a biclique of
G0 with A ⊆ V1 and B ⊆ V2. We shall ﬁnish the proof by showing that
|A| ∗ |B| < K0 in this case. Without loss of generality W ⊆ B. Let a := |A|
and b := |B| − |W|.
The b elements of B ∩ E correspond with edges in G whose endpoints
are not in A. There are 2K − a vertices of G that are not in A so b ≤
1
2(2K − a)(2K − a − 1), with equality if and only if V − A is a clique with
edge set B ∩ E.
We consider two cases:
1. Suppose a > K, so |V − A| = K − c with c := a − K (So 0 < c ≤ K).
Then b ≤ 1
2(K − c)(K − c − 1), so
|A| ∗ |B| ≤ [K + c] ∗ [
1
2
K
2 − K +
1
2
(K − c)(K − c − 1)]
This reduces to
|A| ∗ |B| − (K
3 −
3
2
K
2) ≤
1
2
c(c
2 − (K − 1)c − 2K)
Now c2 − (K − 1)c − 2K is negative for 0 ≤ c ≤ K, so |A| ∗ |B| < K0
for 0 < c ≤ K.
2. Suppose a ≤ K, so |V − A| = K + c with c := K − a (So 0 ≤ c ≤ K).
Since G has no cliques with K vertices, the number of edges in the
subgraph of G induced by V − A, and consequently b, is strictly less
than 1
2(K + c)(K + c − 1) − c. This leads to
|A| ∗ |B| < [K − c] ∗ [
1
2
K
2 − K +
1
2
(K + c)(K + c − 1) − c]
6which reduces to
|A| ∗ |B| − (K
3 −
3
2
K
2) <
1
2
c
2(−c + 3 − K)
Since we may assume that K ≥ 4, the right hand side is negative for
1 ≤ c ≤ K and zero for c = 0. So |A| ∗ |B| < K0.
2
References
[1] M. Dawande, P. Keskinocak and S. Tayur, On the biclique problem in
bipartite graphs, GSIA Working Paper 1996-04, Carnegie Mellon Uni-
versity, Pittsburgh, PA 15213, USA, 1996.
[2] M.R. Garey and D.S. Johnson, Computers and Intractability, A guide
to the theory of NP-completeness (Freeman, San Francisco, 1979).
[3] W.H. Haemers, Bicliques and eigenvalues, J. Comb. Theory B 82 (2001)
56-66.
[4] D.S. Hochbaum, Approximating clique and biclique problems, J. Algo-
rithms 29(1) (1998) 174-200.
[5] D.S. Johnson, The NP-completeness column: an ongoing guide, J. Al-
gorithms 8(3) (1987) 438-448.
[6] L. Lov´ asz, On the Shannon capacity of a graph, IEEE Trans. Inform.
Theory 25 (1979) 1-7.
7[7] D.V. Pasechnik, Bipartite sandwiches: bounding the size of a maximum
biclique, preprint, 1999.
[8] M. Yannakakis, Node- and edge-deletion NP-complete problems, Proc.
10th Ann. ACM Symp. on Theory of Computing, Association for Com-
puting Machinery, New York, (1978) 253-264.
8

The maximum edge biclique problem is NP-complete.

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