www.elsevier.com/locate/dsw
Maximum dispersion problem in dense graphs
A. Czygrinow
Department of Mathematics, Arizona State University, Tempe, AZ 85287-1804, USA
Received 26 April 1999; received in revised form 9 July 2000
Abstract
In this note, we present a polynomial-time approximation scheme for a “dense case” of dispersion problem in weighted graphs, where weights on edges are integers from {1; : : : ; K} for some xed integer K. The algorithm is based on the algorithmic version the regularity lemma. c 2000 Published by Elsevier Science B.V. All rights reserved.
Keywords:Dispersion problem; Algorithm; Regularity lemma
1. Introduction
LetG= (V; E; !), where !:E → {1; : : : ; K} and forS⊂V letE(S) denote the set of edges which have
both endpoints in S. The dispersion problem can be
formulated as follows. Given a weighted graphG, and
s∈ {2;3; : : : ;|V|}nd a subsetS ofV such that|S|=s
and the sumP
{x; y} ∈E(S)!({x; y}) is maximized. The
problem received some attention recently. In [10], the authors present a polynomial-time algorithm which nds a solution whose value is at least14 of the optimal assuming the weights satisfy the triangle inequality.
Better bound of 12, under the same assumptions, was
obtained in [7]. In [5] (improving on previous results of [9]), authors obtain a O(n1=3) — approximation for
the densek-subgraph problem (unweighted version of
the dispersion problem) and they conjecture that for
some ¿0, it is NP-hard to obtain an approximation
ratio of O(n).
E-mail address:andrzej@math.la.asu.edu (A. Czygrinow).
In this paper, we present an approximation algo-rithm for a dense case of the dispersion problem, that is in the case when the optimal value is at leastcn2,
for some constantc. In addition, we assume that the
weights on edges are integers from {1; : : : ; K} for
some xedK. The method is based on the remarkable
lemma of Szemeredi [11] and its algorithmic version due to Alon et al. [1]. For other algorithmic applica-tions of the regularity lemma we refer mainly to [6], which contains applications to the partitioning prob-lems (like MAX–CUT) as well as to [3,4] where some other combinatorial problems are considered. Arora et al. [2] presented a randomized polynomial-time ap-proximation scheme (PTAS) for a dense case of the
dense k-subgraph problem. The algorithm is
deran-domized in [2]; however, the running time of the derandomized algorithm is O(nO(1=2)
). Using the
reg-ularity lemma we will get rid of the 1=2 in the
ex-ponent of n and give a O(n2:4) time PTAS for the
dispersion problem. It should be emphasized though, that our approach follows the general framework de-veloped in [3,6].
For a setS⊂V denoted by disp(S) =P
{x; y} ∈E(S) !({x; y}) and let SOPT be a set of cardinality s for
which the value of disp(·) is maximal. We will show
that one can nd a set S∗ whose value is “close” to
the value ofSOPT.
Theorem 1. For every0¡ ¡1and integerKthere is an algorithm which when given a weighted graph
(V; E; !)onnvertices and with weights in{1; : : : ; K}
nds inO(n2:4)time a setS∗⊂V such that
disp(S∗)¿disp(SOPT)−n2:
Clearly, the theorem produces a meaningful output only if disp(SOPT)¿cn2 for some constant c. In this
case, it can be easily transformed to a polynomial-time
approximation scheme (apply the theorem with′=
c). It is also worth mentioning that the algorithm
from Theorem 1 is polynomial inn=|V|, and is not
polynomial inandK.
The rest of the note is organized as follows. In Section 2, we formulate the regularity lemma of Sze-meredi, Section 3 contains the proof of Theorem 1.
2. The regularity lemma
First, let us introduce necessary denitions
and formulate the regularity lemma of Szemeredi
[11]. Let G = (V; E) be a graph on n vertices.
For nonempty subsets V1⊂V and V2⊂V with
V1∩V2 =∅ dene the density of the pair (V1; V2)
asd(V1; V2) =e(V1; V2)=|V1kV2|, wheree(V1; V2)
de-notes the number of edges with one endpoint in V1
and the other inV2.
called -regular if both of the following conditions
are satised.
1. kVi| − |Vjk61, for alli; j.
2. All but at mostt2of pairs (V
i; Vj) are-regular.
The rst condition simply states that all of the par-tition classes are “almost” equal, the second expresses the fact that at mostfraction of all pairs is-irregular. Thus, for all i= 1; : : : ; t, |Vi| ∈ {⌈n=t⌉;⌊n=t⌋}. Since
these oors and ceilings are clearly insignicant in our analysis we will simplify the exposition and assume that for everyi,|Vi|=n=t.
The regularity lemma of Szemeredi asserts that
every graph which is large enough admits an-regular
partition into a constant number of classes.
Theorem 4. For every ¿0; and every integer l there existN(; l)and L(; l)such that every graph with at least N(; l) vertices admits an -regular partitionV1∪ · · · ∪Vt withl6t6L(; l).
Recently, Alon et al. [1] showed how the above partition can be found algorithmically in O(|V|2:4)
time.
We need to adopt the above notation to weighted
graphs. Let G = (V; E; !) be a weighted graph on colored version of the regularity lemma follows easily from the original proof of Szemeredi (see [8]).
Theorem 5. For every ¿0 and integers l and K there existN=N(; l; K)andL=L(; l; K)such that for any collection ofKgraphs on a vertex setV with
|V|¿Nthere is a partitionV1∪V2∪ · · · ∪Vtwhich is
-regular in all of theKgraphs and for whichl6t6L.
One can nd such a partition in O(n2:4) time using
the Alon et al. [1] algorithm.
3. Algorithm
partition V1∪V2∪ · · · ∪Vt of V dene a function
2. Check all subsets S of size at most s which are
the unions of some of the Vi’s and take S∗ that
maximizes disp(·).
Since t depends only on and K in the second
step of the algorithm we check a constant number of sets. Consequently, the complexity of the algorithm is O(n2:4). Next, in Lemmas 6 and 7, we establish the
Proof. Assume rst that there is exactly one 16i6t
such thatS∩Vi 6=∅andSc∩Vi6=∅. Then for S=S\Vi toS with the contribution ofVi2 and show that there exist a set S′ with|S′|=|S|such that the number of
In both cases, we reduce the number of Vi’s that
in-tersect bothS andScby at least one. Then
disp(S) = X
Now, if there is a singleVithat intersects bothS′and
(S′)c, we subtractV
the proof to obtain S with|S|¡ sand
disp( S)¿disp(S)−2Kn2:
In our next lemma, we show that disp(·) is a “good” approximation to the dispersion function disp(·).
Lemma 7. For anyS⊂V;|disp(S)−disp(S)|64K2n2:
Proof. For everyi= 1; : : : ; t, we have|S∩Vi|6|Vi|=
n=t and so the number of edges that have both
end-points in S ∩Vi, for some i = 1; : : : ; t, is at most
tn2=t26n2. Thus, the total weight of the edges of the
above form is at mostKn2. Therefore,
disp(S)6 X
the weight on edges between the irregular pairs is at
mostKn2. Thus, combining (1) with above arguments
shows that
disp(S)6X!(Vi∩S; Vj∩S) + 3Kn2; (2)
where the summation is taken over {i; j} such that
(Vi; Vj) is -regular in all G1; : : : ; GK and both |Vi ∩
S|¿|Vi| as well as|Vj∩S|¿|Vj|. For such {i; j},
accordingly to the denition of an-regular pair,
dl(Vi∩S; Vj∩S)6dl(Vi; Vj) +: (3)
whereel(A; B) denotes the number of edges between
AandBthat have weightl. Therefore, combining (3)
and (4) yields
In a similar way, one can show that
disp(S)6disp(S) + 4K2n2:
We are now ready to prove Theorem 1.
Proof. LetSOPTdenote a set of sizeswith maximal
dispersion and letS be a set found by the algorithm.
Then by Lemma 7 algorithmic aspects of the regularity lemma, J. Algorithms 16 (1994) 80–109.
[2] S. Arora, D. Krager, M. Karpinski, Polynomial time approximation schemes for dense instances of NP-hard problems, Proceedings of the 27th Annual ACM Symposium on Theory of Computing, 1995, pp. 284 –293.
[4] R.A. Duke, H. Lefmann, V. Rodl, A fast algorithm for computing frequencies in a given graph, SIAM J. Comput. 24 (3) (1995) 598–620.
[5] U. Feige, G. Kortsarz, D. Peleg, The dense k-subgraph problem, manuscript, 1999.
[6] A. Frieze, R. Kannan, The regularity lemma and approximation schemes for dense problems, Proceedings of the 37th FOCS, 1996, pp. 12–20.
[7] R. Hassin, S. Rubinstein, A. Tamir, Approximation algorithms for maximum dispersion, Oper. Res. Lett. 21 (1997) 133–137. [8] J. Komlos, M. Simonovits, Szemeredi’s regularity lemma and its applications in graph theory, in: D. Miklos,
V.T. Sos, T. Sz˝onyi (Eds.), Combinatorics, Paul Erd˝os is Eighty, Vol. 2, Janos Bolyai Math. Soc., Budapest, 1996, pp. 295 –352.
[9] G. Kortsarz, D. Peleg, On choosing a dense subgraph, Proceedings of the 34th IEEE Symposium on Foundations of Computer Science, 1993, pp. 692–701.
[10] S.S. Ravi, D.J. Rosenkrantz, G.K. Tayi, Heuristic and special case algorithms for dispersion problems, Oper. Res. 42 (1994) 299–310.
