Directory UMM :Data Elmu:jurnal:O:Operations Research Letters:Vol27.Issue5.2000:
Operations Research Letters 27 (2000) 223–227
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
c 2000 Published by Elsevier Science B.V. All rights reserved.
algorithmic version the regularity lemma.
Keywords: Dispersion problem; Algorithm; Regularity lemma
1. Introduction
Let G = (V; E; !), where ! : E → {1; : : : ; K} and
for S ⊂ V let E(S) denote the set of edges which have
both endpoints in S. The dispersion problem can be
formulated as follows. Given a weighted graph G, and
s ∈ {2; 3; : : : ;P
|V |} nd a subset S of V such that |S|=s
and the sum {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 least 14 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 dense k-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: [email protected] (A. Czygrinow).
In this paper, we present an approximation algorithm for a dense case of the dispersion problem, that
is in the case when the optimal value is at least cn2 ,
for some constant c. In addition, we assume that the
weights on edges are integers from {1; : : : ; K} for
some xed K. The method is based on the remarkable
lemma of Szemeredi [11] and its algorithmic version
due to Alon et al. [1]. For other algorithmic applications of the regularity lemma we refer mainly to [6],
which contains applications to the partitioning problems (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 approximation scheme (PTAS) for a dense case of the
dense k-subgraph problem. The algorithm is derandomized in [2]; however, the running time of the
2
derandomized algorithm is O(nO(1= ) ). Using the regularity lemma we will get rid of the 1=2 in the exponent 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 developed in [3,6].
c 2000 Published by Elsevier Science B.V. All rights reserved.
0167-6377/00/$ - see front matter
PII: S 0 1 6 7 - 6 3 7 7 ( 0 0 ) 0 0 0 5 8 - 4
224
A. Czygrinow / Operations Research Letters 27 (2000) 223–227
P
For a set S ⊂ V denoted by disp(S) = {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 of SOPT .
Theorem 1. For every 0 ¡ ¡ 1 and integer K there
is an algorithm which when given a weighted graph
(V; E; !) on n vertices and with weights in {1; : : : ; K}
nds in O(n2:4 ) time a set S ∗ ⊂ 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 in n = |V |, and is not
polynomial in and K.
The rest of the note is organized as follows. In
Section 2, we formulate the regularity lemma of Szemeredi, Section 3 contains the proof of Theorem 1.
2. The regularity lemma
First, let us introduce necessary de nitions
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 = ∅ de ne the density of the pair (V1 ; V2 )
as d(V1 ; V2 ) = e(V1 ; V2 )=|V1 kV2 |, where e(V1 ; V2 ) denotes the number of edges with one endpoint in V1
and the other in V2 .
De nition 2. A pair (V1 ; V2 ) is called -regular if for
every V1′ ⊂ V1 with |V1′ |¿|V1 | and every V2′ ⊂ V2 with
|V2′ |¿|V2 | we have
|d(V1 ; V2 ) − d(V1′ ; V2′ )|6:
The rst condition simply states that all of the partition classes are “almost” equal, the second expresses
the fact that at most fraction of all pairs is -irregular.
Thus, for all i = 1; : : : ; t, |Vi | ∈ {⌈n=t⌉; ⌊n=t⌋}. Since
these
oors and ceilings are clearly insigni cant in our
analysis we will simplify the exposition and assume
that for every i, |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 exist N (; l) and L(; l) such that every graph
with at least N (; l) vertices admits an -regular
partition V1 ∪ · · · ∪ Vt with l6t6L(; 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
n vertices with !: E → {1; : : : ; K} where K is constant independent of n. De ne graphs G1 ; : : : ; GK
as Gl = (V; !−1 (l)), that is, Gl is an unweighted
graph on V with {x; y} ∈ E(Gl ) if !({x; y}) = l.
We use dl (U; W ) to denote the density of the
pair (U; W ) in Gl and we de nePthe total denK
sity of (U; W ) as d(U;
l=0 ldl (U; W ).
PW ) =
Thus if !(U; W ) =
u ∈ U; w ∈ W !(u; w) then
d(U; W ) = !(U; W )=|U kW |. The following multicolored 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 exist N = N (; l; K) and L = L(; l; K) such that
for any collection of K graphs on a vertex set V with
|V |¿N there is a partition V1 ∪ V2 ∪ · · · ∪ Vt which is
-regular in all of the K graphs and for which l6t6L.
One can nd such a partition in O(n2:4 ) time using
the Alon et al. [1] algorithm.
De nition 3. A partition V1 ∪ V2 ∪ · · · ∪ Vt of V is
called -regular if both of the following conditions
are satis ed.
3. Algorithm
1. kVi | − |Vj k61, for all i; j.
2. All but at most t 2 of pairs (Vi ; Vj ) are -regular.
In this section, we present the algorithm for the
dispersion problem and prove Theorem 1. Given a
A. Czygrinow / Operations Research Letters 27 (2000) 223–227
partition V1 ∪ V2 ∪ · · · ∪ Vt of V de ne a function
disp : 2V → {1; : : : ; Kn2 } as follows:
X
disp(S) =
d(Vi ; Vj )|Vi ∩ SkVj ∩ S|:
16i¡j6t
225
least one. Indeed, let
X
c1 =
d(Vi1 ; Vi )|Vi ∩ S|
i; i6=i1 ; i2
and
c2 =
X
d(Vi2 ; Vi )|Vi ∩ S|:
i; i6=i1 ; i2
Algorithm
If c1 ¿c2 then let S ′ be the set obtained from S as
follows:
1. Fix = =10K 2 and nd partition V1 ∪ V2 ∪ · · · ∪ Vt
with t¿1= which is -regular with respect to all
graphs G1 ; : : : ; GK .
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
correctness of the algorithm.
Lemma 6. For any set S ⊂ V with |S| = s there is
S which is the union of some of the Vi ’s and that
satis es the following conditions:
• If |Vi1 ∩ S c |6|Vi2 ∩ S| then add Vi1 ∩ S c to S and
subtract any subset of Vi2 ∩ S of size |Vi1 ∩ S c |.
• If |Vi1 ∩ S c |¿|Vi2 ∩ S| then subtract Vi2 ∩ S and add
any subset of Vi1 ∩ S c of size |Vi2 ∩ S|.
In both cases, we reduce the number of Vi ’s that intersect both S and S c by at least one. Then
X
d(Vi ; Vj )|Vi ∩ SkVj ∩ S|
disp(S) =
16i¡j6t; i; j6=i1 ; i2
+ d(Vi1 ; Vi2 )|Vi1 ∩ SkVi2 ∩ S|
+c1 |Vi1 ∩ S| + c2 |Vi2 ∩ S|
X
d(Vi ; Vj )|Vi ∩ SkVj ∩ S|
6
16i¡j6t; i; j6=i1 ; i2
1. |S|6s;
− 2Kn2 .
2. disp(S)¿disp(S)
Proof. Assume rst that there is exactly one 16i6t
such that S ∩Vi 6= ∅ and S c ∩Vi 6= ∅. Then for S =S \Vi
¡ s and
we have |S|
n!
disp(S)¿disp(S)
− K t
2
X
d(Vj ; Vi )|Vj ∩ SkVi ∩ S|
+
j;j6=i
¿ disp(S) − K
n2
n2
+
t2
t
n2
+ c1 |Vi1 ∩ S| + c2 |Vi2 ∩ S|:
t2
Note that to obtain S ′ we replace some subset of Vi2 ∩S
with a subset of Vi1 ∩S c of the same size. Since c1 ¿c2
we have
+K
:
Suppose now that there exist 16i1 6= i2 6t such
that S ∩ Vi1 6= ∅, S c ∩ Vi1 6= ∅ and S ∩ Vi2 6= ∅,
S c ∩ Vi2 6= ∅. We will compare the contribution of Vi1
to S with the contribution of Vi2 and show that there
exist a set S ′ with |S ′ | = |S| such that the number of
Vi ’s that intersect both S ′ and (S ′ )c is reduced by at
c1 |Vi1 ∩ S| + c2 |Vi2 ∩ S|6c1 |Vi1 ∩ S ′ | + c2 |Vi2 ∩ S ′ |:
Therefore,
n2
:
t2
The case c2 ¿ c1 is analogous. In this way, we reduce
the number of Vi ’s that intersect both S ′ and (S ′ )c by
at least one and keep |S ′ | = s. By repeating the above
process at most t − 1 times, we obtain S ′ such that at
most one of Vi ’s intersect both S ′ and (S ′ )c , |S ′ | = s,
and
n2
disp(S ′ )¿disp(S) − K(t − 1) 2 :
t
Now, if there is a single Vi that intersects both S ′ and
(S ′ )c , we subtract Vi as described in the beginning of
disp(S)6disp(S ′ ) + K
226
A. Czygrinow / Operations Research Letters 27 (2000) 223–227
¡ s and
the proof to obtain S with |S|
and (4) yields
disp(S)¿disp(S)
− 2Kn2 :
!(Vi ∩ S; Vj ∩ S)
In our next lemma, we show that disp(·) is a “good”
approximation to the dispersion function disp(·).
2
6
K
X
l=0
Lemma 7. For any S ⊂V; |disp(S)−disp(S)|64K n :
disp(S) 6
16i¡j6t
If |Vi ∩ S| ¡ |Vi | then the number of edges incident
to Vi ∩ S is at most n2 =t. Thus, the number of edges
incident to Vi ∩ S for which |Vi ∩ S| ¡ |Vi |, over all
16i6t, is at most n2 and the total weight on these
edges is at most Kn2 .
For any i 6= j, we have !(Vi ∩ S; Vj ∩ S)6Kn2 =t 2 .
Since there are at most t 2 , -irregular pairs in all Gi ’s,
the weight on edges between the irregular pairs is at
most Kn2 . Thus, combining (1) with above arguments
shows that
X
(2)
disp(S)6
!(Vi ∩ S; Vj ∩ S) + 3Kn2 ;
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 de nition of an -regular pair,
dl (Vi ∩ S; Vj ∩ S)6dl (Vi ; Vj ) + :
(3)
Thus,
K
XX
ldl (Vi ; Vj )|Vi ∩ SkVj ∩ S|
l=0
2 2
6
+ 4K n
K
X X
ldl (Vi ; Vj )|Vi ∩ SkVj ∩ S|
16i¡j6t l=0
+ 4K 2 n2 = disp(S) + 4K 2 n2 :
In a similar way, one can show that
disp(S)6disp(S) + 4K 2 n2 :
We are now ready to prove Theorem 1.
Proof. Let SOPT denote a set of size s with maximal
dispersion and let S be a set found by the algorithm.
Then by Lemma 7
disp(SOPT )6disp(SOPT ) + 4K 2 n2 ;
which by Lemma 6 is less than disp(S) + 6K 2 n2 .
Applying Lemma 7 again yields
disp(SOPT )6disp(S) + 10K 2 n2 :
Since = =10K 2 , we found S that satis es
disp(S)¿disp(SOPT ) − n2 :
Finally, if |S| ¡ s then we add arbitrarily s − |S| vertices to obtain S ∗ such that |S ∗ | = s and
disp(S ∗ )¿disp(S)¿disp(SOPT ) − n2 :
References
!(Vi ∩ S; Vj ∩ S) =
K
X
lel (Vi ∩ S; Vj ∩ S)
l=0
=
n2
:
t2
Consequently, (as K6K 2 )
2
Proof. For every i = 1; : : : ; t, we have |S ∩ Vi |6|Vi | =
n=t and so the number of edges that have both endpoints in S ∩ Vi , for some i = 1; : : : ; t, is at most
tn2 =t 2 6n2 . Thus, the total weight of the edges of the
above form is at most Kn2 . Therefore,
X
disp(S)6
!(Vi ∩ S; Vj ∩ S) + Kn2 :
(1)
ldl (Vi ; Vj )|Vi ∩ SkVj ∩ S| + K 2
K
X
ldl (Vi ∩ S; Vj ∩ S)|Vi ∩ SkVj ∩ S|;
(4)
l=0
where el (A; B) denotes the number of edges between
A and B that have weight l. Therefore, combining (3)
[1] N. Alon, R.A. Duke, H. Lefmann, V. Rodl, R. Yuster, The
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.
[3] A. Czygrinow, S. Poljak, V. Rodl, Constructive quasi-Ramsey
numbers and tournament ranking, SIAM J. Discrete Math.
12(1) (1999) 48– 63.
A. Czygrinow / Operations Research Letters 27 (2000) 223–227
[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,
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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.
[11] E. Szemeredi, Regular Partitions of Graphs, Colloques
Internationaux C.N.R.S., Problemes Combinatories et Theorie
des Graphes, 1978, pp. 399 – 402.
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
c 2000 Published by Elsevier Science B.V. All rights reserved.
algorithmic version the regularity lemma.
Keywords: Dispersion problem; Algorithm; Regularity lemma
1. Introduction
Let G = (V; E; !), where ! : E → {1; : : : ; K} and
for S ⊂ V let E(S) denote the set of edges which have
both endpoints in S. The dispersion problem can be
formulated as follows. Given a weighted graph G, and
s ∈ {2; 3; : : : ;P
|V |} nd a subset S of V such that |S|=s
and the sum {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 least 14 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 dense k-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: [email protected] (A. Czygrinow).
In this paper, we present an approximation algorithm for a dense case of the dispersion problem, that
is in the case when the optimal value is at least cn2 ,
for some constant c. In addition, we assume that the
weights on edges are integers from {1; : : : ; K} for
some xed K. The method is based on the remarkable
lemma of Szemeredi [11] and its algorithmic version
due to Alon et al. [1]. For other algorithmic applications of the regularity lemma we refer mainly to [6],
which contains applications to the partitioning problems (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 approximation scheme (PTAS) for a dense case of the
dense k-subgraph problem. The algorithm is derandomized in [2]; however, the running time of the
2
derandomized algorithm is O(nO(1= ) ). Using the regularity lemma we will get rid of the 1=2 in the exponent 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 developed in [3,6].
c 2000 Published by Elsevier Science B.V. All rights reserved.
0167-6377/00/$ - see front matter
PII: S 0 1 6 7 - 6 3 7 7 ( 0 0 ) 0 0 0 5 8 - 4
224
A. Czygrinow / Operations Research Letters 27 (2000) 223–227
P
For a set S ⊂ V denoted by disp(S) = {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 of SOPT .
Theorem 1. For every 0 ¡ ¡ 1 and integer K there
is an algorithm which when given a weighted graph
(V; E; !) on n vertices and with weights in {1; : : : ; K}
nds in O(n2:4 ) time a set S ∗ ⊂ 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 in n = |V |, and is not
polynomial in and K.
The rest of the note is organized as follows. In
Section 2, we formulate the regularity lemma of Szemeredi, Section 3 contains the proof of Theorem 1.
2. The regularity lemma
First, let us introduce necessary de nitions
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 = ∅ de ne the density of the pair (V1 ; V2 )
as d(V1 ; V2 ) = e(V1 ; V2 )=|V1 kV2 |, where e(V1 ; V2 ) denotes the number of edges with one endpoint in V1
and the other in V2 .
De nition 2. A pair (V1 ; V2 ) is called -regular if for
every V1′ ⊂ V1 with |V1′ |¿|V1 | and every V2′ ⊂ V2 with
|V2′ |¿|V2 | we have
|d(V1 ; V2 ) − d(V1′ ; V2′ )|6:
The rst condition simply states that all of the partition classes are “almost” equal, the second expresses
the fact that at most fraction of all pairs is -irregular.
Thus, for all i = 1; : : : ; t, |Vi | ∈ {⌈n=t⌉; ⌊n=t⌋}. Since
these
oors and ceilings are clearly insigni cant in our
analysis we will simplify the exposition and assume
that for every i, |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 exist N (; l) and L(; l) such that every graph
with at least N (; l) vertices admits an -regular
partition V1 ∪ · · · ∪ Vt with l6t6L(; 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
n vertices with !: E → {1; : : : ; K} where K is constant independent of n. De ne graphs G1 ; : : : ; GK
as Gl = (V; !−1 (l)), that is, Gl is an unweighted
graph on V with {x; y} ∈ E(Gl ) if !({x; y}) = l.
We use dl (U; W ) to denote the density of the
pair (U; W ) in Gl and we de nePthe total denK
sity of (U; W ) as d(U;
l=0 ldl (U; W ).
PW ) =
Thus if !(U; W ) =
u ∈ U; w ∈ W !(u; w) then
d(U; W ) = !(U; W )=|U kW |. The following multicolored 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 exist N = N (; l; K) and L = L(; l; K) such that
for any collection of K graphs on a vertex set V with
|V |¿N there is a partition V1 ∪ V2 ∪ · · · ∪ Vt which is
-regular in all of the K graphs and for which l6t6L.
One can nd such a partition in O(n2:4 ) time using
the Alon et al. [1] algorithm.
De nition 3. A partition V1 ∪ V2 ∪ · · · ∪ Vt of V is
called -regular if both of the following conditions
are satis ed.
3. Algorithm
1. kVi | − |Vj k61, for all i; j.
2. All but at most t 2 of pairs (Vi ; Vj ) are -regular.
In this section, we present the algorithm for the
dispersion problem and prove Theorem 1. Given a
A. Czygrinow / Operations Research Letters 27 (2000) 223–227
partition V1 ∪ V2 ∪ · · · ∪ Vt of V de ne a function
disp : 2V → {1; : : : ; Kn2 } as follows:
X
disp(S) =
d(Vi ; Vj )|Vi ∩ SkVj ∩ S|:
16i¡j6t
225
least one. Indeed, let
X
c1 =
d(Vi1 ; Vi )|Vi ∩ S|
i; i6=i1 ; i2
and
c2 =
X
d(Vi2 ; Vi )|Vi ∩ S|:
i; i6=i1 ; i2
Algorithm
If c1 ¿c2 then let S ′ be the set obtained from S as
follows:
1. Fix = =10K 2 and nd partition V1 ∪ V2 ∪ · · · ∪ Vt
with t¿1= which is -regular with respect to all
graphs G1 ; : : : ; GK .
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
correctness of the algorithm.
Lemma 6. For any set S ⊂ V with |S| = s there is
S which is the union of some of the Vi ’s and that
satis es the following conditions:
• If |Vi1 ∩ S c |6|Vi2 ∩ S| then add Vi1 ∩ S c to S and
subtract any subset of Vi2 ∩ S of size |Vi1 ∩ S c |.
• If |Vi1 ∩ S c |¿|Vi2 ∩ S| then subtract Vi2 ∩ S and add
any subset of Vi1 ∩ S c of size |Vi2 ∩ S|.
In both cases, we reduce the number of Vi ’s that intersect both S and S c by at least one. Then
X
d(Vi ; Vj )|Vi ∩ SkVj ∩ S|
disp(S) =
16i¡j6t; i; j6=i1 ; i2
+ d(Vi1 ; Vi2 )|Vi1 ∩ SkVi2 ∩ S|
+c1 |Vi1 ∩ S| + c2 |Vi2 ∩ S|
X
d(Vi ; Vj )|Vi ∩ SkVj ∩ S|
6
16i¡j6t; i; j6=i1 ; i2
1. |S|6s;
− 2Kn2 .
2. disp(S)¿disp(S)
Proof. Assume rst that there is exactly one 16i6t
such that S ∩Vi 6= ∅ and S c ∩Vi 6= ∅. Then for S =S \Vi
¡ s and
we have |S|
n!
disp(S)¿disp(S)
− K t
2
X
d(Vj ; Vi )|Vj ∩ SkVi ∩ S|
+
j;j6=i
¿ disp(S) − K
n2
n2
+
t2
t
n2
+ c1 |Vi1 ∩ S| + c2 |Vi2 ∩ S|:
t2
Note that to obtain S ′ we replace some subset of Vi2 ∩S
with a subset of Vi1 ∩S c of the same size. Since c1 ¿c2
we have
+K
:
Suppose now that there exist 16i1 6= i2 6t such
that S ∩ Vi1 6= ∅, S c ∩ Vi1 6= ∅ and S ∩ Vi2 6= ∅,
S c ∩ Vi2 6= ∅. We will compare the contribution of Vi1
to S with the contribution of Vi2 and show that there
exist a set S ′ with |S ′ | = |S| such that the number of
Vi ’s that intersect both S ′ and (S ′ )c is reduced by at
c1 |Vi1 ∩ S| + c2 |Vi2 ∩ S|6c1 |Vi1 ∩ S ′ | + c2 |Vi2 ∩ S ′ |:
Therefore,
n2
:
t2
The case c2 ¿ c1 is analogous. In this way, we reduce
the number of Vi ’s that intersect both S ′ and (S ′ )c by
at least one and keep |S ′ | = s. By repeating the above
process at most t − 1 times, we obtain S ′ such that at
most one of Vi ’s intersect both S ′ and (S ′ )c , |S ′ | = s,
and
n2
disp(S ′ )¿disp(S) − K(t − 1) 2 :
t
Now, if there is a single Vi that intersects both S ′ and
(S ′ )c , we subtract Vi as described in the beginning of
disp(S)6disp(S ′ ) + K
226
A. Czygrinow / Operations Research Letters 27 (2000) 223–227
¡ s and
the proof to obtain S with |S|
and (4) yields
disp(S)¿disp(S)
− 2Kn2 :
!(Vi ∩ S; Vj ∩ S)
In our next lemma, we show that disp(·) is a “good”
approximation to the dispersion function disp(·).
2
6
K
X
l=0
Lemma 7. For any S ⊂V; |disp(S)−disp(S)|64K n :
disp(S) 6
16i¡j6t
If |Vi ∩ S| ¡ |Vi | then the number of edges incident
to Vi ∩ S is at most n2 =t. Thus, the number of edges
incident to Vi ∩ S for which |Vi ∩ S| ¡ |Vi |, over all
16i6t, is at most n2 and the total weight on these
edges is at most Kn2 .
For any i 6= j, we have !(Vi ∩ S; Vj ∩ S)6Kn2 =t 2 .
Since there are at most t 2 , -irregular pairs in all Gi ’s,
the weight on edges between the irregular pairs is at
most Kn2 . Thus, combining (1) with above arguments
shows that
X
(2)
disp(S)6
!(Vi ∩ S; Vj ∩ S) + 3Kn2 ;
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 de nition of an -regular pair,
dl (Vi ∩ S; Vj ∩ S)6dl (Vi ; Vj ) + :
(3)
Thus,
K
XX
ldl (Vi ; Vj )|Vi ∩ SkVj ∩ S|
l=0
2 2
6
+ 4K n
K
X X
ldl (Vi ; Vj )|Vi ∩ SkVj ∩ S|
16i¡j6t l=0
+ 4K 2 n2 = disp(S) + 4K 2 n2 :
In a similar way, one can show that
disp(S)6disp(S) + 4K 2 n2 :
We are now ready to prove Theorem 1.
Proof. Let SOPT denote a set of size s with maximal
dispersion and let S be a set found by the algorithm.
Then by Lemma 7
disp(SOPT )6disp(SOPT ) + 4K 2 n2 ;
which by Lemma 6 is less than disp(S) + 6K 2 n2 .
Applying Lemma 7 again yields
disp(SOPT )6disp(S) + 10K 2 n2 :
Since = =10K 2 , we found S that satis es
disp(S)¿disp(SOPT ) − n2 :
Finally, if |S| ¡ s then we add arbitrarily s − |S| vertices to obtain S ∗ such that |S ∗ | = s and
disp(S ∗ )¿disp(S)¿disp(SOPT ) − n2 :
References
!(Vi ∩ S; Vj ∩ S) =
K
X
lel (Vi ∩ S; Vj ∩ S)
l=0
=
n2
:
t2
Consequently, (as K6K 2 )
2
Proof. For every i = 1; : : : ; t, we have |S ∩ Vi |6|Vi | =
n=t and so the number of edges that have both endpoints in S ∩ Vi , for some i = 1; : : : ; t, is at most
tn2 =t 2 6n2 . Thus, the total weight of the edges of the
above form is at most Kn2 . Therefore,
X
disp(S)6
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(1)
ldl (Vi ; Vj )|Vi ∩ SkVj ∩ S| + K 2
K
X
ldl (Vi ∩ S; Vj ∩ S)|Vi ∩ SkVj ∩ S|;
(4)
l=0
where el (A; B) denotes the number of edges between
A and B that have weight l. Therefore, combining (3)
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