www.elsevier.com/locate/orms
On the
k
-cut problem
Francisco Barahona
IBM T.J. Watson Research Center, P.O. 218, Yorktown Heights, NY 10598, USA
Received 1 June 1998; received in revised form 1 September 1999
Abstract
Given a graph with nonnegative edge-weights, let f(k) be the value of an optimal solution of the k-cut problem. We studyfas a function ofk. Letgbe the convex envelope off. We give a polynomial algorithm to computeg. In particular, iffis convex, then it can be computed in polynomial time for allk. We show some experiments in computingg. c2000 Elsevier Science B.V. All rights reserved.
Keywords:k-cut problem; Submodular functions; Minimum cut; Clustering
1. Introduction
LetG= (V; E) be a graph with nonnegative weights
w(e) for each edgee. Given a partitionS1; : : : ; Sk of
V, the set of edges with endnodes in dierent sets of the partition is called ak-cut. This is denoted by
(S1; : : : ; Sk). We usew(T) to denotePe∈Tw(e). We
usento denote|V|. For a given value ofk thek-cut problemconsists of nding a partition S1; : : : ; Sk that
minimizes
w((S1; : : : ; Sk)): (1.1)
Thek-cut problem is a clustering problem. The edge weights represent similarity between objects. Then the problem can be stated as nding k clusters that maximize the similarity between objects in the same cluster.
E-mail address:[email protected] (F. Barahona)
The k-cut problem is NP-Hard if k is part of the input. Ifk is xed, Goldschmidt and Hochbaum [8] showed that this reduces to O(nk2
) minimum cut prob-lems. Karger and Stein [12] gave an algorithm with expected running time O(n2klog2n), this has been
de-randomized by Karger and Motwani [11]. Approxi-mation algorithms have been given in [16,14].
Other problems dealing with multicuts wherek is not xed have also been studied. Chvatal [5] consid-ered
minimize w((S1; : : : ; Sp))=p; (1.2)
wherepis variable. He showed that the minimum is achieved forp= 2, i.e. by a minimum cut.
The closely related problem
minimize w((S1; : : : ; Sp))=(p−1); (1.3)
reduces to n parametric min-cut computations (see [4,6]). Although the two problems look very similar,
the solution of (1.3) is much more involved than the one for (1.2).
Consider now
minimize w((S1; : : : ; Sp))−p; (1.4)
wherepis variable, andp¿2. This is calledmulticut problem. It was shown in [1] that this reduces to O(n3)
minimum cut problems. The methods used for (1.4) are very much related to the ones for (1.3). One can think of (1.4) as a relaxation of (1.1), we explore that idea in this paper.
Letf(k) be the value of an optimal solution of the
k-cut problem. We say thatfis convex if
f(k+ 1)−f(k)6f(k+ 2)−f(k+ 1)
for 26k6n−2. The convex envelope g of f, is the largest convex function such thatg(k)6f(k), for 26k6n. In this paper we show thatg can be com-puted in polynomial time, using (1.4) as a subroutine. This paper is organized as follows. In Section 2 we sketch the solution approach for the multicut problem. In Section 3 we show how to compute the convex envelopeg. In Section 4 we present some experiments with this approximation.
2. The multicut problem
Problem (1.4) reduces to minimizing a submodular function as shown in [1]. In this section we sketch that reduction.
We need rst a few denitions. A functionf: 2E → R is called submodular if
f(A∪B) +f(A∩B)6f(A) +f(B) for allA; B⊆E:
The concept of a submodular function in discrete op-timization is in many respects analogous to that of a convex function in continuous optimization. The only known polynomial algorithm to nd the minimum of a submodular function is based on the ellipsoid method (cf. [9]). However, Queyranne [15] gave recently a simple combinatorial algorithm for minimizing sym-metric submodular functions. A function f is sym-metriciff(A) =f(E\A), for allA⊆E.
GivenA⊆V, we use(A) to denote the set of edges with exactly one endnode inA, and(A) is the set of
edges with both endnodes inA. It is well known that the functionf(S)=w((S)), forS⊆V, is submodular.
ferent setsTi. Then problem (1.4) reduces to
minimiz-ingm(S) for∅ 6=S⊂V. Now we need the following two Lemmas.
Lemma 2.1(Cunningham [6]). Let F=(S); and r be the rank function of the graphic matroid. One can computeh(S)by adding1− |S|to the value of
Given a vector yfeasible for (2.2), thegreedy algo-rithmcf. [7] picks any component y(e) and increases its value until it reaches its upper boundw(e), or one of the other inequalities becomes tight. In the latter case we say that a set istight. Let ˆybe obtained by applying the greedy algorithm. Then for any e ∈ F
either ˆy(e) =w(e) oreis in a tight set. Let Abe the union of the tight sets. This is also tight, by submod-ularity. Thus,
ˆ
y(F) = ˆy(A) + ˆy( A) =w(A) +r( A):
It is easy to see that it is enough to take setsT=(B) in (2.2). So Ais the union of sets of type(B). We are going to obtain the minimum of
l
Lemma 2.3. The function h is submodular.
Proof We have to consider (2.2) forF=(S); F=
(T); F=(S∪T) andF=(S∩T). Suppose that y
was obtained after applying the greedy algorithm for
F=(S∩T). We can extend yto a solution forF=
(S∪T), denote this by y. Now denote yS the vector obtained from yby setting to zero all components not in(S). We can extend yS to a solution forF=(S). We can proceed similarly forF=(T). This shows that
h(S∪T) +h(S∩T)6h(S) +h(T):
The functionmis not symmetric, however we can dene
m′
(S) =12w((S))−1 +h(S)
and minimize m′(S) + m′( S) that is symmetric.
Queyranne’s algorithm requires O(n3) evaluations of
m′. In our case one evaluation of m′ requires O(n)
minimum cut problems (see [2]). So the straightfor-ward implementation would require O(n4) minimum
cuts. However, one can use the solution of a previous step to start the next one, this improves the bound to O(n3) minimum cut problems (see [1]).
is piecewise linear and concave, for¿0. To evaluate it we divide all edge-weights byand solve (1.4). Its slope takes values in{−2; : : : ;−n}. Now, we describe how to obtainl′
for¿0. Given 061¡ 2, we can
evaluatel′
by solving two multicut problems, letp1
andp2 be the sizes of the partitions obtained. Ifp1=
p2, there is no breakpoint in [1; 2], and we have for the number of multicut problems that we have to solve. The rst interval should be given by 1= 0,
and a large number like2=Pw(e).
Assume now that we know all breakpoints
{1; : : : ; r}ofl′, and slopes{−p1; : : : ;−pr+1}. Then
for anyk, it is easy to computeg(k). Sincel(·; k) is concave and piecewise linear, its maximum is at a point with zero slope, or at a breakpoint where the slope changes from positive to negative. So we have to ndpiandpi+1 withpi6k6pi+1. Then
g(k) =l′
(i) +ik:
Suppose now that we use linear interpolation to ex-tend g to noninteger arguments. We obtain a con-vex piecewise linear function whose breakpoints are
{p2; : : : ; pr}, with slopes{1; : : : ; r}.
Now let us see how wellg approximates f. We have thatg(2) =f(2); g(n) =f(n), andg(k)6f(k) for any other valuek∈ {2; : : : ; n}. Alsog is convex, piecewise linear, and if k is a breakpoint of g then
g(k) =f(k). Sog is the best convex approximation off, i.e. its convex envelope. Now we can state our main result.
Fig. 1. A network with 41 nodes.
Fig. 2. A 47 nodes network.
Moreover; if f is convex then f can be computed in polynomial time.
4. Some experiments
We decided to compute g for some practical instances. The main issue here was to see how this
function looks, other aspects like eciency of the implementation or CPU time are beyond the scope of this paper.
Fig. 3. A 64 nodes network.
Fig. 4. A graph with 7% density.
network is designed to connect them. Finally thelocal networks are designed to connect cities within each cluster. (see [13]).
In each case we computed the function g and an upper bound forf. For the upper bound we used the three heuristics below.
• Min-cut heuristic: Given ak-cut, delete its edges. Find a minimum cut in each component. Find the minimum value among all these min cuts. Break this component to derive a (k+ 1)-cut.
• Largest edge heuristic: Start with the n-cut consisting of each individual node. Find an edge of maximum weight. Contract this edge to obtain an (n−1)-cut. Continue with the shrunken graph.
Fig. 5. A graph with 50% density.
We rst tried a network with 41 cities. We plot in Fig. 1 the convex envelope and the upper bound for each value ofk. The breakpoints ofgare marked by a bullet (•). We also plot the gap (in percentage) between g and the upper bound. The largest gap in this case was 10%.
We did a similar experiment with a 47 cities net-work (see Fig. 2). This time the largest gap was 24%. The function f seems to be concave in the interval [2;43]. This is reected in the larger gap.
We also tried the 64 cities network of [3] (see Fig. 3). This time the largest gap was 2.5%. This seems to reect the “more convex shape” of the fun-ctionf.
Finally, we tried two graphs randomly generated. In Fig. 4 we present the results given by a graph with 50 nodes, 7% density and edge weights taking integer values in [0;100]. Fig. 5 comes from a graph with 50 nodes, 50% density and similar edge weights. These two instances seem to indicate that this approximation works better for sparse graphs. If the graph is a tree then clearly f is convex. It would be very interest-ing to characterize other classes of graphs for which
f is convex, or where the gap between f andg is bounded.
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