Dilworth's theorem establishes a link between a minimal path cover and a maximal antichain in a digraph.A new proof for Dilworth's theorem is given. Moreover an algorithm to find both the path cover and the antichain, as considered in the theorem, is presented.

Pijls, W.H.L.M.

Potharst, R.

English

Research Papers in Economics

Dilworth's theorem revisited, an algorithmic proof
Wim Pijls Rob Potharst¤
Econometric Institute Report EI 2011-13
Abstract
Dilworth's theorem establishes a link between a minimal path cover and a maximal
antichain in a digraph. A new proof for Dilworth's theorem is given. Moreover an algorithm
to ¯nd both the path cover and the antichain, as considered in the theorem, is presented.
1 Introduction
Dilworth's theorem establishes a link between a minimal path cover and a maximal antichain
in a digraph. There are multiple proofs of Dilworth's theorem[3, 4, 8, 11]. Those proofs do not
show how an optimal path cover and optimal antichain are obtained for a given graph.
In the current paper we give a new proof, based upon an algorithm that constructs a minimal
path cover along with a maximal antichain. The key of the proof is the following observation.
Any path cover is a °ow in an associated network and vice versa, so ¯nding a minimal path
cover can be reduced to ¯nding a minimal °ow. The algorithm for a minimal °ow generates
a cut (analogously to the famous max°ow/mincut connection), which is closely related to the
maximal antichain. So an algorithm is presented generating a path cover and an antichain which
are certi¯cates of the correctness of Dilworth's theorem.
For a variant of Dilworth's theorem a similar proof was given in [5, section 4.9]. In [6] a
direct proof was presented for the proposition that any maximal cut corresponds to a maximal
antichain. The minimal °ow construction was not involved in that proof.
One application of the maximal antichain problem is found in the ¯eld of data mining, see [9].
2 Path covers and °ow networks
Path cover and antichain. A path cover in a digraph D(V;A) is a collection of paths such
that every node of V is included in at least one path. The number of paths is called the size of
the path cover.
In a digraph a starting node is a node without incoming arcs. Likewise, an end node is a node
without outgoing arcs. We assume that all paths of a path cover run from a starting node to an
end node. If this property is not satis¯ed in a given path cover, it can be introduced without
changing the size of the path cover. An antichain in D(V;A) is a set of nodes, no two of which
are included in any path of D(V;A). The core of this paper is the proof of Dilworth's theorem:
¤Econometric Institute, Erasmus University Rotterdam, P.O.Box 1738, 3000 DR Rotterdam, The Netherlands,
e-mail: fpijls,potharstgese.eur.nl
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Figure 1: A digraph.
Dilworth's theorem: The minimal size of a path cover in a digraph D equals the maxi-
mal size of an antichain in D.
Flow network. A °ow network is a digraph N(V;A) containing two special nodes s and t
being the single starting node and the single end node, and two functions ` and u, the lower and
the upper capacity respectively, mapping the arc set A into R. We call s and t the source and
the sink respectively. A °ow f in a given network is a function f from A into R which obeys the
capacity constraint given by: `(a) · f(a) · u(a) for all a 2 A. Moreover, a °ow has to meet
the balance constraint de¯ned by the equality
P
(j;i)2A f(j;i) =
P
(i;k)2A f(i;k) for any node
i, i 6= s;t. The balance constraint actually says that the incoming °ow (in°ow) must be equal
to the outgoing °ow (out°ow) in each node, except in the source and the sink. If the balance
constraint is met in every node, the out°ow of s equals the in°ow of t. This value is called the
value of the °ow.
Next to a °ow, the pre°ow is a known notion. For a pre°ow, the balance constraint is relaxed:
the in°ow is larger than or equal to the out°ow in each node (except the source).
Given a path cover C in a network, we de¯ne the function f(i;j) for an arc (i;j) as the number
of paths in C that go through arc (i;j). If the capacity constraint holds, f is a °ow and its
value equals the size of C. Conversely, an integer-valued °ow f in an acyclic network is easily
decomposed into a collection of paths (not necessarily constituting a path cover).
An associated °ow network. From the theory of network °ows it is known that any al-
gorithm generating a max°ow ends with a cut, inducing a set of arcs. Since Dilworth's theorem
deals with nodes, we make an extra step. Each node v 2 V is split up into two nodes v1 and v2
with an arc in between. To be more speci¯c, we give the following de¯nition.
De¯nition 1 Given a digraph D(V;A), an associated °ow network N(V 0;A0) is constructed as
follows:
a) for each node v 2 V , there are two nodes v1 and v2 in V 0 and an arc (v1;v2) in A0;
b) for each arc (v;w) 2 A, there is an arc (v2;w1) in A0;
c) in addition to the node pairs (v1;v2) de¯ned in a) the set V 0 contains a source node s and
a sink node t; there is an arc (s;v1) in V 0 for each starting node v 2 V and an arc (v2;t)
2for each end node v 2 V ;
d) the lower capacity `(a0); a0 2 A0 equals 1 for the arcs a0 de¯ned in a) and equals 0 for the
arcs a0 de¯ned in b) or c); the upper capacity of each arc a0 equals +1.
See Figures 1 and 2 for an example of a digraph and its associated °ow network (ignore the
numbers of Figure 2 as yet). Since a (minimal) path cover corresponds to a (minimal) °ow, the
minimal °ow problem is studied in section 3.
3 Finding a minimal network °ow
Min°ow reduced to max°ow. The problem of ¯nding a °ow of minimal value (min°ow
for short) does not receive much attention in the mathematical literature. Only a few refer-
ences can be given, e.g. [2] or [9, section 2.2]. On the other hand there is a vast literature
on max°ow algorithms. Almost any textbook on graph theory or combinatorial optimization
treats this problem, e.g. [1, Chapter 6-7], [7, Chapter 6] and [10, Chapter 10]. For ¯nding a
min°ow, we apply a method di®erent from the above references. A given °ow network N(V;A)
can be transformed into a network ~ N(V;A) by establishing new capacities ~ u and ~ ` as follows:
~ u(i;j) = ¡`(i;j) and ~ `(i;j) = ¡u(i;j) for every arc (i;j) 2 A. At any time the °ow ~ f in the
transformed network ~ N(V;A) corresponds to a °ow f in the original network N(V;A) according
to the equation ~ f = ¡f. Similarly, a max°ow in ~ N(V;A) corresponds to a min°ow in N(V;A).
It is also possible to design a min°ow algorithm in its own right. The operations of a max°ow
algorithm can easily be translated into operations working on the original min°ow instance. The
augmenting paths of a max°ow algorithm can be converted into decreasing paths.
An initial °ow. Two types of max°ow algorithms are distinguished: path augmenting al-
gorithms and pre°ow-push algorithms. Either type requires an initial °ow or an initial pre°ow
respectively. Most of the literature assumes that all lower capacities are equal to 0 and in that
case establishing an initial (pre)°ow is trivial.
However, the above transformation converts the capacities of the associated °ow graph N(V 0;A0)
as de¯ned in section 2 into negative upper and lower bounds. In that case, a di®erent approach
for ¯nding an initial °ow is necessary. Constructing a path cover in an acyclic graph is straight-
forward. This path cover generates a °ow f and the related °ow ~ f = ¡f can be used as the
initial °ow for the max°ow algorithm.
Note. In the context of Dilworth's theorem we only consider networks that do not include
cycles. Suppose we have a network which does include cycles and some arcs (i;j have `(i;j) > 0
or u(i;j) < 0. In that case, the construction of an initial °ow is not straightforward. See [1,
section 6.7] for this problem.
A cut in a network. Starting from the initial °ow, the °ow is repeatedly changed during
the running time of any max°ow algorithm. All algorithms work with an auxiliary graph, the
so-called residual graph. For a °ow network N(V;A) with °ow function f, the residual graph
Rf(V; ¹ A) is de¯ned as follows. The node set remains the same. The arc set ¹ A consists of arcs
(i;j) (forward arcs) such that (i;j) 2 A and f(i;j) < u(i;j), and arcs (j;i) (backwards arcs)
such that (i;j) 2 A and f(i;j) > `(i;j). Any max°ow algorithm ends when the residual graph
Rf contains no path from s to t. As long as the residual graph contains such a path, the value
of the °ow can be increased.
A cut (S;T) in a °ow network is a pair of sets S ½ V and T = V nS, such that s 2 S and
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Figure 2: The °ow network associated to the graph of Figure 1. The numbers denote an arbitrary
°ow function f.
t 2 T. When the algorithm stops, one constructs a cut (S;T) in the °ow network using the ¯nal
residual graph. S is the set of nodes v 2 V such that the ¯nal residual graph has a path from
s to v. Since there is no path from s to t, S does not include t and hence, the collection (S;T)
with T = V nS is a cut.
An arc (i;j) is called a crossing arc if i 2 S and j 2 T; it is called an anti-crossing arc if i 2 T
and j 2 S. Given the de¯nition of the residual graph and the construction of the cut, it is easily
derived that a max°ow f has the property:
f(i;j) = u(i;j) (1)
for each crossing arc (i;j) and f(i;j) = `(i;j) for each anti-crossing arc (i;j).
No anti-crossing arcs. The network N(V 0;A0) de¯ned in section 2 has u(i;j) = +1 for
every arc (i;j). So the corresponding max°ow instance has `(i;j) = ¡1. Since each max°ow
algorithm terminates after a ¯nite number of iterations (apart from some pathological excep-
tions), f(i;j) = `(i;j) = ¡1 cannot happen. Consequently there are no anti-crossings arcs in
the ¯nal °ow.
This property implies an important result for the ¯nal path cover: every path starting in S and
ending in T has exactly one crossing arc. Indeed, if a path contained two crossing arcs, an
anti-crossing arc would be somewhere in between.
Example. The °ow in the network of Figure 2 has value 7, or, equivalently, the corresponding
path cover comprises seven paths from s to t. Notice that the path s;a1;a2;e1;e2; h1;h2;t is
counted twice. The ¯nal min°ow with value 5 is shown in Figure 3. The above mentioned
path is counted once in this min°ow. Furthermore, there is a new path in the induced path
cover: s;c1;c2;f1;f2;j1;j2;t. Due to this new path the number of paths covering the pieces
s;a1;a2;e1;e2;j1 and c2;g1;g2;`1;`2 has been reduced. The piece c2;f1;f2;j1 is contained in an
extra path. The reduction of the °ow value from 7 to 5 is achieved by two steps in the related
max-°ow algorithm using augmenting paths.
The resulting cut is given by S = fs;a1;a2;e1;e2;h1;i1;b1;c1;d1g and T = V nS.
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Figure 3: The minimal °ow or minimal path cover in the °ow network of Figure 2. (The nodes
and arcs inside the S set are drawn in bold.)
4 Proof of Dilworth's theorem
In this section Dilworth's theorem is proved. The essential part of this proof is covered by
Lemma 1.
Lemma 1 The minimal °ow in N(V 0;A0) induces a path cover and an antichain of equal size.
Proof The minimal °ow f in N(V 0;A0) is integer-valued, since all capacities are integers. This
°ow can be decomposed into a path cover C and the number of paths through arc (i;j) equals
f(i;j).
We mentioned in section 3 that any max°ow of min°ow algorithm ends with a cut (S;T). As
argued in section 3, every path starting in S and ending in T has exactly one crossing arc. So
the size of C, say n, equals the number of paths in the crossing arcs, or in formula:
n =
X
i2S;j2T
f(i;j) (2)
The counterpart for min°ow in N(V 0;A0) of (1) says that f(i;j) = `(i;j). De¯ne Q0 as the set
of crossing arcs of the form (v1;v2). These arcs have `(v1;v2) = 1 and they are the only crossing
arcs with `-value 6= 0. Thus, (2) may be extended to
n =
X
i2S;j2T
f(i;j) =
X
i2S;j2T
`(i;j) =
X
(v1;v2)2Q0
1: (3)
De¯ne Q = fv1 j (v1;v2) 2 Q0g. The number of elements in Q0 and hence also in Q is equal to
n, due to (3). So the size of Q equals the size of a path cover.
We show by contradiction that Q is an antichain. Any path through a node v1 2 Q also con-
tains the crossing arc (v1;v2). If a path contained two nodes from Q, it would also contain two
crossing arcs, which is impossible as shown in section 3. 2
Proof of Dilworth's theorem. According to the de¯nition of an antichain, a path in a
path cover cannot include two nodes of an antichain. So the size of any path cover is larger than
or equal to the size of any antichain and hence:
minimum size of a path cover ¸ maximum size of a antichain. (4)
5Equality in this relation is obtained thanks to Lemma 1. Now we have proved Dilworth's the-
orem in N(V 0;A0). Due to the lower capacity `(v1;v2) = 1 in N(V 0;A0) for each arc associated
to a node v 2 V , any path cover in N(V 0;A0) induces a path cover D(V;A). It is clear that an
antichain in N(V 0;A0) corresponds to an antichain in D(V;A). Hence, Dilworth's theorem also
holds in D(V;A). 2
Example (continued). There are six crossing arcs: (b1;b2), (c1;c2), (d1;d2), (h1;h2), (i1;i2)
and (e2;j1), ¯ve of which make up the set Q0. So Q = fb1;c1;d1;h1;i1g and fb;c;d;h;ig is the
obtained maximal antichain in Figure 1.
5 Summary
Constructing a path cover and an antichain of equal size is the crux in the proof of Dilworth's
theorem. The construction of such a path cover and antichain in a digraph D(V;A) is summa-
rized in the following steps.
1. Construct the associated network N(V 0;A0).
2. Construct in N(V 0;A0) an arbitrary path cover, which induces an initial °ow f by super-
position.
3. Transform N(V 0;A0) into network ~ N(V 0;A0) having new upper and the lower capacities
(see section 3). The initial °ow in ~ N(V 0;A0) is ~ f = ¡f.
4. Execute any max°ow algorithm in ~ N(V 0;A0); a derived product is a cut along with a set
of crossing arcs. Convert the max°ow into a min°ow.
5. Decompose the min°ow into a path cover for N(V 0;A0) and next for D(V;A).
6. Select the arcs of the type (v1;v2) from the crossing arcs obtained in step 4. The set of
corresponding nodes v is an antichain in D(V;A).
As mentioned in section 3, we can design a min°ow algorithm in its own right. Hence, the above
steps 3 and 4 can be replaced with the statement: execute a min°ow algorithm in N(V 0;A0).
Like a max°ow algorithm, a min°ow algorithm starts with an initial °ow. Notice that the initial
°ow f of step 2 obeys the capacity constraints of N(V 0;A0).
References
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7

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