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matrix (over V) of $ related to a tree %h :I $, and let

o)* be the (only) path’ between terminals contained in So.

Let Cm be the cyclomatic matrix of Uh8,,. Then it results

c= c,

(13)

C@

with /.i = c p,, + p.

8 W. Mayeda, ‘Synthes is of switching functions by linear graph

theory,”

IBM J.

of

Res.

and

Dev., vol. 4,

pp. 321-329; July, 1960.

3) Let $&(&) be the subgraph (eventually noncon-

netted) formed by the zero impedance branches of S(G).

It is clear that 6” = $. We shall prove that if 3 5 5 9

such that 2” C 5, then 3 $2 6 : G” C 5. Note that such

a tree & can be constructed by the union of the following

subgraphs:

1) S”

2) the trees 5h C $, : ai e 5 - 9”

3) the subgraphs $ -

a:, where s”, is a tree of

Minimum Feedback Arc Sets for a Directed Graph*

D. H. YOUNGER?

Summary-A minim um feedback arc set is, for a directed graph, a

minim um set of arcs which if removed leaves the resultant graph

free of directed loops. This paper establishes a relationship between

these feedback arcs and order; in particular, such a minim um set of

arcs is shown to be determined by a sequential ordering of the

nodes which minim izes the number of arcs, each of which enters a

node that precedes in the ordering the node it leaves. From this

relationship are developed some simple characteristic s of such sets,

as well as properties of the sequential orderings by which these

minimum sets are determined. These properties

form the basis of

an algorithm for finding minimu m feedback arc sets.

PROBLEM

in the topology of directed graphs

that has attracted some interest in recent years

is the following: to determine, for an arbitrary

directed graph,

a minimum

set of arcs which, if removed,

leaves the resultant graph free of directed loops. The

problem was originally suggested by Runyon,’ who

observed that the analysis of sequential switching circuits

with feedback pa ths would be simplified by the knowledge

of such a minimum set. Increased interest in this problem

is in large measure due to Moore, who has encouraged

attempts to find a solution. It was at first h.oped that a

simple and efficient algorithm might be found, perhaps

even an algorithm such that the number of operations

* Received Novembe r 26, 1962; revised manuscript received

February 4, 1963. This work was supported by the National Science

Foundation under Grant G-15078. Publication of this paper is

supported by the Marcellu s Hartley Fund.

t Dept. of Electrica l Engineering, Columbia University , New

York, N. Y.

1 Seshu and Reed [3] include this among a list of research prob-

lems given in the appendix.

required wou ld increase linearly with the number of

nodes in the graph. However, the problem has turned

out to be difficult and suggested algorithms have fallen

far short of that goal. One of the more noteworthy ap-

proaches is that of Tucker [6] who has formulated the

problem as an Integer Program. More recently, some

necessary conditions on such a minimum set of arcs have

been found [2].

Two names that have been suggested for such a set of

arcs are “minimal feedback cutset” and “minimal chord

set.” Much can be said for each, and yet each may convey

an erroneous connotation to some readers. Since the

terminology has not yet become standard, we suggest the

term “minimum feedback arc set.”

Since directed graphs are finding wide use in many

varied disciplines, a method of finding such a funda-

mental quantity for a directed graph is certain to be useful.

For example, consider a mathematical system of n vari-

ables a,, ...

, a,, upon which some set of order conditions

have been imposed, e.g., a, < a,, u3 < a,, a, < u3, a, < a,,

etc. Now assuming that these conditions are transitive,

it may be that some are inconsistent with the rest. Suppose

one wishes to determine a minimum set of conditions

that must be removed from the original set in order that

the subset remaining be consistent. The original set of

conditions may be represented as a directed graph of n

nodes by representing the variables as nodes and the

order conditions as arcs between pairs of nodes; any set of

conditions which corresponds to a minimum feedback

arc set for that graph should be removed.

In engineering, a good deal of success has been achieved

in analyzing complicated systems without feedback. This

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has led to methods for the analysis of a system containing

feedback which reduce the system to one without feed-

back by breaking an appropriate set of feedback arcs [7].

The complexity of this analysis increases drastically with

the number of arcs which must be broken; hence a knowl-

edge of a minimum feedback arc set would be useful.

As suggested above, a linear directed graph is a de-

scription of any set of order relations involving the vari-

ables represented as nodes of the graph. The object of this

paper is to establish the connection between minimum

feedback arc sets and possible sequential orderings of the

nodes. Consequently, some basic properties of such sets

as well as properties of the sequential orderings related to

them are developed.

DEFINITIONS AND NOTATIONS

A linear directed graph is shown in Fig. 1 . The nodes

have been labeled a through e so that they may be referred

to conveniently. Given a labeling of the nodes, the arcs

(directed edges) may be expressed in terms of these labels;

an arc comiecting node i to j is denoted by the ordered

pair (i, j). If more than one arc connects i to j, these may be

distinguished by subscrip t: (i, j),, (i, j),, . . . , (i, j),.

Not only may the graphs to be considered have multiple

connections, they may have arcs connecting a node

to itself; the graph of Fig. 1 has such a self-loop at node

a as well as a multiple connection from node a to c.

Except for such basic terms as node, arc and connected

graph [l], [3] other concepts needed are defined below.

Dejkition 1: A (directed) loop is a connected subgraph

such that each node has one arc leaving and one arc

entering.

DeJnition W: A self-loop is a loop which consists of a

single node and an arc from and to itself.

De$nition 3: A (simple) path is a connected subgraph

including a pair of nodes called the initial and final nodes,

such that the subgraph, augmented by an arc from the

final node to the initial node if these nodes are distinct,

forms a loop.

The notation xPy is used to indicate that a simple path

exists from node x to y.

Dejkition 4: For a directed graph, a feedback arc set

is a set of arcs which, if removed, leaves the resultant graph

free of directed loops.

Desnition 6: A feedback arc set is minimum if no other

feedback arc set for that graph consists of a smaller

number of arcs.

A directed graph may be presented in any of three

forms: as a geometrical figure, as a set of ordered pairs,

or as a matrix; if as a matrix, it is assumed to be the

comlection matrix. In any matrix form, the nodes must

be listed in some order. Although the order generally

chosen is that which agrees with the alphabetic order of

the labels, this is certainly not necessary. The following

definitions make explicit the dependence of the connection

matrix upon order.

Dejkition 6: A sequential ordering of a graph of n nodes

c

Q

b

Fig. 1-A directed graph.

is a l-l function from the nodes of the graph to the

integers 1, * . . , n.

DeJinition 7: For a directed graph of n nodes with se-

quential ordering R, the connection matrix

c = b-ki~R~i~l”.”

has one row and one column for each node of the graph,

and the entry at row R(i), column R(j), denoted

CR(i)R(i)

= number of arcs from node i to node j.

An example o f a sequential ordering of the nodes of the

graph of Fig. 1 is given by

R(a) = 1,

R(c) = 2, R(e) = 3, R(b) = 4, R(d) = 5

which may also be written

(1, 2, 3, 4, 5) = R(a, c, e, b, d).

The corresponding connection matrix is

(1)

a c e b d

al 2 0 10

c00110

C=el 0 0 0 1.

(2).

b0 0 0 0 1

d-0 1 1 0 O-

The subgraph which consists of one of the arcs from a to c,

the arc from c to e, and that from e to a, form a loop which

is written as (a, c),(c, e) (e, a). When a sequential order is

given, a loop or path may be alternatively expressed in

terms of the order symbols; the loop just given becomes

(1, 2),(2, 3)(3, 1 ) under the ordering given by (1). The

self-loop at node a is denoted by (a, a). One simple path

from node a to d is given by (a, b)(b, d), another by

(a, c),(c, e)(e, d). Arcs (a, a), (e, a), (d, c) and (d, e) form

a feedback arc set (which may be shown to be minimum).

Often a graph contains some special characteristics that

simplify the determination of minimum feedback arc sets.

If a graph contains a source (or sink), i.e., a node for

which the arcs incident upon it are all directed away from

(toward) that node, none of these arcs can be contained

in any minimum feedback arc set. If a node has but one

arc entering and one leaving, any minimum feedback arc

set which contains one of these arcs must not contain the

other; however, another minimum set may always be

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obtained by replacing the one arc by the other. If a

graph is not connected, a minimum feedback arc set for

the over-all graph is obtained by combining the minimum

feedback arc sets for each connected component. These

statements are almost self-evident and may be easily

proved. The discussion to follow does not depend on such

special characteristics; hence, though the discussion is valid

for all graphs, it is most pertinent to graphs that are con-

nected, have no sources or sinks, and have at least three

arcs touching each node.

THE RELATION BETWEEN FEEDBACK ARC SETS AND

SEQUENTIAL OBDERINGS

Lemma 1: A graph G has no directed loops if and only

if the nodes can be given a sequential ordering R such

that for each arc (i, j) of the graph, R(i) < R(j).

Proof: Let xPy be the relation that there exists a simple

directed path from node x to y of G. If G is free of directed

loops, then xPy is a strict partial ordering of the nodes,

i.e.,

xPy is irreflexive, asymmetric and transitive [5].

xPy is irreflexive since xPx asserts the existence of a di-

rected loop involving node x, a contradiction. If xPy and

yPx, then some loop must exist involving node x, a con-

tradiction; hence xPy is asymmetric. If z:Py and yPx,

then either XPZ or the paths from x to y and from y to z

involve some common node other than y, i.t:., there exists

a node w such that wPy and yPw, a contradiction; hence

xPy is transitive.

A .partial ordering may always be extended to a total

ordering;’ thus the nodes can be sequentially ordered so

that for each pair of nodes x, y such that xPy, R(x) < R(y).

For each arc (i, j), iPj and hence R(i) < R(j).

Conversely, any sequential ordering R such that for

each arc (i, j), R(i) < R(j) holds, is a strict total ordering

containing the ordering given by xPy. If G contains a

loop, involving node x say, then xPx, contradicting

irreflexivity. Hence G is free of directed loops,

Q.E.D.

The condition that R(i) < R(j) for each arc (i, j) of a

graph is equivalent to the condition that the connection

matrix has only zero entries below and on the major

diagonal. That the nodes of a graph can be ordered so that

this condition for the connection matrix holds, is equiva-

lent by Lemma 1 to the condition that the graph be free of

loops.

It follows from Lemma 1 that for any sequential ordering

R of the nodes of ‘a graph G, the removal of those arcs

(i, j) for which R(i) >_ R(j) must eliminate all directed

loops. That is, such a set of arcs must constitute a feed-

back arc set. Let F, designate the set

{(i, j), R(i) 2 R(j); i, j E G},

(3)

which may be expressed also as

IW), R(j)),R(i)2 R(j), , je GJR.

(4)

2 See Sierpihski [4], p. 189.

ON CIRCUIT THEORY

June

For the graph of Fig. 1 with the sequential ordering

given by (l),

FR = {(a, a), (e, 4, (d,c), and (d,e )

= ((1, 11, (3, 11, (5,2) and (5, 3>}R

constitutes a feedback arc set. Note that the arcs of F,

correspond to those nonzero entries of the connection

matrix on and below the major diagonal.

It is also true that, for any feedback arc set F, some

sequential ordering R must exist such that F, is a subset

of F. To see this, consider removing all arcs of F from

graph G. The resultant graph contains no directed loops

and hence th nodes may be ordered R so that R(i) < R(j)

for each arc (i, j). If the arcs of F are put back into the

graph, those arcs (i, j) of F which satisfy R(i) > R(j)

constitute a feedback arc set FE which is a subset of F.

Hence, in looking for minimum feedback arc sets, only

those sets given by 1 i, j), R(i) >_ R(j)} for some R,

need be considered. These remarks suffice to establish

the following theorem:

Theorem 1: A minimum feedback arc set for an ar-

bitrary directed graph is given by a minimum set o

arcs ( (i, j), R (i) 2 R(j) } , where the number of elements

of this set has been minimized over all possible sequential

orderings of the nodes of the graph.

Rephrased in terms of connection matrix, the theorem

states: a minimum feedback arc set is given by a minimum

set of arcs corresponding to nonzero entries on and below

the major diagonal of the connection matrix, the mini-

mum taken over all possible orderings of the rows and

corresponding columns.

Corollary to Theorem 1: Given that a set of arcs consti-

tutes a minimum feedback arc set, a sequential ordering

can be found such that {(i, j), R(i) 2 R(j)] is that set

If, as suggested by the above theorem, attention is

confined to feedback arc sets that correspond to sequential

orderings, a new interpretation may be given to the term

feedback arc. For a sequential ordering R, a feedbac

arc is any arc (i, j) for which R(i) 2 R(j). The remaining

arcs are called forward arcs.

PROPERTIES OF MINIMUM FEEDBACK ARC SETS

Theorem W: Any subgraph of a directed graph con-

sisting of the arcs of a min@um feedback arc set has no

directed loops except self-loops.

Proof: Let R be a sequential ordering such that

F, = {(i, j), R(i) > R(j)] is a minimum feedback arc

set. The arcs in FB for which R(i) = R(j) are self-loops;

the remaining arcs (i, j) have the property R(i) > R(j) and

hence, as in Lemma 1, form no directed loops.

It follows from this theorem that an arbitrary directed

graph without self-loops can always be partitioned into

two subgraphs such that each is itself free of directed

loops. This remark can be elaborated to prove that the

edges of any nondirected graph without connection from

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a node to itself can be oriented so that the resultant

directed graph is free of directed loops.

Theorem 3: The removal of the arcs of any minimum

feedback arc set of a connected n-node graph leaves a

subgraph of n nodes that is still connected.

Proof: Let F, be a minimum feedback arc set for an

n-node graph G. Suppose the removal of the arcs of F,

from G leaves a subgraph G’ which contains a part (or

node) isolated from the remaining portion of G’. Since the

over-all graph G is taken to be connected, one or more of

its arcs that are in F, must connect the part isolated in G’

to the remaining portion. The addition of any one of these

arcs to G’ certainly adds no loop to that subgraph, con-

tradicting the minimality of F,. Hence, the assumption of

an isolated part or node in G’ is untenable.

The subgraph G’ obtained by removing the arcs of F,

can thus be described as a maximum connected subgraph

of graph G which contains all the nodes of G and is free

of directed loops. Omitting the word “directed,” this

description is suitable for a tree of a nondirected graph.

The closeness of this analogy has prompted the use of the

term “minimal chord set” for the set of arcs to be de-

leted [Z].

I’ROPERTIES OF THOSE SEQUENTIAL ORDERINGS THAT

CORRESPOND TO MINIMUM FEEDBACK ARC SETS

Dejkition 8: An optimum ordering R is a sequential

ordering of the nodes of a directed graph for which

{ (i, j), R(i) >_ R(j) } is a minimum feedback arc set.

DeJnition 9: For a sequential ordering R of a directed

graph, a consecutive subgraph is any (nonempty) sub-

graph composed of nodes consecutively ordered by R

and the arcs connecting these in the over-all graph.

For the graph of Fig. 1, the feedback arc set F, corre-

sponding to the sequential ordering given by (1) has

already been asserted to be minimum; hence, R is an

optimum ordering for that graph. Examples of consecu-

tive subgraphs under R are the single node c (indeed,

any single node), the subgraph framed on nodes c, e, and

b (which consists of the edges (c, e) and (c, b)), and the

over-all graph.

Theorem ~$1The set of optimum orderings for a given

graph is invariant under the removal of self-loops and

loops involving two arcs.

In the case of multiple connections between nodes,

e.g., say these are p arcs from node i to j, and Q arcs from

node j to i, with p > Q, the theorem should be interpreted

as permitting the removal of up to 4 arcs from i to j

and the same number of arcs from j to i.

Proof: An a rc corresponding to a self-loop is of the

form (i, i) and so will be included in the set FE = { (i, j),

R(i) > R(j)] for any R. A loop involving two arcs is one

of the form (i, j)(j, i), i # j. If R(i) > R(j), the arc (i, j)

is in F, but (j, i) is not; if R(i) < R(j) the arc (j, i) is in

F, but (i, j) is not. In either case one and only one of these

is a feedback arc for any R. Hence the choice of sequential

orderings which minimize F, cannot depend on loops

consisting of only one or two arcs.

In accordance with Theorem 4, two directed g raphs

are said to be order equivalent if the removal of all self-

loops and loops of two arcs from each graph results in

graphs with the same connection matrix. A graph without

such loops is termed reduced and is the representative of

all those to which it is order equivalent. The reduced

graph and hence the class it represents may be expressed

in a simpler form than the connection matrix, as follows:

Definition 10: For a reduced graph of n nodes with

sequential ordering R, the connection triangle

C’ = Niww,

1 I R(i) < R(j) I n}

has one row and one column for each node of the graph,

and GcijRtil =

number of arcs from node i to j minus

number of arcs from node j to i.

For the (set of graphs order equivalent to the) graph

of Fig. 1 with the sequential ordering given by (I), the

connection triangle is

a

L

-11 0

c 1 1 -1

C’=

(5)

d

From this array, the reduced graph may immediately

be obtained as a figure: in this case it differs from Fig. 1

in that it lacks the self-loop at node a and the loop

(e, 4 (4 e).

Theorem 5: Given an optimum ordering R of a directed

graph G, let Gl be any consecutive subgraph:

a) G, must have as minimum feedback arc set those

arcs of G, that are feedback arcs according to R;

b) similarly, the subgraph H obtained from G by

deleting all arcs and coalescing all nodes of G, must have as

minimum feedback arc set those arcs of H that are feed-

back arcs by R.3

Proof: Partition the feedback arc set F, into two

subsets:

Fi = ((i, j), R(i) 2 R(j); i, j EG},

(6)

Fi = {(i, j), R(i) 2 R(j); i or j# G,].

(7)

F$ is the set of feedback arcs by R that are in G,, and

F,” is the set of feedback arcs that are in H. Suppose,

first, that Fi is not minimum: that there exists a set F

containing fewer members whose deletion also leaves G,

free of directed loops. Reorder G, by R’ so that

{(i, j), R ’(i) 2 R’(j); i, j E G,)

(8)

3 Part (b) of this theorem WBS llggested by one of the reviewers;

this part helps to simplify the proof of Theorem 6.

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is equal to F. This reordering of G, does not affect the

membership of F,2 since the nodes of G, are consecutively

numbered by R. It follows that F U F,2 is a feedback arc

set for G with fewer members than F,, a contradiction.

Hence F$ is minimum for G,. Assuming F,” not minimum

for H leads to a similar contradiction.

As an illustration, consider once again the R given by

(l), asserted to be optimum for the graph of Fig. 1.

Using order symbols rather than node labels, the feed-

back arcs are

FR = ((1, I>, (3, 11, (5, 2) and (5, 3) lR.

(9)

If G, is the consecutive subgraph framed on nodes 2

through 5, Fi is {5, 2) and (5, 3) } R = { (d, c) and (d, e) }

and F,” is ((1, 1) and (3 , l))R = ((a, a) and (e , a)}. These

are, by the theorem, the minimum feedback arc sets for

G, and H, respectively.

It follows from Theorem 5 a) that for an optimum

ordering R for a graph G, the number of arcs from node i

to node j, where R(j) = R(i) + 1, is greater than or

equals the number of arcs from j to i. That is, in the con-

nection triangle, ck(i),R(i)+l 2 0. In fact, a stronger

statement that includes this one is made by the theorem

which follows.

Notation: For subgraphs G, and G, of a dnected graph

G which. contain no common node, let ccXc, denote the

number of arcs each of which leaves a node in G, and

enters a node in G,.

Theorem 6: Given an optimum ordering R for a directed

graph G, let G, and G, be consecutive subgraphs of nl

and n, nodes such that the highest numbered node in G,

is one less than the lowest numbered node in G,; then:

a)

CG,G~

2 CC,C~,

b) if cG,,& =

CG,G,, then the ordering R’ is also opti-

mum, where for each node i of G

R’(i) = R(i),

i e’ G,

or G,

R’(i) = R(i) - n,, i E G,

(10)

R’(i) = R(i) + n,,

i e G,.

A diagram illustrating the theorem is shown in Fig. 2.

The number of arcs from nodes of G, to nodes of G, is

CG,G.

= 4, whereas CG~G~ 3. The condition cG,G, >_ CG,G,

is necessary for R to be optimum; of course, this condition

must also hold for all other possible G,, Gz as well.4

Proof: a) Let G’ be the consecutive subgraph which

contains all nodes of G, and G,. The subgraph H’ ob-

tained from G’ by first deleting the arcs and coalescing

the nodes of G, and then deleting the arcs and coalescing

the nodes of G, is a subgraph of two nodes whose forward

arcs are those contributing to cc,& and whose feedback

arcs correspond to CG,G,.

By Theorem 5 a) followed by

4An interesting result by Hakimi [2] states that for any cutset

of a directed graph, the number of arcs cut which are elements of a

minimum feedback arc set is not greater than the minimum of the

number of arcs cut which agree with the cutset orientation and the

number of arcs which oppose the cutset orientation.

5 b), the feedback arc set corresponding to cGSG, s mini-

mum for H’ and hence, cc,,& 2 cGZGI,

b) For CG>G, = cG,G1,

the feedback arc set under R

contains the same number of members as the feedback

arc set under R’, and since R is optimum, so is R’,

Q.E.D.

An examination of the graph of Fig. 1 reveals that the

ordering given by (1) does indeed satisfy this theorem.

It is simple enough to choose for this graph an ordering

which does not, but instead consider the graph of Fig. 3

with the sequential ordering

(1, 2, 3, 4, 5, 6 , 7) = R(f, d, b, a, c, e, g).

(11)

This ordering cannot be optimum; for taking GI the single

node d (with R(d) = 2) and G, the single node b (with

R(b) = 3) gives

CO,G,(=CZ = 0) < CG,G~(=C.Q 1);

see Fig. 4(a). Alternatively, consider G, the single node a

(with R(u) = 4) and G, the consecutive subgraph c, e, g

[with (5, 6, 7) = R(c, e, g)] for which

cG,G,(=c45 + c46 + c47 = 1)

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Fig. 2-Illustration of a property of an optimum ordering.

b

Fig. 3-A directed graph used to illustrate Thcorcm 6.

(a)

(b)

Fig. 4-Applica tion of Theorem 6 to the graph of Fig. 3.

the sequential ordering obtained is

(1, 2, 3, 4, 5, 6, 7) = R’(f, b, 4 c, e, g, 4

(13)

which does satisfy CG,G, 2 cG*G, for all G,, Gz. That an

ordering satisfies this condition is not sufficient for it to

be optimum, but any such ordering appears, intuitively at

least, to be a fairly good one.

DEFINITION ANU A PROPERTY OF AIIMISSIBLE

ORDERINGS

The viewpoint in this section will be broadened to in-

clude feedback arc sets and sequential orderings which,

while possessing the properties discussed in the previous

two sections, may be less than optimum.

DeJkition 11: A feedback arc set for a directed graph

is minimal if it contains no proper subset that is also a

feedback arc set for this graph.

To test a feedback arc set F, of a directed graph G for

minimality, the arcs of F, are removed from G leaving a

subgraph G’ free of directed loops. The arcs of FE are

then added individually to G’; if any of these arcs with

G’ form no directed loops then F, is not minimal; how-

ever, if all such arcs are deleted from F,, the resultant

feedback arc set is minimal. A sequential ordering of the

nodes which determines this resultant feedback arc set

can then be found. In case it is not clear whether an arc

with G’ forms any loop, a simple conclusive test for the

nonexistence of dir&ted loops in a graph may be used;

this test is implied by the lemma which follows, a re-

statement of a result of Hakimi [2].

Lemma 2: A graph is free of directed loops if, and

only if, successively removing source nodes and the arcs

leaving each, exhausts the graph.

In this statement, which follows from Lemma 1, the

order in which sources are removed when a choice exists

does not matter. An equally valid test for the lack of loops

in a graph is the successive removal of sink nodes.

DeJinition lb: An ordering R for a directed graph is

admissible if

a) the graph ordered by R satisfies cG,G. > CG,G, or

all appropriate consecutive subgraphs G,, Gz;

b) the feedback arc set determined by R is minimal.

The first criterion should generally be the more critical,

but does not imply the second. The test of each criterion,

if not satisfied, indicates a simple means to develop an

ordering which does meet the test. Hence, the securing of

an admissible ordering is really an easily accomplished

task and is the starting point from which to search for

optimum orderings. It is convenient to call this admissible

ordering first obtained the admissible reference ordering

and to relabel the graph a’, b’, c’, . . . in accordance with

this reference ordering.

As a further bit of terminology, call an n-tuple of nodes

(i, i . . . ,

Ic) for which (1, 2, . . . , n) = R(i, j, . . . , k)

the n-tuple representation of an ordering R. It is often

convenient to order a set of n-tuples lexicographically,

with the left-most symbol of greatest significance. Lexico-

graphic order depends directly on the labeling used; the

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most suitable labels in this work are those which corre-

spond to the admissible reference ordering.

DeJinition 1s: Two sequential orderings of a graph are

identical with respect to feedbaclc, or F identical if they

determine the same feedback arc set. An F- identical class

of orderings is a set of sequential orderings identical with

respect to feedback. The ordering in an F-identical class

whose representation as an n-tuple is lexicographically

the smallest with respect to the admissible reference

labeling is the F representative.

Definition 1.4: Given a sequential ordering R, a sequent

derived from R is an ordered pair of nodes [i, j] for which

R(j) = R(i) + 1. If node i precedes node j according to

the reference labeling, then [i, j] is an up sequent; if i

succeeds j then [i, j] is a down sequent.

An F-identical class for the graph of Fig. 3 is given by

I

(f, 4 b, a, c, e, d

(f, d, a, b, c, e, d

UN ClKCUl’I’ ‘L’HlCUEY J me

ordering R’ with smaller Q (R’) is sought. If such a ” better”

ordering is obtained it becomes the new reference; this

process is iterated until a reference is obtained which

cannot be improved; such an ordering is optimum. During

this process, those orderings R’ for which Q(R’) = Q(R)

for reference R are noted since if the reference is estab-

lished as optimum, they becom e with the reference the

set of optimum orderings.

I

(f, a, 4 b, c, e, d ‘.

(f, 4 a, c, b, e, d

(f, a, d, 6, b, e, cd,

The third of these tuples is the smallest lexicographically

and hence is the F representative. Using the labeling

given, this third tuple has down sequents

[f,

a] and [d, b]

and up sequents [a, d], [b, c], [c, e] and [e, g].

Theorem 7: To every down sequent [i, 91 derived from

an admissible F representative R,, there corresponds one

or more arcs (i, j) in the reduced graph G.

Note: These arcs are feedback arcs according to the

admissible reference ordering.

Proof: By definition of reduced graph, there cannot

be arcs both from i to j and from j to i in G; suppose that

G contains one or more arcs from j to i. Modifying R,

by reversing the order of nodes j and i alters the corre-

sponding feedback arc set only by removing these arcs,

and thereby contradicts the admissibility of R,. The

second alternative, that no arcs connect nodes i and j,

contradicts (since [i, j] is a down sequent) the designation

of R, as F representative. Hence one or more arcs leave

node i and enter node j.

AN

ALGORITHM FOR FINDING MINIMUM FEEDBACK

AX

SETS

For a directed graph G with ordering R, let Q(R) repre-

sent the number of arcs in F,, the feedback arc set

determined by R. It has been shown that determining a

minimum feedback arc set for G is equivalent to finding

an R for which Q(R) is minimum. In this section, a more

general version of this basic problem is considered: that

of finding all minimum feedback arc sets, or equivalently,

all F representatives R for which Q(R) attains its absolute

minimum.

The search for optimum orderings begins with any

ordering which is admissible; with this as reference, an

In this search, the orderings that are examined are

modifications of the reference ‘ordering. The scope of

modifications which may lead to an improvement are

restricted by Theorem 7 to those orderings whose down

sequents are feedback arcs under the reference ordering.

This restriction leaves, potentially at least, a wide region

of search. But any such modification can be reduced to a

series of perturbations in the ordering, each of which

involves the shift of a single node, other nodes remaining

fixed except to accommodate such a shift. These single

node perturbations are the only direct modificat’ions of the

reference ordering that are considered. A further re-

striction is in fact made that the perturbation must

establish an allowable down sequent. Under this single

node perturbation, some feedback arc under the reference

ordering becomes an arc which enters the node immediately

succeeding the node it leaves.

Since each perturbation of the reference ordering must

establish some feedback arc as a down sequent, there are

at most Q(R) such perturbations for the reference ordering

R. Not all Q(R) possible perturbations need always be

considered since some down sequents corresponding to

feedback arcs under the reference ordering can sometimes

be rejected as inferior to other sequents. In fact, if les

than a complete search is demanded, down sequents can

be rejected as unlikely if they cause a large net change of

forward arcs to feedback arcs.

If one of these initial perturbations does not establish

a new reference, then each of these is further perturbed

in the same way. The search thus branches out and, were

it not for an effective limiting mechanism, would become

unwieldy. This limitation arises from the nature of the

single node perturbations permitted.

A single node perturbation of the type permitted can be

describe as a cyclic shift by one order position of the nodes

of a consecutive subgraph, where the subgraph has,

before shift, a feedback arc connecting its two extreme

nodes. This feedback arc becomes by this cyclic shift a

forward arc between nodes adjacent in order. Of the two

possible directions for this cyclic shift, it is convenient to

choose the one changing fewer forward arcs to feedback

arcs.

Because of the branching nature of the search, once a

down sequent is established, the pair of nodes comprising

it are permanently linked in this relationship: the pair

defines a consecutive subgraph. By Theorem 5, a con-

secutive subgraph can be considered a single node with

respect to the remaining portion of the graph. The pair

may be in fact united to produce a single node; the arc or

arcs between them (forward arcs under the perturbed

8/18/2019 Feedback Arc Set

8/8

Younger: M inimum Feedback Arc Se ts for a Directed Graph

ordering) may then be deleted. Hence each perturbation

reduces by one the number of nodes. This reduction,

favorable in itself, has a significant auxiliary effect.

Uniting two nodes of a graph generally creates loops

comprised of two arcs. By Theorem 4, arcs forming such

loops may be deleted. Each loop deleted inevitably con-

tributes exactly one of its two arcs to any minimum

feedback arc set; hence the number of loops deleted should

be subtracted from Q(R) (where R is the reference order-

ing) to thus make more stringent the requirement for a

perturbed ordering to be perferable to the reference. In

fact, if the number of loops deleted exceeds Q(R), then

the down sequent leading to this reduction must be re-

jected. This provides the mechanism for limiting this

branching search.

Perturbing an ordering generally upsets its admissibility.

Hence after uniting the nodes of the down sequent and

canceling loops comprised of two arcs, rearrangement

sufficient to reestablish admissibility should be made.

Then, un less the rearranged ordering is better than the

reference, further perturbations are considered.

The organization of the algorithm is made explicit

in the outline which follows. The operations on the graph

should be interpreted in terms of the connection triangle

representation.

1) Preliminary: Obtain an admissible ordering R,

relabeling the graph according to this ordering. Find the

feedback arc set determined by R and note the number

of elements in this set Q(R).

2) Branching operation: For each feedback arc, perform

a single node perturbation on R to establish the down

sequent corresponding to this arc. Unite the nodes of this

down sequent into a single node, eliminating loops of

two arcs created by this union. Add the number of such

loops to an index number I, where I indicates for a given

perturbed ordering the total number of such cancellations.

3) Evaluation and transfer:

a) If Q(R) - I < 0, reject the down sequent under

consideration. Return to the next feedback arc in step 2,

resetting I to the number it was before consideration of

this rejected down sequent. If all feedback arcs at a given

branching point have been examined, then proceed back

to the next arc of the branching point of the next higher

level; if at the first level, proceed to step 4.

b) If Q(R) - I 2 0, rearrange the perturbed ordering

until it is admissible. If the number of elements in the

feedback arc set determined by the perturbed ordering

Q(R,,,,) < Q(R) - I, consider it the new reference

ordering, and begin the program again at step 1. If

Q(R,,,,) = Q(R) - I, list R,,,, on a list of potentially

optimum F representatives. If Q(R,,,,) 2 Q(R) - I,

note the feedback arcs determined by this ordering and

for this new level of branching, proceed back to step 2.

4) Final processing: From the reference ordering and all

others on the potentially optimum list, find all optimum

F representatives by the equivalences indicated in

Theorem 6 b). For each optimum ordering, enumerate

the corresponding feedback arc set.

CoNCLUsIoN

A minimum feedback arc set has been shown to be

determined by a sequential ordering of the nodes of a

directed

graph which minimizes the number of arcs

each of which enters a node that precedes in the ordering ’

the node it leaves. If feedback arc sets for all n possible

sequential orderings of an n-node graph are scanned, all

optimum orderings (and hence, all minimum feedback arc

sets) can be identified. For large n (say 30) the number

of possible orderings exceeds reasonable bounds; to analyze

these larger graphs, the domain to be scanned must be

made smaller. This has led to a study of the properties of

sequential orderings which correspond to minimum feed-

back arc sets. These properties have, in turn, led to a

search algorithm. While the efficiency of this algorithm is

not easy to estimate, it appears to be suitable for graphs

of fairly large size; the author found in a few hours by

hand calculation the minimum feedback arc sets for a

12-node 41-arc graph, for which the arcs were chosen by a

random process.

For a graph too large for a complete search, a partial

search can be made by being more selective about the

down sequents to be investigated. For still larger graphs,

one may be content with an admissible ordering. But for

any graph, an ordering as good as time limitations and

other resources permit can be developed.

Although the primary purpose of the paper is to lay a

foundation for an algorithm, it is hoped that the properties

of minimum feedback arc sets and the sequential orderings

which determine them will prove to be useful in them-

selves, and serve to emphasize the fundamental role of

order in the analysis of directed graphs.

ACKNOTVLEDGMENT

A discussion with S. L. Hakimi,’ J. P. Runyon and

E. F. Moore of the Bell Telephone Laboratories helped

to deepen the author’s understanding of the problem.

The author is particularly thankful for the guidance and

support of his doctoral advisor P rofessor 0. Wing.

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Dl

121

131

[41

[51

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[71

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