Crossing Properties of Multiterminal Cuts

Robert F. Easley, David Hartvigsen

College of Business Administration, University of Notre Dame, P.O. Box 399, Notre Dame,Indiana 46556-0399

Received 13 May 1997; accepted 8 February 1999

Abstract: Gomory and Hu proved the following classical result: For any graph with nonnegative edgeweights, there exists a collection of noncrossing cuts that contains a minimum cut for every pair of nodes.In this paper, we show how this result generalizes for a natural multiterminal cut problem. We also showthat our result is “best possible,” for k 5 3, by using a computer to find feasible solutions to several largesystems of linear inequalities. © 1999 John Wiley & Sons, Inc. Networks 34: 215–220, 1999

1. INTRODUCTION

Consider a simple undirected graphG 5 (V, E) withnonnegative edge weights. If (A1, B1) and (A2, B2) are thenode partitions ofV corresponding to two cuts ofG, thenwe say that these cutscross if and only if Ai ù Bj isnonempty, for 1# i , j # 2. Gomory and Hu first provedthe following classical “noncrossing” result about the struc-ture of minimum weight cuts:

Theorem 1 (Gomory and Hu[9]; see also[14]). Thereexists a collection of noncrossing cuts in G that contains aminimum weight cut for every pair of nodes in G.

Two important applications of this result (also due toGomory and Hu [9]) are that there is a “small” number ofdifferent minimum cuts and that there is an efficient algo-rithm for finding these minimum cuts. We discuss theseresults in more detail at the end of this section.

In this paper, we study the extent to which Theorem 1can be generalized. In particular, we consider the following

natural generalization of cuts: Given a setV9 of k nodes ofG, a k-cut of G for V9 is a set of edges whose deletionresults in a graph such that each node ofV9 is in a differentcomponent. Hence, 2-cuts are the usual “cuts” in Theorem1. We also define (in the next section) a natural generaliza-tion of “crossing.”

Unfortunately, we find that the obvious version of The-orem 1 with “cut” replaced by “k-cut” is not true. However,we do find that there always exists a collection ofk-cuts thatcontains a minimum weightk-cut for every set ofk nodes inG where two special types of crossing are ruled out. Go-mory and Hu’s result is a special case of our result. Fur-thermore, we show that our result is the “best possible” inthe sense that no other type of crossing can be ruled out fork 5 3.

An interesting aspect of this work is the crucial roleplayed by a computer. Our first objective in this project wasto discover those types of crossings that can be ruled out fork . 2. As we show in Section 2, even fork 5 3, there area large number (729) of different types of crossings toconsider. To formulate a conjecture, we decided to use acomputer. Toward this end, we showed, fork 5 3, that aparticular type of crossing can be ruled out if and only if anCorrespondence to:D. Hartvigsen

© 1999 John Wiley & Sons, Inc. CCC 0028-3045/99/030215-06

215

associated system of linear inequalities has no feasiblesolution. By checking the feasibility of a number of suchsystems, we identified two simple types of crossing that canbe ruled out fork 5 3. We were then able to construct aproof that these two types of crossing can be ruled out for allvalues ofk. This is our main result (Theorem 4 in Section2). By effectively checking the feasibility of a fairly largenumber of linear systems, we were also able to show, fork5 3, that all types of crossing not ruled out in our maintheorem can occur. Thus, our main theorem is “best possi-ble” for k 5 3 (see Theorem 6 in Section 2).

The remainder of the paper is organized as follows:Details of our main results are given in Section 2 and theproofs are given in Section 3. We state some open problemsin Section 4.

Let us end this section with a quick review of somerelated work in the literature. We also note an easy appli-cation of our result that generalizes another well-knownresult of Gomory and Hu.

Define ak-cut problemto be the problem of finding aminimum weightk-cut for a givenV9. Closely related tothis is thek-split problem, which is the problem of findinga minimum weight set of edges whose deletion splits thegraph intok components. The 2-cut problem has numerousapplications, many of which were reported in [17] and [1].The k-cut and k-split problems have applications to thetraveling salesman problem, VLSI chip design, networkreliability, and multiprocessor scheduling (see [8] and [5]).The complexity of solvingk-cut andk-split problems hasalso been studied. For the casek 5 2, of course, bothproblems are polynomially solvable (see [1]). Fork . 2, itis shown in [5] that the 3-cut problem is NP-hard even onplanar graphs, although thek-cut problem can be solved inpolynomial time on planar graphs ifk is fixed. A moreefficient algorithm for the planark-cut problem (whichmakes use of Gomory and Hu’s results [9]) appears in [10].Surprisingly, it is shown in [8] that thek-split problem ispolynomially solvable for fixedk, although it, too, is NP-hard if k is not fixed. An efficient, randomized, polynomial(for fixed k) algorithm for finding ak-split with high prob-ability appears in [16]. Polyhedral aspects ofk-cuts werestudied in [3, 4, 6]. Note that, in the literature,k-cuts andk-splits are sometimes referred to asmultiterminalor mul-tiway cuts.

Next, we refer to some different but closely relatedgeneralizations of Gomory and Hu’s work [9]. The follow-ing adds to Theorem 1 some interesting information on thesizeof the collection of cuts. Hence, Theorem 1 implies thefollowing, also well-known, result of Gomory and Hu:

Theorem 2(Gomory and Hu[9]). There exists a collectionof uVu 2 1 noncrossing cuts that contains a minimum weight

2-cut for each of theS uVu2 D pairs of nodes in G.

This result is easy to prove as follows: Consider the cutsin the Theorem 1 collection in increasing order by weight.Each cut considered must be a minimum cut for a new pairof nodes (otherwise, delete it from the collection). Since thecuts do not cross, induction yields that there areuVu 2 1 ofthem.

Hassin [13] proved the following partial generalizationof Theorem 2:

Theorem 3 (Hassin [13]). There exists a collection of at

mostSuVu21k21 D (or O(uVuk21)) k-cuts that contains a minimum

weight k-cut for each of theSuVuk D (or O(uVuk)) sets of k nodes

in G.

It is interesting to note that there is the same reduction bya factor ofuVu for generalk-cuts that there was for 2-cuts.Hassin also showed that his result is the “best possible.”Also note that the word “noncrossing” does not appear inTheorem 3; Hassin does not consider this notion in hisresults. Our main result allows us to immediately strengthenHassin’s result by adding a special type of “noncrossing”(see Section 2). Gomory and Hu’s Theorem 2 is then aspecial case. Let us also point out that the proof techniqueused by Hassin is interesting in that it uses vector spaces andis completely different from Gomory and Hu’s approach.This suggests the possibility of finding a new proof ofHassin’s result that makes use of our noncrossing result (seeSection 5).

Another well-known result of Gomory and Hu, whichfollows from Theorem 1, is an algorithm to find the set ofuVu 2 1 minimum 2-cuts referred to in Theorem 2. Inparticular, they showed that the amount of work needed tofind these cuts is bounded by the work of solvinguVu 2 1minimum cut problems (see [9]). Hassin [12] presented an

approach for the analogous problem of finding theS uVu21k21 D

k-cuts described in Theorem 3. Fork . 2, his algorithm

requires the solution of 2S uVu21k21 D k-cut problems plus an

exponential amount of extra work. Hartvigsen [11] showedhow to eliminate the factor of 2 and how the extra work canbe done in polynomial time for fixedk.

Let us point out that the result in Theorem 3 as well asthe algorithmic results of Hassin and Hartvigsen apply to amore general class of problems where the weights ofk-cutsneed not derive from nonnegative weights on the edges of agraph. Hassin showed that, with general weights andk 5 2,his algorithm can be implemented so that the amount ofextra work is polynomial. Also, for the case of generalweights andk 5 2, Cheng and Hu [2] found a differentalgorithm that requires the solution of onlyuVu 2 1 2-cutproblems plus a polynomial amount of extra work. We notethat for general weights even minimum 2-cuts may cross.

216 EASLEY AND HARTVIGSEN

Finally, Kapoor [15] demonstrated a natural noncrossingproperty between the minimum 3-split in a graph and theminimum 3-cuts for certain triples of nodes. His notion ofnoncrossing is different from the one used in this paper. Healso presented an algorithm for finding a 3-split that usesGomory and Hu’s notion [9] of a “cut tree.” This algorithmhas lower complexity than that of the algorithm in [8].

2. MAIN RESULTS

We begin this section with a few definitions. We then stateour main result, which is a generalization of Gomory andHu’s Theorem 1 fork-cuts.

Let G 5 (V, E) be an undirected, simple graph. Fork$ 2, let A1, . . . , Ak be a partition of the node setV, thatis, A1, . . . , Ak are nonempty,A1 ø . . . ø Ak 5 V, andAi

ù Aj 5 B, for all 1 # i , j # k. Let ­( A1, . . . , Ak)denote the set of edges ofG with each end node in adifferent set ofA1, . . . , Ak. A set of edges of this form iscalled a k-cut of G. Two k-cuts ­( A1, . . . , Ak) and­(B1, . . . , Bk) are said tocrossif Ai ù Bj is nonempty, forall 1 # i , j # k. Otherwise, they are callednoncrossing.

Let w : E 3 R1 define nonnegativeweightson theedges ofG. Theweightof a k-cut is the sum of the weightsof its edges.

­( A1, . . . , Ak) is said to be ak-cut for a set of nodes V9# V if eachAi contains exactly one node ofV9. ­( A1, . . . ,Ak) is called aminimum k-cutfor V9 if it is a minimumweight k-cut for V9.

Unfortunately, as we show in the next section, this notionof crossing is not strong enough to use in a generalizedstatement of Gomory and Hu’s Theorem 1, that is, mini-mum weightk-cuts can cross. However, with the followingdefinition, we can strengthen our definition to a type ofcrossing that can always be ruled out:

Consider two crossingk-cutsA 5 ­( A1, . . . , Ak) andB5 ­(B1, . . . , Bk). For every choice ofVA andVB, whereA andB arek-cuts forVA andVB, respectively, we definean incidence diagram, denotedD(VA, VB), as follows:D(VA, VB) is a square array withk rows, labeledA1, . . . ,Ak, and k columns, labeledB1, . . . , Bk. Each of thek2

entries orcellsof the array has a label of the form (Ai, Bj)determined by the row and column in which it is contained.Also, each cell contains zero, one, or two symbols from theset {a, b}. In particular, cell (Ai, Bj) contains the symbola if and only if (Ai ù Bj) ù VA Þ B, and cell (Ai, Bj)contains the symbolb if and only if (Ai ù Bj) ù VB Þ B.In general, an incidence diagram is characterized by theproperty that every row contains exactly onea and everycolumn contains exactly oneb. Hence, for a given pair ofcrossingk-cuts, there arekk different ways of placing theasymbols andkk different ways of placing theb symbols.Thus, there arek2k different incidence diagrams. For thecasek 5 3, Figure 1 contains 16 of the possible 729

different incidence diagrams. For examples for the casek5 2, see [14].

We next define our stronger type of crossing. Note thatthis definition depends on the weights of the graph. We saythat two k-cuts A and B strongly crossif they cross andthere exists setsVA andVB such that

● A is a minimumk-cut for VA;● B is a minimumk-cut for VB;● D 5 D(VA, VB) satisfies either of the following two

conditions:

Condition 1. There exists a row and column ofD such thatevery symbol occurs in this row or column (or both);

Condition 2. Every row and column ofD contains exactlytwo symbols.

We next state our main theorem:

Theorem 4. There exists a collection of nonstrongly cross-ing k-cuts that contains a minimum weight k-cut for each setof k nodes in G.

Corollary 5. Theorem 1.

Proof of Corollary 5. Choose any collection of non-strongly crossing 2-cuts that contains a minimum weight2-cut for each pair of nodes and that is minimal with respectto this property (i.e., such that every cut in the collection isa minimum 2-cut for some pair of nodes). Suppose two cuts

Fig. 1.

CROSSING PROPERTIES OF MULTITERMINAL CUTS 217

A andB in the collection cross. LetVA andVB be pairs ofnodes such thatA andB are minimum 2-cuts forVA andVB,respectively. Observe thatD(VA, VB) must satisfy Condi-tion 1 or 2. This is a contradiction. ■

This result can be strengthened using Hassin’s Theorem3 to give a generalization of Gomory and Hu’s Theorem 2.We state this result next and prove it at the beginning of thenext section. The result follows immediately from Hassin’sTheorem 3 and our Theorem 4.

Theorem 6. There exists a collection of at mostSuVu21k21 D

(or O(uVuk21)) non-strongly crossing k-cuts that contains a

minimum weight k-cut for each of theSuVuk D (or O(uVuk)) sets

of k nodes in G.

Theorem 4 is the “best possible” fork 5 3. Roughlyspeaking, this means that for all types of crossing not ruledout there is a weighted graph such that this type of crossingoccurs in all collections of 3-cuts that contain a minimumweight 3-cut for each set of three nodes. The followingtheorem makes this precise. LetKn 5 (V, E) denote the(undirected, simple) complete graph withn nodes:

Theorem 7. In K9, let A and B be crossing3-cuts for VA

and VB, respectively. Suppose the incidence diagram vio-lates both Conditions1 and 2. Then, there exist weightssuch that A is the unique minimum3-cut for VA and B is theunique minimum3-cut for VB.

Example. Theorem4 tells us that there exists a collectionof nonstrongly crossing3-cuts that contains a minimum3-cut for every set of three nodes. Let A and B be crossingminimum3-cuts for VA and VB, respectively, from such acollection. Consider the16 incidence diagrams in Figure1.The definition of strongly crossing says that the incidencediagram D(VA, VB) cannot be among diagrams1, 2, 9 (i.e.,Condition1) or 14, 15,or 16 (i.e., Condition2). Theorem7says that D(VA, VB) can be among any of the other types ofdiagrams.

In the next section, we prove Theorems 4, 6, and 7. It iswell known that Gomory and Hu’s result can be provedusing the “submodular” properties of 2-cuts. One interestingaspect of our work is that Theorem 4 is proved using a“k-cut version” of this submodular property. Another inter-esting aspect is that the “best possible” result, Theorem 7, isproved by using a computer.

3. PROOFS

We prove Theorems 4, 6, and 7 in this section. Each proofrelies on the following simple but useful definition:

Let us call the weights onKn perturbedif no two k-cutshave the same weight. (If this is not initially the case, wecanperturb the weights by adding multiples of some smallpositive number to each weight. This works because thegraph is complete.) Observe that, when the weights areperturbed, the minimumk-cut for any set ofk nodes isunique and, hence, the collection of all minimumk-cuts isunique. This fact appears to simplify the following proofs.

Although the proof of Theorem 6 depends upon Theorem4, we present it first due to its simplicity:

Proof of Theorem 6. Add edges to the graph to make itcomplete, set the weights on the new edges to be zero, andperturb the weights. Then, the collection of minimum k-cuts,because it is unique, satisfies the size condition by Hassin’sTheorem3 and the nonstrongly crossing condition by The-orem4. ■

We next introduce a notion closely related to the well-known notion of submodularity. For anyk-cut ­( A1, . . . ,Ak), let f( A1, . . . , Ak21) denote its weight. (To simplifythe subsequent notation, we have definedf to have onlyk2 1 arguments.) For any two crossingk-cuts A5 ­( A1, . . . , Ak) and B 5 ­(B1, . . . , Bk), consider thefollowing four terms:

T~ A, B! ; f~ A1 ù B2, A1 ù B3, . . . , A1 ù Bk!

T~B, A! ; f~B1 ù A2, B1 ù A3, . . . , B1 ù Ak!

T~ A! ; f~ A1, . . . , Ak21!

T~B! ; f~B1, . . . , Bk21!.

Proposition 8. Consider two crossing k-cuts A5 ­(A1, . . . ,Ak) and B5 ­(B1, . . . ,Bk) in a graph G. Then,

T~ A, B! 1 T~B, A! # T~ A! 1 T~B!. (1)

Proof. To show that (1) holds, we show that if an edgeof G contributes to the left-hand side of (1), then it contrib-utes to the right-hand side. Observe that each edge thatcontributes toT( A, B) and/orT(B, A) in (1) is of one of thefollowing five mutually exclusive types:

Type 1: Endnodes inA1 ù Bi andAj ù Bl, wherei , j , l$ 2. Type 2: Endnodes inB1 ù Ai andAj ù Bl, wherei ,j , l $ 2. Type 3: Endnodes inA1 ù Bi andA1 ù Bj, wherei Þ j . Type 4: Endnodes inB1 ù Ai andB1 ù Aj, where

218 EASLEY AND HARTVIGSEN

i Þ j . Type 5: Endnodes inA1 ù Bi andB1 ù Aj, wherei , j $ 2.

Every edge of Type 1 contributes to termsT( A, B),T( A), and, possibly,T(B). Every edge of Type 2 contrib-utes to termsT(B, A), T(B), and, possibly,T( A). Everyedge of Type 3 contributes to precisely termsT( A, B) andT(B). Every edge of Type 4 contributes to precisely termsT(B, A) andT( A). Finally, every edge of Type 5 contrib-utes to all four terms. Hence, the inequality holds. ■

Remark 1. For the case k5 2, if we do not require that the2-cuts A and B cross, then a general function f:2V 3 R1

that satisfies the inequality1 is called asubmodular func-tion in the literature(see[7]). Our definition of f is slightlyless general for k5 2, because we require A and B to cross.

Proof of Theorem 4.Add edges to the graph to make itcomplete, set the weights on the new edges to be zero, andperturb all the weights. Note that minimumk-cuts in thenew graph are minimumk-cuts in the original graph. Con-sider the unique collection of minimumk-cuts. We showthat no two cuts in this collection are strongly crossing.

By contradiction, assume that we have two cutsA andBin this collection that are the minimumk-cuts for VA andVB, respectively. Also, assume that they satisfy Condition 1and, without loss of generality, that the symbols are allcontained in rowA1 and columnB1 of D. It follows that

VB ù $A1 ù Bl% Þ B and

VA ù $B1 ù Al% Þ B for l $ 2.

Hence,

­~ A1 ù B2, A1 ù B3, . . . , A1 ù Bk, V \$A1\$A1 ù B1%%!

is well defined and is ak-cut for VB. Similarly,

­~B1 ù A2, B1 ù A3, . . . , B1 ù Ak, V \$B1\$B1 ù A1%%!

is well defined and is ak-cut for VA. SinceA andB are theunique minimumk-cuts forVA andVB, respectively, usingour notation from above, we have

T~ A! , T~ A, B! and

T~B! , T~B, A!.

But summing these strict inequalities contradicts (1) inProposition 8.

Next, assume thatA andB satisfy Condition 2. Clearly,each row and column ofD must contain precisely onesymbola and one symbolb. Thus,B is ak-cut for VA and

A is a k-cut for VB. Because the weights are perturbed, wemay assume, without loss of generality, that the weight ofAis strictly less than the weight ofB. But thenB cannot be aminimum k-cut for VB. The result follows. ■

Let us outline our strategy for proving Theorem 7. Wefirst define a system of linear inequalitiesS(D) for eachincidence diagramD for k 5 3. S(D) has the property thatit is feasible if and only if there exist weights such thatA isthe unique minimum 3-cut forVA and B is the uniqueminimum 3-cut forVB. Thus, it suffices to generate eachdiagramD that violates Condition 1 and Condition 2 andshow thatS(D) is feasible. This is accomplished by writinga computer program that, essentially, generates each ofthese systems and then checks their feasibility with calls toan LP solver. We next describe this system of inequalities:

In K9 5 (V, E), consider two crossing 3-cuts­( A1, A2,A3) and­(B1, B2, B3) for VA andVB, respectively. LetDdenote the associated incidence diagram and letV 5 {( Ai,Bj)} 1#i , j#3 (i.e., the nine nodes ofK9 are the cells ofD).Let a1, a2, a3 denote the cells ofD that contain the symbola and letb1, b2, b3 be the cells ofD that contain the symbolb. Let

Ri 5 $~ Ai, B1!, ~ Ai, B2!, ~ Ai, B3!% for i 5 1, 2, 3

Ci 5 $~ A1, Bi!, ~ A2, Bi!, ~ A3, Bi!% for i 5 1, 2, 3.

Hence,­(R1, R2, R3) is a 3-cut ofK9 associated with therows of D and ­(C1, C2, C3) is a 3-cut ofK9 associatedwith the columns ofD.

We associate a variablexe with eache [ E. For E9 # E,we let x(E9) [ ¥e[E9 xe. Then,S(D) is defined to be thefollowing set of inequalities, plus nonnegativity on thevariables, wherec . 0 is an arbitrary constant:

x~­~R1, R2, R3!! # x~­~X1, X2, X3!! 2 c (2)

for all 3-cuts­(X1, X2, X3) of K9 for { a1, a2, a3}, except­(R1, R2, R3), plus

x~­~C1, C2, C3!! # x~­~Y1, Y2, Y3!! 2 c (3)

for all 3-cuts­(Y1, Y2, Y3) of K9 for { b1, b2, b3}, except­(C1, C2, C3).

We next make three remarks concerning a key propertyof these systems, the size of these systems, and how manysuch systems we need to consider. This is followed by atheorem whose validity immediately implies Theorem 7.

Remark 2. Consider K9. Let A and B be crossing3-cuts forVA and VB, respectively, with incidence diagram D. Byconstruction, S(D) is feasible if and only if there exist

CROSSING PROPERTIES OF MULTITERMINAL CUTS 219

weights such that A is the unique minimum3-cut for VA andB is the unique minimum3-cut for VB.

Remark 3. There are728 inequalities of form2 and 728inequalities of form3. To see this, consider the inequalitiesof form2. Each3-cut ­(X1, X2, X3) of K9 that separates{ a1,a2, a3} can be viewed as an assignment of the six cells inV\{ a1, a2, a3} to a1, a2, or a3. Excluding the assignment thatyields R1, R2, R3 results in36 2 1 5 728 assignments.

Remark 4. With the aid of a computer, we explicitly gen-erated the729 diagrams for k5 3. We found that exactly612of these diagrams violate Condition1 and Condition2.

Theorem 9. Let k 5 3 and let D be an incidence diagramthat violates Condition1 and Condition2. Then, S(D) isfeasible.

Proof. We effectively generated and checked the feasi-bility of the systems corresponding to the 612 diagrams inRemark 4 using the software package Maple [18]. Actually,we reduced this work significantly by expoliting some sym-metries in the incidence diagrams. In the end, it sufficed tocheck only the 10 diagrams in Figure 1 that violate Condi-tion 1 and Condition 2. ■

Proof of Theorem 7.The result follows immediatelyfrom Remark 2 and Theorem 9. ■

4. OPEN PROBLEMS

The notion of strongly crossing introduced in this paper fork-cuts suggests the following three open problems, whichfollow naturally from the work of Gomory and Hu [9]:

1. Use the nonstrongly crossing result in Theorem 4 toconstructively prove Hassin’s Theorem 3.

2. Use the nonstrongly crossing result in Theorem 4 as the

basis of an algorithm for finding theS uVu21k21 D non-

strongly crossing minimumk-cuts described in Theo-rem 6, where this algorithm requires the solution of

S uVu21k21 D minimum k-cut problems and the additional

work is polynomial, for fixedk.

3. Can “3-cut” in Theorem 7 be replaced with “k-cut”?

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220 EASLEY AND HARTVIGSEN