disconnecting sets in single and two-terminal-pair networks

Disconnecting Sets in Single and Two-Terminal-Pair Networks *

Frieda Granot

Faculty of Commerce and Business Administration, The University of British Columbia, Vancouver, B.C., Canada V6T 122

Michal Penn

Faculty of Industrial Engineering and Management, Technion-Israel Institute of Technology, Haifa 32000, Israel

Maurice Queyranne

Faculty of Commerce and Business Administration, The University of British Columbia, Vancouver, B.C., Canada V6T 122

We consider mixed networks, which may include both directed and undirected edges. For a nontrivial vertex subset S, an S-disconnecting set is a set of edges and vertices which intersects every path from any vertex in S to any vertex not in S. Given nonnegative edge and vertex costs, we show that the minimum cost of an S-disconnecting set defines a submodular function. This implies that the set of all S inducing minimum- cost disconnecting sets is the set of closures of a binary relation, thus extending Picard-Queyranne’s (1 980) result on ordinary minimum cuts. We apply this result to two-pair multicommodity problems in undirected networks, extending Hu’s (1 963) result to disconnecting sets that may include vertices as well as edges. These results and a result of Provan and Shier (1 994) may be used for generating all sets S that induce such minimum-cost disconnecting sets and ranking such sets in order of corresponding costs, for both one-pair problems in mixed networks and two-pair problems in undirected networks. 0 7996 John Wiley & Sons, Inc.

1. INTRODUCTION

In a communication network, vertices (nodes) as well as edges (links) may be subject to failures or may be disabled by an opponent, with the effect of interrupting all communications from some vertex or vertices, the emitter( s), to some other vertex or vertices, the receiver( s). Some edges in a communication network may be directed, e.g., a single channel, and traffic can only follow a prescribed direction along the edge. Other (multichannel) edges are undirected

*This research was partially supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada and by the B. and G. Greenberg Research Fund (Ottawa).

and may be used in either direction. We consider mixed networks, composed of directed and undirected edges, in which vertices as well as edges may fail. For any proper subset of vertices S , an S-disconnecting set is a set of vertices and/ or edges whose failure interrupts all communications between any vertex in S and any vertex not in S. A disconnecting set is an S-disconnecting set for some proper vertex subset S. A mixed cutset is a disconnecting set which is minimal, i.e., any proper subset of which is not a disconnecting set [ 6 ] . We study properties of such disconnecting sets in mixed networks with vertex and edge failures. In particular, we consider problems of finding all minimum cost subsets of vertices and edges that disconnect prescribed pairs (emitter, receiver) of terminals.

NETWORKS, VOl. 27 (1996) 117-1 23 0 1996 John Wiley & Sons, Inc. CCC 0028-3045/96/020117-07

117

118 GRANOT, PENN, AND QUEYRANNE

The study of disconnecting sets in networks has a relatively long history. A project for evaluating the ca- pacity of the eastern European rail network to support a large-scale conventional war and the effort required for interdiction, formulated by U.S. military Generals F. S. Ross and T. E. Harris, motivated the interest of L, R. Ford and D. R. Fulkerson in network flows and led to their discovery of the celebrated Maximum Flow Minimum Cut Theorem [ 61. This result essentially solved the problem of finding one minimum cost mixed cutset, i.e., a minimum cost edge and/or vertex subset disconnecting one pair of terminals in a directed, undirected, or mixed network. See the books by Frank and Frisch [ 61, Shier [ 251, and Colbourn [ 21 for reviews and references on these aspects of communication with respect to networks.

In a network N = (V , E ) with vertex set V and edge set E , an s-t cutset is a minimal set of edges which disconnect vertex s from vertex t . An s-t cut is the set of all edges connecting a vertex subset X with s E X and t @ X to its complement 2 = V\X. We say that vertex set X induces this edge set. Given edge costs (or “weights” or “capacities”), a minimum cost s-t cutset is an s-t cutset with minimum total cost. When all edge costs are positive, a minimum cost s-t cutset is also a minimum cost s-t cut, which we call more simply a minimum cut. Picard and Queyranne [ 201 reviewed a number of (sometimes unexpected) applications of minimum cuts in networks.

The problem of finding all minimum cuts in a network may be difficult, since the output may be of exponential size. Picard and Queyranne [ 191 used the complementary slackness conditions in the Maximum Flow Minimum Cut Theorem to present a characterization of and an ef- fective generation method for all vertex subsets that induce minimum s-t cutsets in a network. For this, they defined a binary relation on the vertex set, such that a vertex subset induces a minimum cut if and only if it is a closure of that relation. See [ 22, 26, 271 and the references in [ 19, 211 for some methods for enumerating all closures of a binary relation. Provan and Shier [ 2 11 extended Picard- Queyranne’s approach and provided a general paradigm for listing various types of cuts in a network. In particular, they showed how to generate, in an n-vertex network all minimum cost cutsets in O( n) time per cutset, whereas Picard-Queyranne’s method may require an exponential time per cutset.

Related problems include the ranking of cuts in non- decreasing order of their costs; multiterminal problems of minimum cuts between all pairs of vertices; and multicommodity problems of cuts simultaneously separating several pairs of terminals. For cut ranking problems, Ha- macher et al. [ l l ] (see also [12, 131) showed, for any integer K , how to efficiently generate a ranked list of K vertex subsets that induce s-t cuts with the smallest costs. They also showed how to generate K s-t cutsets with

smallest cost, but their procedure is exponential in K. The existence of a purely polynomial algorithm for the latter ranking problem is an open question. The multiterminal cut problem is that of finding a minimum cut between every pair of vertices in a network. For an undirected network, this problem is solved by the celebrated “cut tree” result of Gomory and Hu (see, e.g., [9]). Granot and Hassin [ 101 showed how to find a minimum- cost mixed cutset between every pair of vertices in an undirected network using only n - 1 minimum cut com- putations. These methods also apply to determining global minimum cuts, i.e., cuts that separate some (unspecified) pair of terminals. In addition to these flow-based ap- proaches, new methods based on contraction and/or probabilistic ideas have recently been developed for global minimum cut problems; see David Karger’s dissertation [ 161 for a detailed presentation and discussion. Based on Provan and Shier’s paradigm, Kanevsky [ 15 ] presented an algorithm for finding, in an undirected unweighted graph, all minimum cardinality separating vertex sets, i.e., sets of vertices whose removal disconnects the graph. His algorithm runs in O( n) time per separating vertex set, if one ignores an initial preprocessing that is polynomial in the size of the network. Multicommodity problems arise in networks where a number of terminal pairs (s, , t, ) are given. In contrast to multiterminal problems, the failure of all elements in a multicommodity disconnecting set simultaneously interrupts communication between s, and t, for all i . Hu [ 141 showed that, in an undirected network with two terminal pairs, a minimum-cost disconnecting set can be obtained by solving two ordinary minimum cut problems. Provan and Shier [ 2 I] extended their paradigm to the listing of certain types of multicommodity cuts where all terminal pairs (s, , t, ) have a common element s, = s for all i .

In this paper, we show that Picard-Queyranne’s characterization of all minimum cuts extends to mixed cuts in mixed networks and to undirected two-pair multicommodity cut problems. In Section 2, we show that this characterization may be derived from a direct combination of three simple facts: the cut function of a network is submodular (0. Ore, see [28]); the set of minimizers of a submodular function on a lattice is a sublattice thereof (0. Ore, Ibid. ); and a set sublattice is the set of closures of a binary relation (Birkhoff [ I ] ) . In particular, we es- tablish the fundamental submodularity property of the cut function when vertices as well as edges may be included in the cut. Submodularity is one of the deepest and most useful properties in combinatorial optimization, and the characterization of all minimizers is but one of its important consequences. See [ 7,8, 17,28,29] for further discussion of submodularity and related properties. In Section 3 , we briefly review a classic transformation that reduces a minimum mixed cutset problem in a mixed network to a minimum cut problem in a directed network.

DISCONNECTING SETS IN MIXED NETWORKS 119

This transformation may be used to enumerate, as in [ 2 1 1 , all vertex subsets that induce minimum-cost s-t disconnecting sets. In Section 4, we consider two-pair multicommodity cut problems in undirected networks. We show that in undirected networks all minimum-cost two- pair mixed cutsets can be generated in O( n ) time per s- t mixed cutset. We extend Hu’s result [ 141 to the case of mixed cutsets and cuts and show how the methods of Section 3 also apply to two-pair multicommodity cut enumeration problems.

2. DISCONNECTING SETS IN A MIXED NETWORK

A mixed network N (V, E ) consists of a vertex set V and an edge set E. An edge ( i , j ) is a pair of vertices i and j , called its endpoints, and it may be directed or undirected. A directed edge is an ordered pair of vertices i and j , where i is its tail and j is its head; the edge is said to befrom i to j . An undirected edge is an unordered pair of vertices i and j and is said to be from i to j , as well as from j to i . Where S , T E V are two vertex subsets, ( S , T ) denotes the set of all edgesfrom S to T , i.e., those having one endpoint i E S and the other j E T , where i is the tail of the edge if it is directed.

A path P in N is an alternating sequence P - ( vl , e l , v2, . . . , v,) of vertices and edges in N with the following properties: ( i ) p 2 1 and P begins and ends with a vertex; and (ii) for all k = 1, . . . , p - 1, edge ek is from vk to v k + l . We say that P is a path from v1 to v,, or a vI-v, path. If v1 E S and v, E T , we also say that P is a path from S to T , or an S-T path.

V\S, its complement. A set C c V U E is S-disconnecting iff every S-S path contains at least one element of C.

Let S C Vbe a nontrivial vertex subset, and

Lemma 2.1. Let S’ and S” be two vertex subsets in a mixed network N = (V, E ) . Let C‘ = V‘ U E’ and C“ = V” U El’ be an S‘-disconnecting and S”-disconnecting set, respectively, where v’, v‘’ g V and E’, E” c E. Let T & S’ n S” and U S‘ U S”. Let

W - (v’ n vi’) u (v i n sif) u (vif n s’) x- ( v n vV)u(v inS”)u(v i ’nS) F

G = ( U , a) n (E‘ U E”).

( T , T ) f l (E’ U El’)

Then, W U F is a T-disconnecting set if T # kT and X U G is a U-disconnecting set if U # V.

Proof ( i ) Let P be an i- j path with i E T and j E T. Let v k be the last element of T o n path P , starting from i . Thus, we may assume, w.l.o.g., that vk+l 4 S’. We shall

prove that either v k E Wor ek E F , or vk+l E W , showing that W U F intersects P in either case. Indeed, we must have either vk E V’ or ek E E‘ or vk+l E V’, for, otherwise, the S’-p path P‘ = ( v k , ek, v k + l ) would not intersect C‘. If vk E V’, then v k E I/’ n S” c W. If ek E E‘, then ek E ( T , T ) n E‘ c F. The remaining case is where vk+l

E V‘. If vk+l E S”, then vk+l E W. Otherwise, vktl 4 S”. But, then, path P‘ is also an S’-s” path, and it must intersect C“. Now, as above, if vk € V”, then vk € V” n s‘ E W , while if ek E E”, then ek E ( T , T ) n E” E F. Finally, if vk+l E V”, then Vk+l E V’ n V” _c W.

(ii) Observe that a U-disconnecting set in N corresponds to a a-disconnecting set in the network obtained by reversing all the edges of N . Using the above observation and part ( i ) of the proof, it follows that X U G is a U-disconnecting set. This completes the proof.

A real-valued function f : 2’- R is submodular if and only if it satisfies the submodular inequality

for all subsets S‘, S“ E V. Let c : V U E - R define vertex and edge costs, and let c [ A ] = CaEAc[ a ] for any subset A c V U E . Defineg: 2 Y - R by

g[ S ] - min { c[ C] : C is an S-disconnecting set } ,

with g[O] = g[ V ] = 0. The following result generalizes to mixed networks (viz., networks including both directed and undirected edges) and to vertex costs a classic result attributed to 0. Ore (see [ 281, p. 3 1 1 ).

Proposition 2.2. If all edge costs c[e] are nonnegative, then g is submodular.

Prooj Let C‘ = V‘ U E’ and C“ = V‘‘ U E“ be S’- disconnecting and S“-disconnecting sets, respectively, such that g[ S‘] = c[ C‘] and g[ S”] = c[ C”]. First note that C‘ U C“ contains a disconnecting set for S’ n S” and one for S‘ U S”. Therefore,

AS’n S”] I g[S’] + g[S’’] and

g[ S’ u S”] I g[ S’] + g[ Y ] .

This implies that the submodular inequality holds when- ever S’ n S” = 0 or S’ U S” = V . Otherwise, S’ f l S” # J3 and S’ U S“ # V and let W , X , F , and G be as defined in Lemma 2.1. Thus, Lemma 2.1 and the definition of g imply that g[ S’ f l S”] I c[ W U F ] and g[ S’ U S’] s c [ X U GI. Therefore,


( g [ s u s ~ ~ ] + g [ s n ~ ” 1 ) - ( g [ s ] + g [ s ~

I ( c [ W U F ] + c[XU G I )

- (c[V’ U E’] + c[V” U E ” ] )

= - c [ ( (S ‘ \S , S”\s’) u (S”\Sl, SI\SN))

n (E‘ u ~ 7 1 I 0,

where the last inequality follows from the nonnegativity of the edge costs. The proof is complete.

Submodular functions have a large number of properties that can be used for solving a variety of related problems (see [ 17 ] and the monograph [ 8 I ) . In the rest of this paper, we consider problems of characterizing and enumerating disconnecting sets with minimum total cost.

A sublattice L of 2 ’ is a set of subsets of V such that S n T E L a n d S U T E L f o r a l l S , TE&.Following Picard [ 1 81, a closure S for a binary relation R defined on V is a subset of V such that for all i , j E V, iR j and j E S imply i E S. Closures form a fundamental class of combinatorial objects arising in numerous applications [ 2 0 ] . Closures are also called down sets [ 4 ] , ideals [26, 27],* initial sets, hereditary sets, or selections; see [ 201 for additional discussion and references.

Corollary 2.3. I f & is a sublattice of 2 ‘and all edge costs c[e] are nonnegative, then ( i ) the set of all optimal solutions to min { g[ S ] : S E L } is a sublattice of 6, and ( i i ) this set is that of all closures for a binary relation over 6.

Proof: Claim ( i ) follows directly from Proposition 2.2 and the classic result, also attributed to 0. Ore ( [ 281, p. 3 13), that the set of all minimizers of a submodular function on a (sub)lattice L is a sublattice of 6. Claim ( i i ) follows from Birkhoff’s Representation Theorem for Fi- nite Distributive Lattices [ 1 1 , which implies an equiva- lence between subset sublattices and sets of closures of binary relations (see also [ 3 , 231.

The binary relation in part (ii) of Corollary 2.3 depends on the sublattice L. A special case is detailed in the next section.

3. ENUMERATING MINIMUM COST MIXED S-t CUTS IN A MIXED NETWORK

A special case of Corollary 2.3 arises when L is the sublattice of all vertex subsets that contain a given vertex s, called the source, and do not contain another given vertex

*Note that the term ideal has a more specialized meaning in lattice theory [4], where it is also required to be closed under the lattice join operation.

t , called the sink. In reference to the communication application discussed in the Introduction, we are now in- terested in all minimum-cost ways of interrupting all traffic from s to t by disabling vertices and edges. An s-t mixed cutset is an S-disconnecting set C for some S E L. When all edge costs are nonnegative, Corollary 2.3 extends the main result in [ 191 to mixed networks with vertex costs. To simplify the discussion and avoid trivial special cases, we assume in the sequel that all edge and vertex costs are positive. Note that all vertices in S with negative cost must be included in every minimum cost S-disconnecting set. Furthermore, any subset of vertices in S with zero cost and edges with zero cost may be included in a minimum- cost S-disconnecting set.

Classic transformations ( [ 61, pp. 23-25) reduce the problem of finding one minimum-cost s-t mixed cut in a mixed network to that of finding a minimum s-t cut in a directed network fi A ( p, l?) , defined as follows: Let = V’ U V”, where V’ and V” are two disjoint copies of the vertex set V, with elements denoted i‘, j’, . . . E V’ and i“, j”, . . . E V”, respectively. The edge set l? consists of ( i ) an edge (i”, j ’ ) with cost qi”, j’] = c[e] for every directed edge e = ( i , j ) E E ; (ii) two oppositely directed edges (i”, j ‘ ) and ( j” , i’) , with costs qi”, j ’ ] = qj”, i‘] = c[e] , for every undirected edge e = ( i , J ) E E ; and (iii) an edge (i’, i”) with cost c [ i ] for every vertex i E V. An s-t” cut in 6’ is the edge set (3, 5) for some subset 3 c p with s‘ E 3 and t“ 4 3. Given a minimum cost s’-t“ cut (3, s) in fi, it is easily verified that for all i” E 3, also including i‘ into 3 yields a minimum cost s‘-t” cut in fi. Thus, letting S’ (respectively, SN) denote the image in V of 3 fl V’ (respectively, of 3 n V”) , we may assume that S” G S‘. Then, a minimum s-t mixed cut C in N is defined by C A A U B with A S\SN and B = (S”, s). This mixed cut is an S-disconnecting set, where S is the image of S’ in V . Conversely, any minimum s-t mixed cut C = ( A U B ) in N defines a minimum cost s’-tS cut (3 , s) in 3, where = S‘ U S”, S’ is the image of S and S“ that of S\A, and S E L is such that C is an S-disconnecting set. (Note, however, that these transformations do not trivially imply the submodularity of the function g , because of the ‘‘min” operation in the definition of g, which does not preserve submodularity in general.)

An sl-t’lflow x : l?- R in fisatisfies the balance equa- tions

for all v E f \ { s’, t”} . A maximumflow x is an s‘-t” flow x such that 0 I x[ e“] I 821 for all e“ E and the netflow Ce=(sc ,u)E~x[e] = C e = ( u , t ” ) E ~ ~ [ e ] is as large as possible. By the Maximum Flow Minimum Cut Theorem and the main result in [ 191, it follows that a relation R on f m a y be defined as follows, given any maximum flow x in fi:


Then, there is a one-to-one correspondence between the set of all closures 3 for R with s‘ E s and t“ 4 3 and that of all minimum-cost sf-tS cuts ( 3 , s ) . ~ 1 1 minimum s-t mixed cuts C in N are then obtained from such cuts with S” E S’ by the transformation indicated above.

As indicated previously in the Introduction, in the case of directed graphs, there may be Cn exponential number of minimum cost sl-t’’ cuts (3, S ) that represent a particular s’-t” cutset. Provan and Shier [ 2 I] remedied this problem by developing a paradigm for generating all s‘- t” cutsets in O( I V l ) timeper cutset. In their paradigm, they created a unique (3 , S ) s’4‘ cut for each s’-t” cutset. Now, since the complexity of the classic transformation described above is O( I E 1 ) and this transformation is used only once at the initial preprocessing step, one can use the paradigm developed in [ 2 11 to obtain the following:

Theorem 3.1. Let N = (V, E ) be a mixed network and let C : V U E + R, be a nonnegative cost function. Then, all minimum-cost s-t mixed cutsets can be generated in O( 1 V I ) time per s-t mixed cutset.

The ranking methods of [ l l ] also combine in a straightforward manner with the above transformation and allow the ranking of K minimum cost s-t cutsets in a mixed network. The details are left to the reader.

4. DISCONNECTING SETS IN UNDIRECTED TWO-PAIR NETWORKS

Let N = (V, E ) be an undirected network, i.e., edge set E contains no directed edge. Let (sI, t l ) and ( s2 , t 2 ) be two terminal pairs in the network N . A (vertex and edge) subset C E V U E is a two-pair mixed cutset if it intersects every sl-tl path and every s2-t2 path. Even in the case of cutsets composed of edges only, we cannot directly generalize the previous results because the vertex sets that induce two-pair cutsets need not form a sublattice. For example, if the four terminals are all distinct, each of { sI , s 2 ) and { sI, t 2 } induces a two-pair cutset but their inter- section { sI ) induces an sl-tl disconnecting set which need not intersect every s2-t2 path. In the rest of this paper, we show how to use the preceding results indirectly to generate all minimum-cost two-pair mixed cutsets in an undirected network with positive costs.

The following result generalizes Hu’s observation [ 141 that, in an undirected network, a minimum-cost two-pair cutset is of the form ( S , s) with S containing exactly one of sI and t l and exactly one of s2 and tZ . We find it con- venient to express our result by adding edges to enforce requirements that certain vertices remain connected after

the failure of all elements in a disconnecting set. (An alternative, as in Hu’s original result, is to coalesce the corresponding vertices to a single vertex. This alternative implies slightly more complicated changes to the network than the addition of a few edges as below.)

Proposition 4.1. Let C be a minimal subset of V U E which intersects every sl-tl path and every s2-t2path. Then, either

( 2 ) C intersects every sl-tl path in N 1 = (V, E l ) , where

( i i ) C intersects ever-v sl-t, path in N 2 = (V, E 2 ) where E 2 = E U ((31, t 2 ) , ( t i j s 2 ) } . Proof: By contradiction, assume that neither ( i ) nor

(ii) hold. Then, there exists an sl-tl path P in N ’ which does not intersect C. This path P must contain either edge ( S I , SZ), with ( $ 1 , sd 4 E , or edge ( t l , t 2 ) , with ( t l , t 2 ) 4 E , but not both (for, otherwise, P would contain an s2-t2 path in N which does not intersect C, contradicting the assumption that C is a two-pair cut in N ) . W.l.o.g., assume that P contains edge ( sI , s2) , i.e., P = (sl , ( s1 , s2) , s2, . . . , t l ) , so that P contains an sZ-tl path P‘ in N which does not intersect C. Since (ii) does not hold, there exists an sl-tl path Q in N 2 which does not intersect C. As above, this path Q must contain either edge (sI, t2 ) , with (sl, t 2 ) 4 E , or edge ( t , , s2) , with ( t l , s2) 4 E , but not both. In the former case, Q contains a t2-tl path in N which, together with P‘, formst an s2-t2 path in N which does not intersect C. In the latter case, Q contains an sl- s2 path in N which, together with P’, forms an sl-tl path in N which does not intersect C. Thus, in either case, we have a contradiction with the assumption that Cis a two-

This result suggests a simple application of the earlier results to enumerate all minimum-cost two-pair mixed cutsets in an undirected network with positive costs. As- sign a large cost, say M = CeEE c[e] + 1, to each of the edges (sl, s2) and ( t l , t2 ) in N 1 and edges (sI, t 2 ) and ( t l , s2) in N 2 . Find the minimum cost y’ of an sl-tl mixed cutset in N’ for i = 1 and 2, and let y A min{yl , y’}. If yl = y, then enumerate all minimum cost sl-tl cuts in N ’ . If y 2 = y, then enumerate all minimum cost sI- t , cuts in N 2 . This procedure yields all minimum-cost two-pair mixed cutsets in N .

By this transformation and Theorem 3.1, we obtain the following :

Theorem 4.2. Let N = (V, E ) be an undirected network, and C : V U E + R,, a nonnegative cost function; then,

El = E U { (SI, S Z ) , ( t l , t2 )} ; or

pair cut in N . The proof is complete.

+Note that we are using Q in the reverse direction. This is the only place in this proof where we use the assumption that the network N is undirected.


all minimum-cost two-pair mixed cutsets can be generated w in O( I V 1 ) time per s-t mixed cutset.

Another application of Proposition 4.1 is to the ranking of two-pair mixed cutsets and of vertex subsets that induce two-pair mixed cutsets. For either problem, apply in par- allel the corresponding ( one-pair ) cut ranking method from Hamacher et al. [ 1 I ] to each of the two networks in Proposition 4.1 and merge the results on-line to produce the requisite ranked list. The computational time required is about twice the time required for the corresponding one-pair problem.

We can further generalize the results to mixed undirected networks with k terminal pairs ( s,, t, ), i = 1, . . . , k, if there exists a vertex s which is in every pair (s, , t, ) (see Seymour [24]), say, s, = s ( 1 5 i I k ) . Construct G' as follows: Add a super sink f to G, and for each 1 I i I k , add an edge joining f and t, . Give this edge a cost M = C e E E c[e] + 1. Then, to every cutset in G' separating s from f and with cost less than M corresponds a cutset in G separating all pairs (s, , t, ) and with the same cost; and vice versa, to every cutset in G separating all pairs (s, , t, ) corresponds a cutset in G' separating s from f and with the same cost.

We can apply a similar construction to obtain all minimum-cost cutsets when there are two vertices s, s' such that { s, s ' } meets each pair (s,, t , ) once. In this case, we reduce the problem to the two-pair network problem.

We conclude by giving two examples showing that Proposition 4.1 does not generalize directly to networks containing directed edges (Fig. 1 ) or to undirected networks with more than two terminal pairs (Fig. 2 ) , even when cutsets are composed of edges only. In Figure 1, the directed edge ( a , 6 ) forms a two-pair cutset, but it does not intersect an sl-tl path [sl, (sl, sz), s2, ( s 2 , t l ) , t l ] in N1noransl- t l path [ s l , ( s l , t 2 ) , t 2 , ( t2 , t l ) , t l ] i nN2 .1n Figure 2, assume that all edge costs are equal. Then, any two-edge subset intersects all s,-t, paths in this network

Fig. 1. A directed two-pair network.

s1 = tz

9 h

Fig. 2. An undirected three-pair network.

for all i = 1, 2, 3, and it does so at minimum total cost. Yet, such an edge subset cannot be obtained as a minimum cost s,-t, cutset when any set of edges connecting terminals is added to the network.

The authors gratefully acknowledge helpful and constructive remarks made by an anonymous referee on an earlier version of this paper.

REFERENCES

G. Birkhoff, On the combination of subalgebras. Proc. Camb. Philos. Sac. 29 (1933) 441-464.

C. J. Colbourn, The Combinatorics ofNetwork Reliability. Oxford University Press, Oxford ( 1987). W. H. Cunningham, Minimum cuts, modular functions, and matroid polyhedra. Networks 15 (1985) 205-215. B. A. Davey and H. A. Priestley, Introduction to Lattices and Order. Cambridge University Press, Cambridge (1990). L. R. Ford, Jr. and D. R. Fulkerson, Maximal flow through a network. Can. J. Math. 8 (1956) 399-404.

H. Frank and I. T. Frisch. Communication, Transmission and Transportation Networks. Addison-Wesley, Reading, MA (1971). S. Fujishige, Submodular systems and related topics. Math. Program. Study 22 ( 1984) 1 13- 13 1. S. Fujishige, Submodular Functions and Optimization. North-Holland, Amsterdam ( 199 1 ). R. E. Gomory and T. C. Hu, Multi-terminal network flows. SZAM J. Appl. Math. 9 ( 1961 ) 551-570. F. Granot and R. Hassin, Multi-terminal maximum flows in node-capacitated networks. Discr. Appl. Math. 13 (1986) 157-163.


H. W. Hamacher, J.-C. Picard, and M. Queyranne, Ranking the cuts and cut-sets of a network. Ann. Discr. Math. 19 (1984) 183-200. H. W. Hamacher, J.-C. Picard, and M. Queyranne, On finding the K best cuts in a network. Oper. Res. Lett. 2

H. W. Hamacher and M. Queyranne, K best solutions to combinatorial optimization problems. Ann. Oper. Res. 4 (1985) 123-143. T. C. Hu, Multi-commodity network flows. Oper. Res.

A. Kanevskey, Finding all minimum-size separating vertex sets in a graph. Networks 23 ( 1993) 533-541. D. R. Karger, Random sampling in graph optimization problems. PhD Dissertation, Stanford University (Oc- tober 1994). L. LovPsz, Submodular functions and convexity. Math- ematical Programming-The State of The Art (A. Bachem, M. Grotschel, and B. Korte, Eds.). Springer- Verlag, Berlin (1983) 235-257.

J.-C. Picard, Maximum closure of a graph and application to combinatorial problems. Manug. Sci. 22 ( 1976) 1268- 1272.

J.-C. Picard and M. Queyranne, On the structure of all minimum cuts in a network and applications. Math. Program. Study 13 (1980) 8-16. J.-C. Picard and M. Queyranne, Selected applications of minimum cuts in networks. ZNFOR 20 ( 1982) 394-422.

(1984) 303-305.

11 ( 1963 ) 344-360.

J. S. Provan and D. R. Shier, A paradigm for listing (s, t)-cuts in graphs. Algorithmica, to appear. G. Pruesse and F. Ruskey, Generating linear extensions fast. Paper presented at the Sixth SIAM Conference on Discrete Mathematics, Vancouver ( 1992 ).

M. Queyranne, Anneaux achevCld'ensembles et prkordres. Technical report EP77-R- 14, Ecole Polytechnique de MontrCal, QuCbec, Canada ( 1977). P. D. Seymour, Four-terminal flows. Networks 10 ( 1980)

D. R. Shier, Network Reliability and Algebraic Structures. Clarendon Press, Oxford ( 199 1 ) . G. Steiner, An algorithm to generate the ideals of a partial order. Oper. Res. Lett. 5 (1986) 317-320.

G. Steiner, On the complexity of dynamic programming for sequencing problems with precedence constraints. Production Planning and Scheduling ( M. Queyranne, Ed.). Annals of Operations Research, Vol. 26, J. C. Baltzer AG, Basel (1990) 103-123. D. M. Topkis, Minimizing a submodular function on a lattice. Oper. Rex 26 (1978) 305-321. A. F. Veinott, Jr., Latfice Programming, to appear.

79-86.

Received February 24, 1994 Accepted August 15, 1995

disconnecting sets in single and two-terminal-pair networks

Documents