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Most vital links and nodes in weighted networks
Volume I, Number 4
OPERATIONS RESEARCH LETTERS
September 1982
M O S T V I T A L L I N K S A N D N O D E S IN W E I G H T E D N E T W O R K S
H.W. CORLEY and David Y.SHA Department of Industrial Engineering, The University of Texas at Arlington, Arlington, 2~ 76019, U.S.A. Received May 1982 Revised August 1982
The n most vital links (or nodes) in a weighted network are those n links (nodes) whose removal from the network results in the grept~st increase in shortest distance between two specified nodes. Preliminary results are presented for obtaining these entities. Networks, distance algorithms, applications
1. Introduction Consider a conflict situation where there is a transportation network between two points, a defender of the system, and an interdictor. The defender wishes to know which road section is most vital to him so that he can reinforce this road against attack, while the interdictor wants to destroy the road which most increases the shortest distance through the network between the two points. This situation illustrates the class of problems to be considered here. The networks to be considered here will be finite, directed, and connected with nonnegative weights (distances), where st~.ndard terminology is used. Most results also apply to undirected networks; all apply upon replacing an undirected link with two directed ones. The notation G(N; A) will denote a network with set of nodes N and set of links A. The notation (a, b) will sometimes be used to represent a link I E A from node a to node b, and w(a, b) will denote the weight, or distance. associated with (a, b). G(N; A) is said to b e , : ..,arable with respect to nodes, s, t E N if there exists r links in A whose removal eliminates all paths from s to t. One problem here is to find a link in a network whose removal from the network results in the greatest increase in the shortest distance between two specified nodes called the origin node and the destination node. Such a link is called a single A
most vital link in the network. A related problem is to find a single most vital node in the network, which is similarly defined. These problems can obviously be extended to n links or nodes. Explicitly, consider the network G(N; A) and two nodes s, t E N. A set of n links L* C_A is a set of n most vital links of G(N; A) with respect to s and t if the shortest distance between s and t in G(N; A - L*) is greater than or equal to the shortest distance between s and t in G(N; A - L , ) for all link sets L, C_A containing n elements. Similarly, a set of n nodes M* c_ N - {s, t} is a set of n most vital nodes of G(N; A) with respect to s and t if the shortest distance between s and t in G(N - Mr*; At) is greater than or equal to the shortest distance between s and t in G(N - M.; A2) for any set of n nodes M, C_N - {s, t}, where A I consists of those links not incident with M* and A 2 those not incident with M,. Previous work on related problems has been restricted to flow networks [I,4].
2. Results Results directed at developing efficient algorithms for the problems of determining most vital links and node,,~ are next presented. The first gives a relationship between the I st shortest path and the kth shortest path between s and t which can then be used to establish the characterization of a most vital link in Result 2.
0167-6377/82/0000-0000/$02.75 © 1982 North-Holland
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Volume !, Number4
OPERATIONSRESEARCHLETI'ERS
Result I. Let G( N; A) be a network for which link (a, b) appears in all 1st, 2n. 3rd, ..., ( k - l)th shortest ~ from node s to node t but not in some kth shortest path. Then this kth shortest path in G( N; A) from s to t is a 1st shortest path in G( N; .4 - {(a, b)}) f r o m s to t.
Pred. Let Pk ~e some k th shortest path in G(N; `4) from s to t which does not include (a, b) and denote the length of Pt by dl,. Note that Pk is also a path from s to t in G(N; `4 - {(a, b) }). To arrive at a contradiction, assume that Pk is not the shortest path in G(N; A - ((a, b)}), i.e., that there exists a path P from s to t in G(N; A - {(a, b)}) of length d < dr. But such a P is also a path from s to t in G(N; ,4) that does not contain (a, b) and is shorter than Pt, in contradiction to the hypothesis that all 1st, 2rid, 3rd, ..., ( k - - i)th shortest paths in G(N; ,4) contain link (a, b). The result is now established. Result 2. Let G(N; A) be a network which is not l-separable with respect to nodes s and t, and let Q be the set of links common to all 1st shortest paths from s to t in G(N; A). Suppose that (a, b) is a link in Q common to all 2nd, 3rd, ..., ( k - I )th shortest paths from s to t but not in some kth shortest path. Thon (a, b) is a most vital link of G(N; A) if and only if there is no link in Q whose first such disappearance in some jth shortest path from s to t occurs forj > k. Preef, We prove the necessity of the condition by contraposition. Let (x, 3') E Q be a link whose first disappearance in some jth shortest path occurs for j • k. Sincej • k, it follows from Result I that the shortest distance from s to t in G(N; A - {(x, 3' ) }) is greater than in G(~V; ,4 - ((a, b)}). Thus (a, b) is not a most vital link. The sufficiency is again proved by centraposition. Assume that (a, b) is not a most vital link, i.e., there is a link (x, y) E A for which the shortest distance from s to t in G(N; A - {(x, y)}) is 8rearer than in G(N; A - ( (a, b) }). Obviously (x, 3,) E Q. Otherwise the shortest distances in G(N; `4) and G(N; `4 - {(x, 3')}) are the same, and (x, 3") cannot have its assumed property. Also, since G(N; ,4) is not I-separable with respect to s and t, there is a path from s to t not including (x, 3,). Consider a shortest such path. This path is, say, a j t h shortest path in G(N; A) and satisfies the
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hypotheses for Result 1 applied to (x, 3,). Therefore, it is a 1st shortest path of G(N; A - { ( x , 3,)}). Next applying Result 1 to (a, b) it follows that the stipulated kth shortest path of G(N; A) in which (a, b) first disappears is a 1st shortest path of G(N; A - {(a, b)}). Hence j > k from the assumption on (x, y), and the proof is complete. Result 2 does not include the cases when Q is empty or G(N; A) is l-separable with respect to s and t. These obvious cases are considered in Result 3. Result 3. I f Q is empty in Result 2, then euery link in A is a most vital link. I f G( N; A) is I.separable with respect to s and t, then the set of single ,vnost vital links is the set of links m ~ery path from s to t. Proof. If Q is empty, removing any one link does not affect the shortest distance from s and t since there is a shortest parth not containing the link. If G(N; A) is I-separable with respect to s and t, removing any link that is in every path produces an infinite distance from s to t while other links do not. The following algorithm for obtaining a most vital link is based on Results 2 and 3. We abuse notation slightly in sometimes treating a path P, an ordered set of links, as an unordered set. We also use the notation p / t o denote some arbitrary member i of the set ofjth shortest paths from s to t. Algorithm Step 1. Find all 1st shortest paths, say p~, p~, ..., p~,, and construct the set Qj =p~ N p2t N . . . Npnt. If QI = if, terminate; every link in A is a most vital link. Otherwise, set k = 2 and 8o to Step 2. Step 2. Find all k th shortest paths, say, p~, p2~, ..., p~,. If there are none, 80 to Step 4. Step 3. Set Qk = Qk-I ¢3p~ N p~ N . . . Npnk~.If Qk = 9, go to Step 4. If Qk is a singleton, set k = k + 1 and 8o to Step 4. If Qk is neither the empty set nor a singleton, set k = k + !, and go to Step 2. Step 4. Terminate. Any element of Qk-~ is a most vital link. The algorithm is obviously a finite procedure as
Volume I, Number 4
OPERATIONS RESEARCHLETTERS
a consequence of Step 2 and the fact that there are only a finite number of paths in a finite network. In Step 2, the set of k th shortest paths can be obtained using any standard procedure. The optimality of the algorithm is justified as follows. The termination criteria in Step 3, i.e., Qk being either the null set or a singleton, yield the set of single most vital links when G(N; A) is not l-separable with respect to s and t as a consequence of Result 2. Similarly, Result 3 validates Step I when Q t = g. The case where G(N; A) is l-separable with respect to s and t is considered in Step 2. If there are no k th shortest paths in that step, Qk-t contains the set of links in every path from s to t so that Result 3 applies. It should also be mentioned that a slight modification of the algorithm accomodates the situation where certain links cannot be re. moved from G(N; A), i.e., are not candidates for being a single most vital link. Simply exclude such links from the sets Qk in Steps 1 and 3. Unfortunately the n most vital links problem is not equivalent to the sequential process of finding a single most vital link, removing it, finding a single most vital link in the remaining network, removing it, etc., for n iterations (even when G(N; A) is not n-separable) as simple examples for n = 2 reveal. Moreover, the combinatorial difficulties of an enumeration are significantly more severe. While a completely satisfactory solution procedure has not been developed, some insight into the problem is gained in Result 4 below which contains necessary conditions for a set of n most vital links. These conditions can substantially reduce the number of combinations of n links which are candidates for being a set of n most vital links Result 4. I f G( 3I; A ) is n.separable with respect to s and t, then any set of n links in A which separate s and t is a set of n most vital links. Suppose that G( N; A) is not n-separable with respect to s and t, and let L* be a set of n most oital links. Let m I denote the maximum number of nonintersecting 1st shortest paths from s to t in G( N; A). I f m I > n, any n links in A form a set of n most vital links. I f mt ~ n , there exists a link i t E L * which lies on some Ist shortest path from s to t of G( N; A). In general for k = 2, ..., n, let m k denote the maximum number of nonintersecling 1st shortest paths from s to t of G(N; A - {I t, ..., It_t}). i f ink > n k + 1, then It, ..., I~-t and any n - k + 1 links of A - {ij, ..., i k_ i} form a set of n most vital links of
September 1982
G ( N ; A). I f m k ~ n - - k + !, there exists a link Ik E L * - - { I t, ..., Ik-t} which lies on some 1st shortest path from s to t of G(N; A - {It, ...,
i,-i}). Proof. The first statement is obvious, so suppose G(N; A) is not n-separable with respect to s and t. If m I > n, removing any n links does not affect the shortest distance from s to t in G(N; A). Thus any n links form a set of, n most vital finks. When m t ~ n, L* would not be a set of n most vital links if no member of L* were part of some 1st shortest path from s to t of G(N; A). Similar arguments apply to the general case. The problem of finding most vital nodes can be reduced to that of finding most vital links through the construction of an augmented network G(N*; A*) as follows, where a node in O(N; A) corresponds to a 'dummy' link in G(N*; A*). For each node x E N, except s and t which are defined as s' and t" respectively, define two nodes x', x" E N* such that (x", x ' ) ~ A * for each x E N and (x', y") E A* for each (x, y) E A. Any link of the form (x", x ' ) E A* will be called a dummy link. The weight of a link in A* is defined as w(x', y") = w(x, y) for all (x, y) E A and w(x", x') - 0 for all x E N. It can then be easily established that an n most vital nodes problem for O(N; A) corresponds to determining n most vital links of G(N*; A*) that are dummy links.
3. Remarks The general problem of n most vital links is in need of substantially more effort. For example, it remains open whether there exist sufficient conditions that would provide a constructive procedure. For the case n - ~ prelL-ninary computations comparing the stated algorithm with some enumeration schemes seem to indicate that the algorithm presented here is more efficient, and increasingly so as the size of the network increases; but more sophisticated computational results are needed. Regardless, it is apparent that for a fixed number n of most vital links even the general problem can be solved in polynomial time in the total number of links m in A since a complete enumeration over all possible combinations of n links would involve solving substantially less than m n shortest distance 159
Volmne !, Number 4
OPERATIONS RESEARCH LETI'ERS
problems, each of which can be solved by a polynomial algorithm [2,3]. Rdegences [I] EI.W, Corloy and H. Chart& "Findin8 the n most vital nodes into a flow network", Management $ci. 21, 362-364
(1974).
160
September 1982
[2] N. Deo, Graph Theory with Applicationsto Engineering and Computer Science, Prentiee-Hall,Enslewood Cliffs(1974). [3] M. Garey and D. Johnson, Computers and Intractability:,4 Guide to the Theory of NP.Completeness, Freeman, San Francisco (1979), [4] H.D. Ratliff, G.T. Si¢ilia and S.H. Lubore, "Finding the n most vital links in flow networks", Management $cL 21, 531-539 (1975).
OPERATIONS RESEARCH LETTERS
September 1982
M O S T V I T A L L I N K S A N D N O D E S IN W E I G H T E D N E T W O R K S
H.W. CORLEY and David Y.SHA Department of Industrial Engineering, The University of Texas at Arlington, Arlington, 2~ 76019, U.S.A. Received May 1982 Revised August 1982
The n most vital links (or nodes) in a weighted network are those n links (nodes) whose removal from the network results in the grept~st increase in shortest distance between two specified nodes. Preliminary results are presented for obtaining these entities. Networks, distance algorithms, applications
1. Introduction Consider a conflict situation where there is a transportation network between two points, a defender of the system, and an interdictor. The defender wishes to know which road section is most vital to him so that he can reinforce this road against attack, while the interdictor wants to destroy the road which most increases the shortest distance through the network between the two points. This situation illustrates the class of problems to be considered here. The networks to be considered here will be finite, directed, and connected with nonnegative weights (distances), where st~.ndard terminology is used. Most results also apply to undirected networks; all apply upon replacing an undirected link with two directed ones. The notation G(N; A) will denote a network with set of nodes N and set of links A. The notation (a, b) will sometimes be used to represent a link I E A from node a to node b, and w(a, b) will denote the weight, or distance. associated with (a, b). G(N; A) is said to b e , : ..,arable with respect to nodes, s, t E N if there exists r links in A whose removal eliminates all paths from s to t. One problem here is to find a link in a network whose removal from the network results in the greatest increase in the shortest distance between two specified nodes called the origin node and the destination node. Such a link is called a single A
most vital link in the network. A related problem is to find a single most vital node in the network, which is similarly defined. These problems can obviously be extended to n links or nodes. Explicitly, consider the network G(N; A) and two nodes s, t E N. A set of n links L* C_A is a set of n most vital links of G(N; A) with respect to s and t if the shortest distance between s and t in G(N; A - L*) is greater than or equal to the shortest distance between s and t in G(N; A - L , ) for all link sets L, C_A containing n elements. Similarly, a set of n nodes M* c_ N - {s, t} is a set of n most vital nodes of G(N; A) with respect to s and t if the shortest distance between s and t in G(N - Mr*; At) is greater than or equal to the shortest distance between s and t in G(N - M.; A2) for any set of n nodes M, C_N - {s, t}, where A I consists of those links not incident with M* and A 2 those not incident with M,. Previous work on related problems has been restricted to flow networks [I,4].
2. Results Results directed at developing efficient algorithms for the problems of determining most vital links and node,,~ are next presented. The first gives a relationship between the I st shortest path and the kth shortest path between s and t which can then be used to establish the characterization of a most vital link in Result 2.
0167-6377/82/0000-0000/$02.75 © 1982 North-Holland
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Volume !, Number4
OPERATIONSRESEARCHLETI'ERS
Result I. Let G( N; A) be a network for which link (a, b) appears in all 1st, 2n. 3rd, ..., ( k - l)th shortest ~ from node s to node t but not in some kth shortest path. Then this kth shortest path in G( N; A) from s to t is a 1st shortest path in G( N; .4 - {(a, b)}) f r o m s to t.
Pred. Let Pk ~e some k th shortest path in G(N; `4) from s to t which does not include (a, b) and denote the length of Pt by dl,. Note that Pk is also a path from s to t in G(N; `4 - {(a, b) }). To arrive at a contradiction, assume that Pk is not the shortest path in G(N; A - ((a, b)}), i.e., that there exists a path P from s to t in G(N; A - {(a, b)}) of length d < dr. But such a P is also a path from s to t in G(N; ,4) that does not contain (a, b) and is shorter than Pt, in contradiction to the hypothesis that all 1st, 2rid, 3rd, ..., ( k - - i)th shortest paths in G(N; ,4) contain link (a, b). The result is now established. Result 2. Let G(N; A) be a network which is not l-separable with respect to nodes s and t, and let Q be the set of links common to all 1st shortest paths from s to t in G(N; A). Suppose that (a, b) is a link in Q common to all 2nd, 3rd, ..., ( k - I )th shortest paths from s to t but not in some kth shortest path. Thon (a, b) is a most vital link of G(N; A) if and only if there is no link in Q whose first such disappearance in some jth shortest path from s to t occurs forj > k. Preef, We prove the necessity of the condition by contraposition. Let (x, 3') E Q be a link whose first disappearance in some jth shortest path occurs for j • k. Sincej • k, it follows from Result I that the shortest distance from s to t in G(N; A - {(x, 3' ) }) is greater than in G(~V; ,4 - ((a, b)}). Thus (a, b) is not a most vital link. The sufficiency is again proved by centraposition. Assume that (a, b) is not a most vital link, i.e., there is a link (x, y) E A for which the shortest distance from s to t in G(N; A - {(x, y)}) is 8rearer than in G(N; A - ( (a, b) }). Obviously (x, 3,) E Q. Otherwise the shortest distances in G(N; `4) and G(N; `4 - {(x, 3')}) are the same, and (x, 3") cannot have its assumed property. Also, since G(N; ,4) is not I-separable with respect to s and t, there is a path from s to t not including (x, 3,). Consider a shortest such path. This path is, say, a j t h shortest path in G(N; A) and satisfies the
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hypotheses for Result 1 applied to (x, 3,). Therefore, it is a 1st shortest path of G(N; A - { ( x , 3,)}). Next applying Result 1 to (a, b) it follows that the stipulated kth shortest path of G(N; A) in which (a, b) first disappears is a 1st shortest path of G(N; A - {(a, b)}). Hence j > k from the assumption on (x, y), and the proof is complete. Result 2 does not include the cases when Q is empty or G(N; A) is l-separable with respect to s and t. These obvious cases are considered in Result 3. Result 3. I f Q is empty in Result 2, then euery link in A is a most vital link. I f G( N; A) is I.separable with respect to s and t, then the set of single ,vnost vital links is the set of links m ~ery path from s to t. Proof. If Q is empty, removing any one link does not affect the shortest distance from s and t since there is a shortest parth not containing the link. If G(N; A) is I-separable with respect to s and t, removing any link that is in every path produces an infinite distance from s to t while other links do not. The following algorithm for obtaining a most vital link is based on Results 2 and 3. We abuse notation slightly in sometimes treating a path P, an ordered set of links, as an unordered set. We also use the notation p / t o denote some arbitrary member i of the set ofjth shortest paths from s to t. Algorithm Step 1. Find all 1st shortest paths, say p~, p~, ..., p~,, and construct the set Qj =p~ N p2t N . . . Npnt. If QI = if, terminate; every link in A is a most vital link. Otherwise, set k = 2 and 8o to Step 2. Step 2. Find all k th shortest paths, say, p~, p2~, ..., p~,. If there are none, 80 to Step 4. Step 3. Set Qk = Qk-I ¢3p~ N p~ N . . . Npnk~.If Qk = 9, go to Step 4. If Qk is a singleton, set k = k + 1 and 8o to Step 4. If Qk is neither the empty set nor a singleton, set k = k + !, and go to Step 2. Step 4. Terminate. Any element of Qk-~ is a most vital link. The algorithm is obviously a finite procedure as
Volume I, Number 4
OPERATIONS RESEARCHLETTERS
a consequence of Step 2 and the fact that there are only a finite number of paths in a finite network. In Step 2, the set of k th shortest paths can be obtained using any standard procedure. The optimality of the algorithm is justified as follows. The termination criteria in Step 3, i.e., Qk being either the null set or a singleton, yield the set of single most vital links when G(N; A) is not l-separable with respect to s and t as a consequence of Result 2. Similarly, Result 3 validates Step I when Q t = g. The case where G(N; A) is l-separable with respect to s and t is considered in Step 2. If there are no k th shortest paths in that step, Qk-t contains the set of links in every path from s to t so that Result 3 applies. It should also be mentioned that a slight modification of the algorithm accomodates the situation where certain links cannot be re. moved from G(N; A), i.e., are not candidates for being a single most vital link. Simply exclude such links from the sets Qk in Steps 1 and 3. Unfortunately the n most vital links problem is not equivalent to the sequential process of finding a single most vital link, removing it, finding a single most vital link in the remaining network, removing it, etc., for n iterations (even when G(N; A) is not n-separable) as simple examples for n = 2 reveal. Moreover, the combinatorial difficulties of an enumeration are significantly more severe. While a completely satisfactory solution procedure has not been developed, some insight into the problem is gained in Result 4 below which contains necessary conditions for a set of n most vital links. These conditions can substantially reduce the number of combinations of n links which are candidates for being a set of n most vital links Result 4. I f G( 3I; A ) is n.separable with respect to s and t, then any set of n links in A which separate s and t is a set of n most vital links. Suppose that G( N; A) is not n-separable with respect to s and t, and let L* be a set of n most oital links. Let m I denote the maximum number of nonintersecting 1st shortest paths from s to t in G( N; A). I f m I > n, any n links in A form a set of n most vital links. I f mt ~ n , there exists a link i t E L * which lies on some Ist shortest path from s to t of G( N; A). In general for k = 2, ..., n, let m k denote the maximum number of nonintersecling 1st shortest paths from s to t of G(N; A - {I t, ..., It_t}). i f ink > n k + 1, then It, ..., I~-t and any n - k + 1 links of A - {ij, ..., i k_ i} form a set of n most vital links of
September 1982
G ( N ; A). I f m k ~ n - - k + !, there exists a link Ik E L * - - { I t, ..., Ik-t} which lies on some 1st shortest path from s to t of G(N; A - {It, ...,
i,-i}). Proof. The first statement is obvious, so suppose G(N; A) is not n-separable with respect to s and t. If m I > n, removing any n links does not affect the shortest distance from s to t in G(N; A). Thus any n links form a set of, n most vital finks. When m t ~ n, L* would not be a set of n most vital links if no member of L* were part of some 1st shortest path from s to t of G(N; A). Similar arguments apply to the general case. The problem of finding most vital nodes can be reduced to that of finding most vital links through the construction of an augmented network G(N*; A*) as follows, where a node in O(N; A) corresponds to a 'dummy' link in G(N*; A*). For each node x E N, except s and t which are defined as s' and t" respectively, define two nodes x', x" E N* such that (x", x ' ) ~ A * for each x E N and (x', y") E A* for each (x, y) E A. Any link of the form (x", x ' ) E A* will be called a dummy link. The weight of a link in A* is defined as w(x', y") = w(x, y) for all (x, y) E A and w(x", x') - 0 for all x E N. It can then be easily established that an n most vital nodes problem for O(N; A) corresponds to determining n most vital links of G(N*; A*) that are dummy links.
3. Remarks The general problem of n most vital links is in need of substantially more effort. For example, it remains open whether there exist sufficient conditions that would provide a constructive procedure. For the case n - ~ prelL-ninary computations comparing the stated algorithm with some enumeration schemes seem to indicate that the algorithm presented here is more efficient, and increasingly so as the size of the network increases; but more sophisticated computational results are needed. Regardless, it is apparent that for a fixed number n of most vital links even the general problem can be solved in polynomial time in the total number of links m in A since a complete enumeration over all possible combinations of n links would involve solving substantially less than m n shortest distance 159
Volmne !, Number 4
OPERATIONS RESEARCH LETI'ERS
problems, each of which can be solved by a polynomial algorithm [2,3]. Rdegences [I] EI.W, Corloy and H. Chart& "Findin8 the n most vital nodes into a flow network", Management $ci. 21, 362-364
(1974).
160
September 1982
[2] N. Deo, Graph Theory with Applicationsto Engineering and Computer Science, Prentiee-Hall,Enslewood Cliffs(1974). [3] M. Garey and D. Johnson, Computers and Intractability:,4 Guide to the Theory of NP.Completeness, Freeman, San Francisco (1979), [4] H.D. Ratliff, G.T. Si¢ilia and S.H. Lubore, "Finding the n most vital links in flow networks", Management $cL 21, 531-539 (1975).