ALGORITHMS AND COMPLETE FORMULATIONS FOR THE NETWORK DESIGN PROBLEM Trilochan Sastry Indian Institute of Management, Ahmedabad November 1997 Abstract

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1 ALGORITHMS AND COMPLETE FORMULATIONS FOR THE NETWORK DESIGN PROBLEM Trilochan Sastry Indian Institute of Management, Ahmedabad November 1997 Abstract We study the multi commodity uncapacitated network design problem, which is known to be NP-complete. We also study the capacitated one facility non-bifurcated version of the problem in which ow of a commodity is restricted to ow on at most two paths. We describe an exact O(n 2 2 K + n3 K ) algorithm for these two versions of the multi commodity problem. This algorithm can also be used to solve the directed version of the Steiner tree problem and performs a little better than the O(n 2 2 K + n3 K + n 3 ) dynamic programming based algorithm due to Dreyfus and Wagner (1971). We also describe the convex hull of integer solutions to the two commodity uncapacitated problem using O(m) variables and O(n) constraints. We show that the capacitated two commodity problem can be solved either by using any standard shortest path algorithm or by the O(n 2 ) algorithm described for the uncapacitated case. We also describe its convex hull. Key words and phrases: network design, algorithm, convex hull. 1

2 1 Introduction We rst study the uncapacitated multi commodity network design problem on a directed graph, which can be described as follows. Consider a directed graph G = (N; A), with node set N, arc set A, and origin destination pairs O k ; D k, with demand of 1 unit between every pair for k = 1; : : : ; K. Capacity can be purchased on each arc (i; j) 2 A at cost w ij 0. Flow costs are assumed to be zero. This problem arises in the telecommunications and transportation industries. The Steiner Tree problem, which is known to be NP-complete, is a special case of this problem in which all commodities have a common origin. Balakrishnan, Magnanti and Wong (1989) have studied the uncapacitated network design problem and solved large instances using a dual ascent based procedure. They also provide detailed references to the literature on this problem. Several aglorithms for the Steiner Tree problem are known. Hwang, Richards and Winter (1992) provide detailed references to the literature on Steiner trees. In particular, Dreyfus and Wagner (1971) describe a dynamic programming based O(n 2 2 K + n3 K + n 3 ) algorithm for the Steiner tree problem. 2 An Exact Algorithm Let m = jaj and n = jnj denote the number of arcs and nodes repectively, and let K = f1; : : : ; Kg be the set of commodities. If a commodity ows on two paths, we can redirect all ow from one path to the other without increasing cost since capacity is unrestricted. Hereafter, we only consider optimal solutions with each commodity k having exactly one path from O k to D k and 0? 1 ows for each commodity on any arc. Let Q K denote a subset of commodities. Let a(i; j) denote the shortest distance from node i to node j using w ij as arc costs. In the algorithm, we implicitly allow the xed charge variables y ij to take any nonnegative integer value. In any feasible solution, we say that an arc (i; j) shares at most q commodities if y ij 2 whenever x k ij = 1 for q + 1 or more commodities. The algorithm nds i (Q), the minimum cost of sending one unit of ow from nodes D k : k 2 Q to node i, using a generalisation of Dijkstra's shortest path algorithm for graphs with nonnegative arc costs. Similarly, it nds i (Q), the minimum cost of sending one unit of ow from node i to nodes D k : k 2 Q. Thus, i (Q) + i (Q) is the minimum cost of sending one unit from O k to D k for all k 2 Q if all commodities ow through node i. By 1

3 varying Q over all subsets of K, the algorithm nds the optimal solution. Algorithm Multi Path Let A(i) denote the set of arcs adjacent to node i Initialise pred(i; k)? O k and succ(i; k)? D k for all i 2 N, k 2 K i () = i () = 0, i (fkg) = a(o k ; i), i (fkg) = a(i; D k ) for all i 2 N OP T? k2k a(o k ; D k ) for q = 2 to K do begin for all subsets Q K such that jqj = q do begin i (Q) = a(o k ; i), i (Q) = a(i; D k ) k2q k2q i (Q) = min f i (Q1) + i (Q2) : Q1 \ Q2 = ; Q1 [ Q2 = Qg i (Q) = min f i (Q1) + i (Q2) : Q1 \ Q2 = ; Q1 [ Q2 = Qg S?, T? while jsj < n or jt j < n do begin choose i; u : i (Q) = min f j (Q) : j 2 Sg, u (Q) = min f j (Q) : j 2 Sg S? S [ fig; S? S? fig, T? T [ fig; T? T? fig for j 2 A(i) if j (Q) > i (Q) + w ij then begin j (Q) = i (Q) + w ij pred(j; k)? i for all k 2 Q end for j 2 A(u) if j (Q) > u (Q) + w uj then begin j (Q) = u (Q) + w uj succ(j; k)? u for all k 2 Q end endfwhileg OP T (Q) = min f i (Q) + i (Q) : i 2 Ng OP T (Q) = min fop T (Q1) + OP T (Q2) : Q1 \ Q2 = ; Q1 [ Q2 = Qg end end Let the iterations in which q varies from 2 to K be the outer iterations. Let the iterations in which we choose all subsets Q of cardinality jqj = q be the 2

4 intermediate iterations, and those in which we vary the cardinality of the sets S and T, the inner iterations. Theorem 1 Algorithm multi path solves the uncapacitated multi commodity network design problem UMC. Proof The result can be established by using the following inductive hypotheses on the cardinality of the sets Q, S and T. For a given value of q, and set Q : jqj = q, and some set S or T in an inner iteration, (i) i (Q) is the mimimum cost of reaching node i from nodes O k : k 2 Q, and i (Q) is the mimimum cost of reaching nodes D k : k 2 Q from node i. (ii) j (Q) ( j (Q)) for any node j =2 S (j =2 T ) is the mimimum cost of reaching node j (nodes D k : k 2 Q) from nodes O k : k 2 Q, (from node j) either using only nodes in S (nodes in T ) as intermediate nodes, or the shortest paths from O k : k 2 Q (from j) to node j (to D k : k 2 Q). (iii) At the start of any intermediate iteration, and before the start of the inner iterations, i (Q) ( i (Q)) equals the minimum cost of reaching node i (nodes D k : k 2 Q) from origins O k : k 2 Q (from node i) if at most q? 1 commodities share an arc. (iv) At the end of any outer iteration with jqj = q, OP T (Q) equals the minimum cost of sending ow from O k to D k for k 2 Q. Lemma 1 Algorithm multi path takes O(n 2 2 K + n3 K ) iterations. Proof For given q and Q, it is clear that there are O(n 2 ) inner iterations. At the start of the intermediate loop we update the i (Q) values over all nodes i 2 N and all subsets of Q. This takes O(n2 q ) time. Therefore, to complete all intermediate iterations for a given value of q takes O( K C q n 2 + K C q n2 q ) iterations, where K C q is the number of ways of choosing q objects from K. The total number of iterations is therefore O( P K q=1( K C q n 2 + K C q 2 q n)). Since (1 + 1) K = P K q=1( K C q ) and (1 + 2) K = P K q=1( K C q 2 q ), the result follows

5 We now consider the capacitated problem in which capacity can be purchased in batches of C units at a cost of w ij. The objective is to minimise the total cost while satisfying demand d k between every origin destination pair. Let d k = k C + r k where we dene r k, the residue as r k = C if d k is a multiple of C, and k C as the full ow. Flows are non-bifurcated and each commodity can ow on at most two paths, with the full ow on one path and the residual ow on another. Algorithm multi path can be modied as follows to solve the problem. Each of the full ows k C is sent on the shortest path from O k to D k. For the residual ows, we use Algorithm Multi Path and modify the inner iterations by setting j (Q) = min f j (Q); i (Q) + w ij d Pk2Q r k C e : i 2 A(j)g to update the values of j (Q). Arguments similar to those used in the proof of Theorem 1 establish that this modied algorithm solves the problem in O(n 2 2 K + n3 K ). 3 The Convex Hull of the Two Commodity Problem Hu(1963), Sakarovitch (1973), Seymour (1979) and Seymour (1980) have studied the two commodity ow problem. We consider the undirected version of the uncapacitated two commodity network design problem U T C. Dene arc (i; j) to be a shared forward arc if x 1 ij > 0 and x2 ij > 0, or x1 ji > 0 and x 2 ji > 0. Arc (i; j) is a shared reverse arc if x1 ij > 0 and x2 ji > 0, or x 1 ji > 0 and x2 ij > 0. Path P is shared if both x1 ij + x1 ji > 0 and x2 ij + x2 ji > 0 for all arcs (i; j) on the path. Simple arguments based on re-routing ows establish the following result. Lemma 2 There exists an optimal solution for U T C with at most one shared path such that all shared arcs are shared forward arcs or all shared arcs are shared reverse arcs. Remark 1 In the case of two commodity ows, there exist optimal ows that are multiples of 0:5. However, in the case of two commodity uncapacitated network design, ows are integral. This result allows us to classify the optimal solution as either a forward or a reverse solution. We describe the so called two-path algorithm to solve 4

6 the problem. For the forward problem we dene s k = O k and t k = D k, while for the reverse problem, we dene s1 = O1, t1 = D1, s2 = D2 and t2 = O2. We use the multi path algorithm to solve the problem. The algorithm has two passes, one for the forward problem, and the other for the reverse problem. Choose the one with the minimum cost. This gives the optimal solution. Algorithm two path takes O(n 2 ) iterations if the distances a(s k ; j) and a(j; t k ) are known. However, these distances can be obtained by nding the simple shortest path trees rooted at nodes s k and t k in at most O(n 2 ) time. The complexity of the two path algorithm is therefore O(n 2 ). Several combinatorial problems can be solved in polynomial time. However, the convex hull of feasible integer solutions, if known, often has an exponential number of constraints. For instance, in the case of the spanning tree problem, for any S N if A(S) P denotes the set of arcs with both end nodes in S, then the inequalities e2a(s) y e jsj? 1 completely describe the convex hull of integer solutions. However, there are O(2 n ) such inequalities. An extended reformulation in a polynomial number of variables and constraints is also known for this problem where we use a multi commodity formulation (see Magnanti and Wolsey (1995)). This extended formulation has O(mn) variables and constraints whereas the original integer formulation has O(m) variables. However, for the two commodity uncapacitated network design problem, we obtain the convex hull with O(m) variables and O(n) constraints. A natural formulation for U T C has O(m) variables and O(m) constraints. We say that path P shared between paths P1 and P2 for commodities 1 and 2 is maximal if any arc (i; j) 2 P1 \ P2 belongs to P. Suppose there is a shared maximal path from node i to node j with commodity 1 owing from i to j. We say that the shared path starts in node i and ends in node j. If k = 1 or = f (k = 2 and = b), then ow of commodity k on arc (i; j) is said to occur before the shared path if the ow has not yet entered node i (node j ) and is said to occur after the shared path if it has left node j (node i ). We dene the following 0-1 variables. For each node j 2 N let: u j () = 1 if the shared path starts in node j v j () = 1 if the shared path ends in node j For each arc (i; j) 2 A let : e k ij = 1 only if commodity k ow occurs before the shared path = 1 only if commodity k ow occurs after the shared path g k ij 5

7 h ij () = 1 if it is a shared arc. The uncapacitated two commodity U T C problem can now be reformulated as follows. Reformulation U T C(R) Min = w ij (e 1 ij + e 2 ij + g 1 ij + g 2 ij + h ij ()) subject to: i i i (i;j)2a (e 1 ij? e 1 ji)? u j (f)? u j (b) = (e 2 ij? e2 ji)? u j (f)? v j (b) = =f;b (?1 j = O 1 0 otherwise (?1 j = O 2 0 otherwise (h ij ()? h ji ()) + u j ()? v j () = 0 (3) ( (g 1 1 j = D ij? g1 ji) + v j (f) + v j (b) = 1 (4) 0 otherwise i ( (g 2 ij? gji) 2 1 j = D + v j (f) + u j (b) = 2 (5) 0 otherwise i e k ij ; gk ij ; h ij() 0 for k = 1; 2; = f; b: It is easy to transform any solution (x; y) to the two commodity problem UT C with at most one shared path to obtain a feasible solution to the reformulation. The reformulation has O(m) variables and O(n) constraints. Theorem 2 The reformulation U T C(R) completely describes the convex hull of integer solutions for the uncapacitated two commodity network design problem. Proof. The dual DUT C(R) of UT C(R) is given below. (1) (2) 6

8 2 Max (D k k? k O k ) k=1 subject to: k j? k i w ij j k? k i w ij j ()? i () w ij j (f)? 1 j? 2 j 0 1 j + 2 j? j(f) 0 j (b) + 2 j? 1 j 0 1 j? 2 j? j(b) 0 Since the outer iterations start with q = 2 in the multi path algorithm, and there are only two commodities, we can drop the sux Q from i (Q). However, we use i () for = f; b to distinguish the values from the forward and reverse problems. Using the j () values from algorithm two path, let k j = a(o k; j), 1 j = a(o 1; D1)? a(j; D1), 2 j = OP T? a(o 1; D1)? a(j; D2), j (f) = j (f), and j (b) = j (b) + a(o1; D1)? OP T. Notice that k O k = 0, 1 D 1 = a(o1; D1) and that 2 D 2 = OP T? a(o1; D1). Hence if the dual variables are feasible, they are optimal. From algorithm two path notice that since j (f) ( j (b)) is the minimum cost of sending one unit of ow from each of the nodes O1 and O2 (O1 and D2) to node j, it follows that j (f) a(o1; j) + a(o2; j) ( j (b) a(o1; j) + a(j; D2)). Further, since a(j; D1) + a(j; D2) is an upper bound on the cost of sending one unit from node j to each of nodes D1 and D2, it follows that j (f) + a(j; D1) + a(j; D2) OP T: Similarly, j (b) + a(j; D1) + a(o2; j) OP T: It is now easy to verify that these values of the dual variables satisfy dual feasibility. 4 The Two Commodity Capacitated Problem We now consider the two commodity one facility capacitated network design problem (TCOF) in which capacity can be purchased in batches of C units on each arc (i; j) 2 A at cost w ij 0. Flow costs are assumed to be zero. Magnanti, Mirchandani and Vachani (1995) and Chopra et.al. (1995) have 7

9 studied the two facility version of the problem, where capacity is available in batches of 1 or C units. Chopra, Gilboa and Sastry (1996) studied the single origin-destination version of the one and two facility problem where they describe an exact algorithm and an extended formulation for the problem. Dene the following two problems associated with T COF. The rst is the full ow problem F F of sending k C units from O k to D k, and the other is the residual ow problem RF of sending r k units from O k to D k. Lemma 3 The full ow problem can be solved by sending k C units of ow on the shortest path from O k to D k for k = 1; 2 using w ij as arc costs. We modify the denition of a shared arc as follows. Arc (i; j) is shared if d x1 ij + x1 ji C e + d x2 ij + x2 ji C e > d x1 ij + x1 ji + x2 ij + x2 ji e: C Notice that according to this denition, if x k ij + xk ji = C for some integer, then arc (i; j) cannot be shared. Thus, an arc is shared only if (? 1)C < x k ij + xk ji < C for both commodities for some integer 1. We dene two paths P1 and P2 as independent if any arc (i; j) 2 P1 \ P2 is not shared. Notice that if two paths are independent, then there is no cost saving even if they have arcs in common. There are two cases to consider for T COF, each of which is easy to solve. The next lemma can be established based on the following result for two commodity ows due to Hu(1963): There is a feasible two commodity ow which does not exceed arc capacities if and only if the capacity of all cuts separating origins O1; O2 from destinations D1; D2 exceeds the total ow d1 + d2. Lemma 4 There is an optimal solution to T COF with : (i) no shared path if r1 + r2 > C, and (ii) at most one shared path, and each residual ow on exactly one path, if r1 + r2 C. Therefore, if r1 + r2 > C, then T COF can be solved by sending d k units along the shortest path between O k and D k for k = 1; 2 using w ij as arc costs. The convex hull for T F OC is therefore represented by the usual network ow constraints for two shortest path problems, one for each commodity, in which we send k + 1 units from O k to D k. Theorem 3 If r1 +r2 C, we can solve T COF by solving the full ow and the residual ow problems separately. Moreover, the residual ow problem reduces to the uncapacitated two commodity problem U T C. 8

10 Proof From Lemma 4, at most one path is shared, and the residual ows are not split. Therefore, the shared path, if any, has only residual ows. The full ows can therefore be sent on the shortest paths. Consider the residual ow problem RF. Since r1 + r2 C, y ij = 1 is a sucient value for y ij on any shared arc. Therefore, there is an optimal solution with exactly one path for each commodity, and at most one shared path. This is precisely the two commodity uncapacitated problem. If r1 + r2 C, then the convex hull is represented by the constraints for the full ow problem F F and the constraints for the uncapacitated problem's reformulation UT C(R). However, the variables y ij need to be split into variables y ij (F F ) and y ij (RF ) for the full ow and the residual ow problems. Similarly, the ow variables x k ij are split into xk ij (F F ) for the full ow problem, and the variables e k ij ; gk ij in reformulation UT C(R). Acknowledgements The author is indebted to Yogesh Agarwal and Mechthild Stoer for several useful discussions. 5 References Balakrishnan, A., T.L.Magnanti and R.T.Wong (1989) "A Dual Ascent Procedure for Large-Scale Uncapacitated Network Design", Operations Research 37, Bondy, J.A., and U.S.R.Murty, "Graph Theory with Applications", North Holland, Amsterdam (1976). Chopra,S., I.Gilboa and T.Sastry (1996) "Algorithms and Extended Formulations for One and Two Facility Network Design", Integer Programming and Combinatorial Optimization, 5th International IPCO Conference, 1996, Springer. Chopra, S., D.Bienstock, O.Gunluck, C.Y.Tsai, "Minimum cost capacity installation for multi commodity network ows", Research Report, Northwestern University, January Dreyfus, S.E., and R.A.Wagner (1971) "The Steiner problem in graphs", Networks 1,

11 Hu, T.C. (1963) "Multi-commodity Network Flows" Operations Research 11, Hwang, F.K., D.S.Richards and P.Winter, (1992) "The Steiner tree problem", Annals of Discrete Math. 53, North-Holland. Magnanti,T.L., P.Mirchandani and R.Vachani (1995) "Modelling and Solving the Two Facility Capacitated Network Loading Problem", Operations Research, 43, Magnanti, T.L. and L.A.Wolsey, "Optimal Trees", Handbooks in OR & MS, vol 7, Elsevier Science. Sakarovitch, M. (1973) "Two Commodity Network Flows and Linear Programming", Math.Prog. 4, Seymour, P.D. (1979) "A Short Proof of the Two-Commodity Flow Theorem", Journal of Combinatorial Theory B 26, Seymour, P.D. (1980) "Four-Terminus Flows", Networks 10,