Goldberg, A., Tarjan, R.: Solving minimum-cost flow problems by successive approximation. In: Proceedings of the nineteenth annual ACM conference on Theory of computing, ACM Press (1987) 7--18  Home/Search   Document Not in Database   Summary   Related Articles   Check

....what is perhaps the generic algorithm of the class, the e relaxation method [7] Section 4 develops some basic serial complexity analysis tools for this algorithm [28, 29, 8] also addressing the special case of maximum flow problems. Section 5 combines this analysis with the notion of scaling [30 32, 19, 24, 41, 5], yielding a polynomial (O(N 3 log NC) serial algorithm for the minimum cost flow problem (N is the number of nodes, and C the largest absolute value of the arc cost coefficients) In Section 6, we introduce the auction algorithm [9] for the assignment problems, and show how it may be regarded ....

....Pi is a value of the ith price that maximizes q(p) with all other prices held fixed. When e is small, it is possible to approach the optimal solution even if each step does not result in a dual cost improvement. The method eventually stays in a small neighborhood of the optimal solution. 30] [32]. This methodology can also be applied directly to solving maximum flow problems with arbitrary initial prices, as opposed to initial prices satisfying p pj q 1 for all arcs (i,j) as in [28] and [29] Another problem is that of flow looping. This is discussed in Section 4 (see Fig. 7) and ....

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A.V. Goldberg and R.E. Tarjan, "Solving minimum cost flow problems by successive approxima- tions," Proc. 19th ACM STOC, May 1987.

....what is perhaps the generic algorithm of the class, the e relaxation method [7] Section 4 develops some basic serial complexity analysis tools for this algorithm [28, 29, 8] also addressing the special case of maximum flow problems. Section 5 combines this analysis with the notion of scaling [30 32, 19, 24, 41, 5], yielding a polynomial (O(N 3 log NC) serial algorithm for the minimum cost flow problem (N is the number of nodes, and C the largest absolute value of the arc cost coefficients) In Section 6, we introduce the auction algorithm [9] for the assignment problems, and show how it may be regarded ....

....e, where Pi is a value of the ith price that maximizes q(p) with all other prices held fixed. When e is small, it is possible to approach the optimal solution even if each step does not result in a dual cost improvement. The method eventually stays in a small neighborhood of the optimal solution. [30] [32] This methodology can also be applied directly to solving maximum flow problems with arbitrary initial prices, as opposed to initial prices satisfying p pj q 1 for all arcs (i,j) as in [28] and [29] Another problem is that of flow looping. This is discussed in Section 4 (see Fig. 7) ....

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A.V. Goldberg, "Solving minimum-cost flow problems by successive approximations," extended abstract, submitted to STOC 87, Nov. 1986.

.... the primal or the dual cost at any iteration, and they are based on a relaxed version of the CS conditions, called # complementary slackness (# CS for short) They have an excellent worst case computational complexity, when properly implemented, as shown in [Gol87] see also [BeE88] BeT89] [GoT90]) Their practical performance is also very good and they are well suited for parallel implementation (see [BCE95] LiZ91] NiZ93] We will extend two such methods, the # relaxation method and the auction sequential shortest path method, to the general convex cost case. One possibility for ....

....A scaling approach in connection with the # relaxation method for linear cost problems, is # scaling. This approach was originally introduced in [Ber79] as a means of improving the performance of the auction algorithm for the assignment problem. Its complexity analysis was given in [Gol87] and [GoT90]. The key idea of # scaling is to apply the # relaxation method several times, starting with a large value of # and to successively reduce # up to a final value that will give the desirable degree of accuracy to our solution. Furthermore, the price and flow information from one application of the ....

Goldberg, A. V., and Tarjan, R. E., "Solving Minimum Cost Flow Problems by Successive Approximation," Mathematics of Operations Research, Vol. 15, 1990, pp. 430-466.

....and provides exponentiation as a single step. In Section 5 we will show how to modify our algorithms to work in the standard RAM model. The question of which minimum cost flow algorithm to use is also deferred to Section 5, where we show that the cost scaling algorithm of Goldberg and Tarjan [5] is a good choice in most instances. 9 4.1 Finding an Initial Solution To find an initial solution, we separately route each commodity i. For commodity i, we find i and a flow f i , such that f i satisfies the demands of this commodity and obeys capacity constraints i Delta u(vw) Let i ....

....the second set of relaxed constraints. In Section 5.2, we discuss which minimum cost flow algorithm to use. We will use different minimum cost flow algorithms in different situations. For general concurrent flow problems, the best choice seems to be either the algorithm of Goldberg and Tarjan [5] or that of Ahuja, Goldberg, Orlin and Tarjan [1] For concurrent flow with uniform capacity, we use Gabow and Tarjan s [3] algorithm for the assignment problem. When both the demands and capacities are uniform, we use the algorithm that iteratively computes shortest paths in the residual graph ....

A. V. Goldberg and R. E. Tarjan. Solving minimum-cost flow problems by successive approximation. Mathematics of Operations Research, 15(3):430--466, 1990.

....cost flow is implemented in a transportation plan x ij j(ij ) 2 A] where ij : lv (k (s) f(i; j) Stage s ends by setting x : x x . Pseudo code can be found in Algorithm 2. For the computation of the minimum cost flows, we use the cost scaling algorithm by Goldberg and Tarjan [GT90] which we implemented using LEDA graphs [MNU96] Lemma 2.2 The MCF heuristic returns a feasible solution of IPA. Proof. The lemma can be proved by induction on the stage number s, where stage 0 denotes the initialisation. Let x denote the value of x at the end of stage s. Our induction ....

....This completes the proof of the lemma. Lemma 2.3 The MCF heuristic can be implemented to work in O(dn C) time, where C = max i;j2V c ij . Proof. The complexity follows from the fact that the heuristic computes d minimum cost flows, each of which can be computed in O(n log C) time [GT90] 2.2 An Implementable Order Let c k = max 2L (k) be the maximal number of times an element of commodity k occurs in any admissible loading configuration. The preferred loading configuration for transporting q units of commodity k, denoted pref k (q) is defined as follows: pref k (q) ....

Goldberg, A. V. and Tarjan, R. E. Solving minimum cost flow problem by successive approximation. Mathematics of Opns. Res. 15 (1990), 430--466.

....O( 4klogklog, TMC. i mlogk) O(k ,mlog ( log( U) Deterministic O(c 3k2logklogt(TMCp mlog k) O(k tmlog ( og(tU) Proof: Combine Lelnma 2.4.1, 2.4.2, Corollary 2.4.16, Corollary 2.4. 18 and the fact that a maximum flow in an z node m edge graph can be computed in O(nmlog(n2 m) time [23]. Note that only in the non silnple case does the tilne for initialization appear in the final tilne bounds. A scaling algorithm The dependence on e given in Theorem 2.4.21 can be reduced somewhat, through a technique we call e scaling. Instead of calling DECONGEST with the value of e given in ....

....log II(MC F Deterministic 0( 6 2 logk)k2 lognMCp) O(k nmlog ( log(nU) 2 log ) 2 log Z(MCF Proof: Combine Theorem 2.4.21, Corollary 2.4.16, Corollary 2.4.18, Lemma 2.4.22, Theorem 2.4. 20 and the fact that a maximum flow in an z node m edge graph can be computed in O(zm log(z2 m) time [23]. Note that only in the non simple case does the time for initialization appear in the final bound. Given an instance of a concurrent flow problem, the running time for Algorithm SCAL INGCONCURRENT is never greater than that of Algorithm CONCURRENT. For the rest of this chapter, we will quote ....

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A. V. Goldberg and R. E. Tarjan. Solving minimum-cost flow problems by successive approximation. Mathematics of Operations Research, 15(3):430 466, 1990.

....q is a maximum 2 route flow in , and j satisfies j W a W j z j m . The proof of correctness of the above algorithm can be found in [11] The maxflow between two nodes in a network can be computed in time ( by the Goldberg Tarjan highest label preflow push algorithm [8]. Since this algorithm solves at most three maxflow problems, the running time of the algorithm is ( The solution of the maximum 2 route flow is now used to compute the set of 2 critical links for the ingress egress pair 9 : The next theorem gives the conditions that a ....

A. V. Goldberg, R. E. Tarjan, "Solving Minimum Cost Flow Problem by Successive Approximation", Proc. of the 19th ACM Symposium on the Theory of Computing, pp.7-18, 1987.

....to place registers on the arc e i,j can be found in [4] 6. A MINIMUM COST NETWORK FLOW FORMULATION In this section, we focus on transforming the dual of the linear program (LP) in Section 5 to a formulation of a minimum cost network flow problem, since this latter can be solved more efficiently [1, 8, 13]. By replacing with . The LP can be written as in Figure 7. For each i in V, let be the set of immediate successors of i and be its set of immediate predecessors. The dual of the linear program in Figure 7 can be written as in Figure 8. In this figure, the variables are: s, s, s and s. ....

....then they can be removed from the constraints. The final form of the transformed dual of the linear program in Figure 7 is presented in Figure 10, which is a formulation of a minimum cost network flow problem, and hence it can be solved more efficiently using, for instance, algorithms provided in [1, 8, 13]. 7. EXPERIMENTAL RESULTS To test the effectiveness of our approach, the MILP in Figure 6 and the corresponding linear program (LP) obtained by ignoring the constraint integer in (19) are experimented on well known benchmarks. Circuits from the ISCAS89 benchmark suite are used to test the ....

A.-V. Goldberg and R.-E. Tarjan, "Solving Minimum-Cost Flow Problems by Successive Approximation," Proceedings of the 19th Annual ACM Symposium on Theory of Computing, New York City, May 25-27, 1987.

....The maximum ow is the maximum amount of material that can ow through the graph from source to sink without exceeding the capacity of the nodes. 1 Finding the maximal ow is a problem which can be solved in O(nm log(n 2 =m) log n) where n is the number of nodes and m is the number of edges [7]. In the ReMove scheduling problem, the topology of the processor can be translated into a ow graph through which the live results can ow to the register les. If the maximal ow through the network is equal to the number of live results that must be written into the register les, then there ....

A.V. Goldberg and R.E. Tarjan. Solving Minimum-Cost Flow Problems by Successive Approximation. Mathematics of Operations Research, 15(3):430-466, 1990.

....f 2 is a maximum 2 route flow in G, and v 2 satisfies v 2 = 2u = 2v 1 v 0 . The proof of correctness of the above algorithm can be found in [11] The maxflow between two nodes in a network can be computed in time O(n 2 # m) by the Goldberg 4 Tarjan highest label preflow push algorithm [7]. Since this algorithm solves at most three maxflow problems, the running time of the algorithm is O(n 2 # m) B. 2 Critical Link Computation The solution of the maximum 2 route flow is now used to compute the set of 2 critical links for the ingress egress pair (a, b) The next theorem gives ....

A. V. Goldberg, R. E. Tarjan, "Solving Minimum Cost Flow Problem by Successive Approximation", Proc. of the 19th ACM Symposium on the Theory of Computing, pp.7-18, 1987.

....not possible to apply the packing algorithm of [15] to this representation of the problem. The desired oracle (2) corresponds to approximately solving k independent single commodity minimum cost flow problems. This can be done by adapting the Goldberg Tarjan cost scaling min cost flow algorithm [4], as in Leighton et al. 13] The adaptation is needed because the costs are exponential. The resulting oracle runs in O(kmn) time. Applying Theorem 2.5, we get an O(ffl Gamma3 k 2 mn) running time for solving the problem. To improve the running time we can use an approach that is ....

A. V. Goldberg and R. E. Tarjan. Solving minimum-cost flow problems by successive approximation. In Proc. 19th Annual ACM Symposium on Theory of Computing, pages 7--18, 1987.

....time for the case of unit capacity links. To be precise, their running time is O(MCF (mn; mn) where MCF(a,b) is the running time of minimum cost flow with O(a) nodes and O(b) edges. The current best running times of minimum cost flow on a graph with O(a) nodes and O(b) edges are all O(ab) GT90, AGOT92, Orl88] where O denotes the running time, without log factors. Hoppe and Tardos developed a polynomial time algorithm for the general problem. The running time of their algorithm is roughly O(m 5 ) Now we analyze a slight modification of our algorithm and show that it runs in ....

A. V. Goldberg and R. E. Tarjan. Solving minimum-cost flow problems by successive approximation. Mathematics of Operations Research, 15(3):430--466, 1990.

....ff sd : Therefore the problem of computing the weights of the arcs is now reduced to determining the set of critical arcs for all ingress egress pairs. The maximum flow between two nodes in a network can be computed in time O(n 2 p m) by the Goldberg Tarjan highest label preflow push algorithm [8] or in time O(nm n 2 log U) using an excess scaling algorithm. There may be several alternate mincuts for a given ingress egress pair. The critical links for the ingress egress pairs are arcs belonging to the union of all these mincuts. We show that the set of critical links for a given ....

A. V. Goldberg, R. E. Tarjan, "Solving Minimum Cost Flow Problem by Successive Approximation", Proceedigns of the 19th ACM Symposium on the Theory of Computing, pp.7-18, 1987.

....for the possible increases in node labels. Canceling flow generating cycles now becomes a bottleneck operation. Each call requires O(mn 2 log B) time; it is performed once after every maximum flow computation. Chapter 6 Push Relabel Method In this chapter we adapt the Goldberg Tarjan [24] push relabel method for the traditional minimum cost flow problem to the generalized maximum flow problem. Tseng and Bertsekas [56] proposed an # relaxation method for solving the more general generalized minimum cost flow problem with separable convex costs. However, the complexity of their ....

....any constant # 0. Finally, we show how to find optimal flows in polynomial time by incorporating error scaling and canceling flow generating cycles. We believe our algorithm will be practical and discuss implementation issues. 58 59 6. 1 Push Relabel Method for Min Cost Flow Goldberg and Tarjan [22, 24] designed the push relabel method for the traditional maximum flow and minimum cost flow problems. Push relabel methods send flow along individual arcs instead of entire augmenting paths. This provides additional flexibility and leads to improvements in both the worst case complexity as well as ....

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A. V. Goldberg and R. E. Tarjan. Solving minimum cost flow problems by successive approximation. Mathematics of Operations Research, 15:430--466, 1990. 113

....ff sd : Therfore the problem of computing the weights of the arcs is now reduced to determining the set of critical arcs for all ingress egress pairs. The maximum flow between two nodes in a network can be computed in time O(n 2 p m) by the Goldberg Tarjan highest label preflow push algorithm [9] or in time O(nm n 2 log U) using an excess scaling algorithm. There may be several alternate mincuts for a given ingress egress pair. The critical links for the ingress egress pairs are arcs belonging to the union of all these mincuts. We show that the set of critical links for a given ....

A. V. Goldberg, R. E. Tarjan, "Solving Minimum Cost Flow Problem by Successive Approximation", Proceedigns of the 19th ACM Symposium on the Theory of Computing, pp.7-18, 1987.

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A. V. Goldberg and R. E. Tarjan. Solving Minimum-Cost Flow Problems by Successive Approximation. In Proc. 19th Annual ACM Symposium on Theory of Computing, pages 7--18, 1987. (Math. of Oper. Res., to appear).

....as m log B, which would make any known capacityscaling algorithm too slow. The classical cost scaling algorithms cannot be used directly because the costs are (irrational) logarithms of rational numbers. A cost scaling algorithm can be constructed based on the idea of ffl optimality, as done in [10, 12]. Note that the gain of a flow generating cycle can be as small as 2 GammaO(B n ) and hence the absolute value of the cost of a negative cycle can be as small as B GammaO(n) Consequently, the required precision seems to be Omega Gamma B Gamman ) and therefore the known ....

....Dynamic Tree data structure. Sleator and Tarjan introduced dynamic trees [29, 32, 3] in order to implement the operations described in Figure 6, where each operation takes amortized O(log n) time. This data structure was used for speeding up several maximum flow and minimum cost flow algorithms [10, 12, 13, 14, 29], including the CycleCanceling algorithm for minimum cost flows. The main idea is that augmenting the flow along a path does not saturate all the arcs along this path. Instead, the path is subdivided into shorter paths. Storing these paths in a dynamic tree data structure allows us to use the ....

A. V. Goldberg and R. E. Tarjan. Solving Minimum-Cost Flow Problems by Successive Approximation. In Proc. 19th Annual ACM Symposium on Theory of Computing, pages 7--18, 1987. (Math. of Oper. Res., to appear).

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Goldberg, A., Tarjan, R.: Solving minimum-cost flow problems by successive approximation. In: Proceedings of the nineteenth annual ACM conference on Theory of computing, ACM Press (1987) 7--18

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A. V. Goldberg and R. E. Tarjan. Solving minimum-cost flow problems by successive approximation. Mathematics of Operations Research, 15(3):430--466, 1990.

No context found.

A. V. Goldberg and R. E. Tarjan. Solving minimum cost flow problems by successive approximation. Mathematics of Operations Research 15, 430--466, 1990.

No context found.

A. V. Goldberg and R. E. Tarjan. Solving minimum-cost flow problems by successive approximation. Mathematics of Operations Research, 15(3):430--466, 1990.

No context found.

A.V. Goldberg and R.E. Tarjan. Solving minimum-cost flow problems by successive approximation. Mathematics of Operation Research, 15:430--466, 1990.

No context found.

A. V. Goldberg, R. E. Tarjan, "Solving Minimum Cost Flow Problem by Successive Approximation", Proceedings of the 19th ACM Symposium on the Theory of Computing, pp.7-18, 1987.

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A.V. Goldberg and R.R. Tarjan. Solving minimum cost flow problems by successive approximation. Proc. 19th ACM STOC., pages 7--18, 1987.

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A.V. Goldberg. Solving minimum cost flow problems by successive approximations. Extended abstract. Submitted to 19th ACM STOC., 1986.

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