Andrew V. Goldberg, Serge A. Plotkin, and Eva Tardos. Combinatorial algorithms for the generalized circulation problem. Mathematics of Operations Research, 16(2):351--381, May 1991.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....maximum generalised ow algorithms are due to Onaga [13] and Truemper [18] Onaga s algorithm uses shortest path computations while Truemper s algorithm uses maximum ow computations, but both algorithms may need in the worst case exponentially many iterations. Goldberg, Plotkin, and Tardos [4] designed the rst two polynomial time combinatorial algorithms for the maximum generalised ow problem, which use shortest path computations and minimumcost (non generalised) ow computations. The running times of the theoretically faster of those two algorithms is O(n m(m n log n) log n log ....

....in next paragraph. Let f be a ow in network G such that the residual network G f is a non gain network and let be a proper labeling of G f . De ne the total relabeled residual excess as TotResEx f; e f; v) The ow decomposition theorem for generalised ows (see, for example, [4] or [16] for details) implies that TotResEx f; e opt (t) e f (t) 7) This inequality implies that if TotResEx f; e f (t) 8) then e f (t) 1 ) e opt (t) so ow f is optimal. 3 Onaga s and Truemper s algorithms The rst and the simplest combinatorial algorithm for the maximum ....

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A. V. Goldberg, S. A. Plotkin, and  E. Tardos. Combinatorial Algorithms for the Generalized Circulation Problem. Math. Oper. Res., 16(2):351-381, 1991. 21 (a) nodes/edges ExSc3 PushRelabel CPLEX-p CPLEX-d

....the combinatorial structures of the underlying network and the ows in this network, and often uses as subroutines combinatorial algorithms for simpler network problems, such as the shortest paths problem, the maximum ow problem, and the minimum cost ow problem. Goldberg, Plotkin, and Tardos [4] designed the rst two polynomial time combinatorial algorithms for the maximum generalised ow problem: algorithm MCF, based on repeated minimum cost ow computations, and the Fat Path algorithm, based on saturating ow generating cycles and sending the supplies from the nodes to the sink along ....

....ow network G is a maximum ow if and only if there are no positive residual supplies is left in the residual network G f . An arbitrary generalised ow network G can be converted into a non gain residual network G f in O(mn 2 log B) time by canceling (saturating) all ow generating cycles [4]. Since the time bound on computing maximum generalised ows which we claim in this paper is O(m 2 n log B) we can assume that the computation always begins with a non gain residual network. We use the following special case of the generalised ow decomposition theorem. Theorem 3. Each ....

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A. V. Goldberg, S. A. Plotkin, and  E. Tardos. Combinatorial Algorithms for the Generalized Circulation Problem. Math. Oper. Res., 16(2):351-381, 1991.

....can be solved exactly via general purpose linear programming techniques, including simplex, ellipsoid, and interior point methods. Researchers have also designed e#cient combinatorial algorithms that exploit the underlying network flow structure of the problem. Goldberg, Plotkin, and Tardos [14] designed the first polynomial time combinatorial algorithms for generalized maximum flow. Their algorithms were refined and improved upon in [15, 16, 29] with the fastest algorithm developed so far by Goldfarb, Jin, and Orlin [16] For generalized maximum flow, researchers have also developed ....

....g. Each iteration captures at least 1 2 of the remaining flow possible, so the optimality gap geometrically decreases to zero. We obtain an # approximate flow after log(1 #) iterations. If # is su#ciently small, say M 3m , then the # approximate flow can be e#ciently rounded to an optimal flow [14]. The main flaw in this approach is that the residual network may contain flow generating cycles, even if the original network did not. Recall, our fast generalized shortest path subroutine does not work if there are flow generating cycles. To overcome this obstacle, before running the packing ....

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A. V. Goldberg, S. A. Plotkin, and E. Tardos. Combinatorial algorithms for the generalized circulation problem. Mathematics of Operations Research, 16:351--379, 1991.

....maximum flow problem. Our algorithm is the first polynomial time push relabel algorithm for generalized flows. We believe that our push relabel algorithm will be quite practical for computing approximate flows. In Chapter 7, we design a new variant of the Fat Path capacity scaling algorithm of [20]. Our variant matches the best known complexity for the problem, and it is much simpler than the variant in [49] In Chapter 8, we discuss a strongly polynomial variant of a procedure of [20] which cancels all flow generating cycles. This is used by many of our algorithm to reroute flow from ....

....approximate flows. In Chapter 7, we design a new variant of the Fat Path capacity scaling algorithm of [20] Our variant matches the best known complexity for the problem, and it is much simpler than the variant in [49] In Chapter 8, we discuss a strongly polynomial variant of a procedure of [20] which cancels all flow generating cycles. This is used by many of our algorithm to reroute flow from their current paths to more e#cient paths. Chapter 2 Preliminaries In this chapter we review several fundamental network flow problems. We formally define the generalized maximum flow problem ....

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A. V. Goldberg, S. A. Plotkin, and E. Tardos. Combinatorial algorithms for the generalized circulation problem. Mathematics of Operations Research, 16:351-- 379, 1991.

....theorem, so it does not fit into any of these categories. The generalized maximum flow problem is a special case of our problem in which there are no costs. In one version of the problem, the goal is to maximize the amount of flow sent through a given arc. Goldberg, Plotkin, and Tardos [13] designed the first combinatorial polynomial algorithms for this problem. Subsequently, researchers [7, 16, 17, 18, 31, 36, 41] proposed new polynomial algorithms, also using flow based techniques. All of these are primal dual style algorithms. Remarkably, the specializations of our algorithms to ....

A. V. Goldberg, S. A. Plotkin, and E. Tardos. Combinatorial algorithms for the generalized circulation problem. Mathematics of Operations Research, 16:351--379, 1991.

....flow problem is a special case of a linear programming problem (LP) it can be solved in polynomial time using the ellipsoid method [7] or Karmarkar s algorithm [6] However, there is no polynomial bound on the worst case complexity of the generalized network simplex algorithm. Goldberg et al. [4] state that no polynomial time bounds are known for combinatorial algorithms for generalized maximal flow problems. This gap is explained by the fact that the integer generalized maximum flow problem is NP complete as shown by Ahuja et al. 1] p. 798. The generalized shortest path problem is a ....

A. V. Goldberg, S. A. Plotkin and E. Tardos, Combinatorial Algorithms for the generalized circulation problem, in, Mathematics of Operations Research, 16 (1991), 351 -- 381.

....can be solved exactly via general purpose linear programming techniques, including simplex, ellipsoid, and interior point methods. Researchers have also designed efficient combinatorial algorithms that exploit the underlying network flow structure of the problem. Goldberg, Plotkin, and Tardos [14] designed the first polynomial time combinatorial algorithms for generalized maximum flow. Their algorithms were refined and improved upon in [15, 16, 29] with the fastest algorithm developed so far by Goldfarb, Jin, and Orlin [16] For generalized maximum flow, researchers have also developed ....

....captures at least 1=2 of the remaining flow possible, so the optimality gap geometrically decreases to zero. We obtain an ffl approximate flow after log(1=ffl) iterations. If ffl is sufficiently small, say M Gamma3m , then the ffl approximate flow can be efficiently rounded to an optimal flow [14]. The main flaw in this approach is that the residual network may contain flow generating cycles, even if the original network did not. Recall, our fast generalized shortest path subroutine does not work if there are flow generating cycles. To overcome this obstacle, before running the packing ....

[Article contains additional citation context not shown here]

A. V. Goldberg, S. A. Plotkin, and ' E. Tardos. Combinatorial algorithms for the generalized circulation problem. Mathematics of Operations Research, 16:351--379, 1991.

....the solution in the same way, we have to bound the determinant of an m k Theta m k matrix in which all entries are 0 or 1, except possibly the column containing the capacities. Hence the claim follows. 8.3 The Generalized Flow Problem. For the definition of the generalized flow problem see [2]. In this problem, flows moving on edges have different gains or losses. To solve the generalized multicommodity flow problem using our algorithm, all we have to change are the entries in B which instead of being 0 and 1, will now depend on the gain loss factor on the edges. Let fl be the ratio ....

A. V. Goldberg, S. A. Plotkin, and ' E. Tardos. Combinatorial Algorithms for the Generalized Circulation Problem. In Proc. 29th IEEE Annual Symposium on Foundations of Computer Science, pages 174--185, 1988.

....[19] and exponential time variants. Truemper [22] observed that the problem is closely related to the minimum cost flow problem, and that many of the early generalized maximum flow algorithms were, in fact, analogs of pseudo polynomial minimum cost flow algorithms. Goldberg, Plotkin and Tardos [7] designed the first two combinatorial polynomial time algorithms for the problem: Fat Path and MCF. The Fat Path algorithm uses capacity scaling and a subroutine that cancels flow generating cycles. The MCF algorithm performs minimum cost flow computations. Radzik [20] modified the Fat Path ....

....In the process, the net flow into every node, including the sink, can only be increased. It also finds node labels so that G g 0 ; is a no gain network. This subroutine is used by all of our algorithms. CancelCycles was designed by Goldberg, Plotkin, and Tardos and is described in detail in [7]. It is an adaptation of Goldberg and Tarjan s [9] cancel and tighten algorithm for the minimum cost flow problem using costs c(v; w) Gamma log b fl(v; w) for some base b 1. Note that negative cost cycles correspond to flow generating cycles. In Section 7 we discuss practical implementation ....

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Goldberg, A. V., Plotkin, S. A., and Tardos, ' E. (1991), "Combinatorial algorithms for the generalized circulation problem," Mathematics of Operations Research, 16, 351--379.

....the order to examine active nodes (e.g. the wave implementation) we can reduce the number of nonsaturating pushes. A dual approach is to use more clever data structures to reduce the amortized time per nonsaturating push. Using a version of dynamic trees specialized for generalized networks [6], we obtain the following theorem. Theorem 5.1. Algorithm RPP computes a optimal flow in O(mn 3 Gamma1 log B) time. Iterative Rounded Preflow Push (IRPP) RPP does not compute an optimal flow in polynomial time, since the precision required is roughly = B Gammam . Like Algorithm ....

Goldberg, A. V., Plotkin, S. A., and Tardos, ' E. (1988), "Combinatorial algorithms for the generalized circulation problem," Technical Report STAN-CS-88-1209, Stanford University.

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A. V. Goldberg, S. A. Plotkin, and E. Tardos. Combinatorial Algorithms for the Generalized Circulation Problem. Technical Report STAN-CS-88-1209, Stanford University, 1988.

....entries per row and c is a vector with at most d non zero coefficients, then for fixed d the linear program maxfcx : Ax bg can be solved in O(n 3 m log n n d 1 m log d m) time. The most important problem of this form is the dual of the (minimum cost) generalized transshipment problem [11, 17, 28]. In this generalization of the transshipment problem we are given a network G, gain factors fl and costs c on the arcs, and demands b on the nodes; there are no capacities. A generalized transshipment is a non negative flow vector f on the arcs, such that if flow f(v; w) enters the arc (v; w) at ....

A. Goldberg, S. Plotkin, and ' E. Tardos. Combinatorial algorithms for the generalized circulation problem. In Proc. 29th IEEE Annual Symposium on Foundations of Computer Science, pages 432--443, October 1988.

....along the cycle. The amount of flow pushed along the arcs of the cycle is different for each arc and depends on the gains of the arcs. Therefore this update can not be done by subtracting the same value from all of them, as it is done in the case of flows. One way to solve this problem (used in [11]) is to extend the Dynamic Trees data structure to support the operation of multiplying all the values stored in a tree by a given number in amortized 29 make tree(v) Make node v into a one node dynamic tree. Node v must be in no other tree. find root(v) Find and return the root of the tree ....

A. V. Goldberg, S. A. Plotkin, and ' E. Tardos. Combinatorial Algorithms for the Generalized Circulation Problem. Technical Report STAN-CS-88-1209, Stanford University, 1988. (Also available as Technical Memorandum MIT/LCS/TM-358, Laboratory for Computer Science, M.I.T., 1988.).

....the determinant of an m k Theta m k matrix in which all entries are 0 or 1, except possibly the column containing the capacities. Hence the claim follows. CHAPTER 3. QUADRATIC PROGRAMMING APPROACH 45 3.8.3 The Generalized Flow Problem. For the definition of the generalized flow problem see [25]. In this problem, flows moving on edges have different gains or losses. To solve the generalized multi commodity flow problem using our algorithm, all we have to change are the entries in B which instead of being 0 and 1, will now depend on the gain loss factor on the edges. Let fl be the ratio ....

A. V. Goldberg, S. A. Plotkin, and ' E. Tardos. Combinatorial Algorithms for the Generalized Circulation Problem. In Proc. 29th IEEE Annual Symposium on Foundations of Computer Science, pages 174--185, 1988.

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Andrew V. Goldberg, Serge A. Plotkin, and Eva Tardos. Combinatorial algorithms for the generalized circulation problem. Mathematics of Operations Research, 16(2):351--381, May 1991.

No context found.

Andrew V. Goldberg, Serge A. Plotkin, and  Eva Tardos. Combinatorial algorithms for the generalized circulation problem. Mathematics of Operations Research, 16(2):351-381, May 1991.

No context found.

Andrew V. Goldberg, Serge A. Plotkin, and Eva Tardos. Combinatorial algorithms for the generalized circulation problem. Mathematics of Operations Research, 16(2):351--381, May 1991.

No context found.

A. V. Goldberg, S. A. Plotkin, and # ETardos. Combinatorial algorithms for the generalized circulation problem. Mathematics of Operations Research, 16:351#379, 1991.

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A. V. Goldberg, S. A. Plotkin, and # ETardos. Combinatorial algorithms for the generalized circulation problem. Technical Report STAN-CS-88-1209, Stanford University, 1988.

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