Finding minimum-cost circulations by canceling negative cycles
TL;DR: It is shown that a judicious choice of cycles for canceling leads to a polynomial bound on the number of iterations in this algorithm, which is comparable to those of the fastest previously known algorithms.
Abstract: A classical algorithm for finding a minimum-cost circulation consists of repeatedly finding a residual cycle of negative cost and canceling it by pushing enough flow around the cycle to saturate an arc. We show that a judicious choice of cycles for canceling leads to a polynomial bound on the number of iterations in this algorithm. This gives a very simple strongly polynomial algorithm that uses no scaling. A variant of the algorithm that uses dynamic trees runs in O(nm(log n)min{log(nC), m log n}) time on a network of n vertices, m arcs, and arc costs of maximum absolute value C. This bound is comparable to those of the fastest previously known algorithms.
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01 Jan 1988
TL;DR: This algorithm improves the best previous strongly polynomial algorithm due to Galil and Tardos, by a factor of m/n, and is even more efficient if the number of arcs with finite upper bounds, say m', is much less than m.
Abstract: We present a new strongly polynomial algorithm for the minimum cost flow problem, based on a refinement of the Edmonds-Karp scaling technique. Our algorithm solves the uncapacitated minimum cost flow problem as a sequence of O(n log n) shortest path problems on networks with n nodes and m arcs and runs in O(n log n(m + n log n)) steps. Using a standard transformation, this approach yields an O(m log n (m + n log n)) algorithm for the capacitated minimum cost flow problem. This algorithm improves the best previous strongly polynomial algorithm due to Galil and Tardos, by a factor of m/n. Our algorithm is even more efficient if the number of arcs with finite upper bounds, say m', is much less than m. In this case, the number of shortest path problems solved is O((m + n) log n).
352 citations
TL;DR: This work develops a new approach to solving minimum-cost circulation problems that combines methods for solving the maximum flow problem with successive approximation techniques based on cost scaling and shows that a minimum- cost circulation can be computed by solving a sequence of On lognC blocking flow problems.
Abstract: We develop a new approach to solving minimum-cost circulation problems. Our approach combines methods for solving the maximum flow problem with successive approximation techniques based on cost scaling. We measure the accuracy of a solution by the amount that the complementary slackness conditions are violated.
We propose a simple minimum-cost circulation algorithm, one version of which runs in On3lognC time on an n-vertex network with integer arc costs of absolute value at most C. By incorporating sophisticated data structures into the algorithm, we obtain a time bound of Onm logn2/mlognC on a network with m arcs. A slightly different use of our approach shows that a minimum-cost circulation can be computed by solving a sequence of On lognC blocking flow problems. A corollary of this result is an On2log nlognC-time, m-processor parallel minimum-cost circulation algorithm. Our approach also yields strongly polynomial minimum-cost circulation algorithms.
Our results provide evidence that the minimum-cost circulation problem is not much harder than the maximum flow problem. We believe that a suitable implementation of our method will perform extremely well in practice.
331 citations
01 Oct 2015
TL;DR: This paper proposes a solution called Simple Lazy Facility Location (SLFL) that optimizes the placement of VNF instances in response to on-demand workload and suggests that SLFL can accept two times more workload while incurring similar operational cost compared to first-fit and random placements.
Abstract: Nowadays, many cloud providers offer Virtual Network Function (VNF) services that are dynamically scaled according to the workload. Enterprises enjoy these services by only paying for the actual consumed resources. From a cloud provider's standpoint, the cost of these services must be kept as low as possible, while QoS is maintained and service downtime is minimized. In this paper, we introduce Elastic Virtual Network Function Placement (EVNFP) problem and present a model for minimizing operational costs in providing VNF services. In this model, the elasticity overhead and the trade-off between bandwidth and host resource consumption are considered together, while the previous works ignored this perspective of the problem. We propose a solution called Simple Lazy Facility Location (SLFL) that optimizes the placement of VNF instances in response to on-demand workload. Our experiments suggest that SLFL can accept two times more workload while incurring similar operational cost compared to first-fit and random placements.
208 citations
Book•
07 Feb 2018TL;DR: Experiments on random graphs suggest that the expected time for finding a minimum mean cycle with the algorithm is O(n log n + m) time, improved from O(nm log n) for the parametric shortest path algorithm of Karp and Orlin.
Abstract: Author(s): Young, NE; Tarjant, RE; Orlin, JB | Abstract: We use Fibonacci heaps to improve a parametric shortest path algorithm of Karp and Orlin, and we combine our algorithm and the method of Schneider and Schneider's minimum‐balance algorithm to obtain a faster minimum‐balance algorithm. For a graph with n vertices and m edges, our parametric shortest path algorithm and our minimum‐balance algorithm both run in O(nm + n2 log n) time, improved from O(nm log n) for the parametric shortest path algorithm of Karp and Orlin and O(n2m) for the minimum‐balance algorithm of Schneider and Schneider. An important application of the parametric shortest path algorithm is in finding a minimum mean cycle. Experiments on random graphs suggest that the expected time for finding a minimum mean cycle with our algorithm is O(n log n + m). Copyright © 1991 Wiley Periodicals, Inc., A Wiley Company
189 citations
Cites methods from "Finding minimum-cost circulations b..."
...For a graph with n vertices and m edges, our parametric shortest path algorithm and our minimum-balance algorithm both run in O(nm + n2 log n) time, improved from O(nm log n) for the parametric shortest path algorithm of Karp and Orlin and O(n2m) for the minimum-balance algorithm of Schneider and…...
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01 Jan 1993
TL;DR: In this paper, the authors gave an O((square root of n)m log N) algorithm for the single-source shortest path problem with integral arc lengths, where n and m is the number of nodes and arcs in the input network and N is essentially the absolute value of the most negative arc length.
Abstract: : We give an O((square root of n)m log N) algorithm for the single-source shortest paths problem with integral arc lengths. (Here n and m is the number of nodes and arcs in the input network and N is essentially the absolute value of the most negative arc length.) This improves previous bounds for the problem.
180 citations
References
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TL;DR: This clearly written, mathematically rigorous text includes a novel algorithmic exposition of the simplex method and also discusses the Soviet ellipsoid algorithm for linear programming; efficient algorithms for network flow, matching, spanning trees, and matroids; the theory of NP-complete problems; approximation algorithms, local search heuristics for NPcomplete problems, more.
Abstract: This clearly written , mathematically rigorous text includes a novel algorithmic exposition of the simplex method and also discusses the Soviet ellipsoid algorithm for linear programming; efficient algorithms for network flow, matching, spanning trees, and matroids; the theory of NP-complete problems; approximation algorithms, local search heuristics for NPcomplete problems, more All chapters are supplemented by thoughtprovoking problems A useful work for graduate-level students with backgrounds in computer science, operations research, and electrical engineering Mathematicians wishing a self-contained introduction need look no further—American Mathematical Monthly 1982 ed
7,221 citations
Book•
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01 Jan 1962
TL;DR: Ford and Fulkerson as mentioned in this paper set the foundation for the study of network flow problems and developed powerful computational tools for solving and analyzing network flow models, and also furthered the understanding of linear programming.
Abstract: In this classic book, first published in 1962, L. R. Ford, Jr., and D. R. Fulkerson set the foundation for the study of network flow problems. The models and algorithms introduced in Flows in Networks are used widely today in the fields of transportation systems, manufacturing, inventory planning, image processing, and Internet traffic. The techniques presented by Ford and Fulkerson spurred the development of powerful computational tools for solving and analyzing network flow models, and also furthered the understanding of linear programming. In addition, the book helped illuminate and unify results in combinatorial mathematics while emphasizing proofs based on computationally efficient construction. Flows in Networks is rich with insights that remain relevant to current research in engineering, management, and other sciences. This landmark work belongs on the bookshelf of every researcher working with networks.
4,341 citations
TL;DR: Using F-heaps, a new data structure for implementing heaps that extends the binomial queues proposed by Vuillemin and studied further by Brown, the improved bound for minimum spanning trees is the most striking.
Abstract: In this paper we develop a new data structure for implementing heaps (priority queues). Our structure, Fibonacci heaps (abbreviated F-heaps), extends the binomial queues proposed by Vuillemin and studied further by Brown. F-heaps support arbitrary deletion from an n-item heap in O(log n) amortized time and all other standard heap operations in O(1) amortized time. Using F-heaps we are able to obtain improved running times for several network optimization algorithms. In particular, we obtain the following worst-case bounds, where n is the number of vertices and m the number of edges in the problem graph: O(n log n + m) for the single-source shortest path problem with nonnegative edge lengths, improved from O(mlog(m/n+2)n);O(n2log n + nm) for the all-pairs shortest path problem, improved from O(nm log(m/n+2)n);O(n2log n + nm) for the assignment problem (weighted bipartite matching), improved from O(nmlog(m/n+2)n);O(mβ(m, n)) for the minimum spanning tree problem, improved from O(mlog log(m/n+2)n); where β(m, n) = min {i | log(i)n ≤ m/n}. Note that β(m, n) ≤ log*n if m ≥ n.Of these results, the improved bound for minimum spanning trees is the most striking, although all the results give asymptotic improvements for graphs of appropriate densities.
2,484 citations
TL;DR: New algorithms for the maximum flow problem, the Hitchcock transportation problem, and the general minimum-cost flow problem are presented, and Dinic shows that, in a network with n nodes and p arcs, a maximum flow can be computed in 0 (n2p) primitive operations by an algorithm which augments along shortest augmenting paths.
Abstract: This paper presents new algorithms for the maximum flow problem, the Hitchcock transportation problem, and the general minimum-cost flow problem. Upper bounds on the numbers of steps in these algorithms are derived, and are shown to compale favorably with upper bounds on the numbers of steps required by earlier algorithms. First, the paper states the maximum flow problem, gives the Ford-Fulkerson labeling method for its solution, and points out that an improper choice of flow augmenting paths can lead to severe computational difficulties. Then rules of choice that avoid these difficulties are given. We show that, if each flow augmentation is made along an augmenting path having a minimum number of arcs, then a maximum flow in an n-node network will be obtained after no more than ~(n a - n) augmentations; and then we show that if each flow change is chosen to produce a maximum increase in the flow value then, provided the capacities are integral, a maximum flow will be determined within at most 1 + logM/(M--1) if(t, S) augmentations, wheref*(t, s) is the value of the maximum flow and M is the maximum number of arcs across a cut. Next a new algorithm is given for the minimum-cost flow problem, in which all shortest-path computations are performed on networks with all weights nonnegative. In particular, this algorithm solves the n X n assigmnent problem in O(n 3) steps. Following that we explore a "scaling" technique for solving a minimum-cost flow problem by treating a sequence of derived problems with "scaled down" capacities. It is shown that, using this technique, the solution of a Iiitchcock transportation problem with m sources and n sinks, m ~ n, and maximum flow B, requires at most (n + 2) log2 (B/n) flow augmentations. Similar results are also given for the general minimum-cost flow problem. An abstract stating the main results of the present paper was presented at the Calgary International Conference on Combinatorial Structures and Their Applications, June 1969. In a paper by l)inic (1970) a result closely related to the main result of Section 1.2 is obtained. Dinic shows that, in a network with n nodes and p arcs, a maximum flow can be computed in 0 (n2p) primitive operations by an algorithm which augments along shortest augmenting paths. KEY WOl¢l)S AND PHP~ASES: network flows, transportation problem, analysis of algorithms CR CATEGOI{.IES: 5.3, 5.4, 8.3
2,186 citations
"Finding minimum-cost circulations b..." refers methods in this paper
...For the special case of the maximum flow problem, Edmonds and Karp [ 9 ] give such an analysis....
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...The idea of scaling is due to Edmonds and Karp [ 9 ], who used this idea to devise the first polynomial-time algorithm for the problem....
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...Namely, the minimum-mean selection rule is a natural generalization of the rule proposed by Edmonds and Karp [ 9 ] and Dinic [8] for selecting augmenting paths in the Ford-Fulkerson maximum-flow algorithm [lo]....
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