Minimum cost flow algorithms for series-parallel networks

Bein, Wolfgang; Brucker, Peter; Tamir, Arie

doi:10.1016/0166-218x(85)90006-x

Cited by 48 publications

(31 citation statements)

References 2 publications

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“…We then apply this latter algorithm to the linear cost network flow problem on series-parallel networks to obtain an O(mlog 2 m) time algorithm for this problem, where m is the number of arcs in the network. This result significantly improves upon the O(nm + m log m) time bound obtained by Bein, Brucker and Tamir [BBT85], where n denotes the number of nodes in the network.…”

Section: Introductionsupporting

confidence: 62%

“…The fastest known algorithm for solving the linear cost network flow problem on series-parallel graphs is that due to Bein, Brucker and Tannir [BBT85] with a time bound of O(nm + m log m), where n is the number of nodes and m is the number of arcs. Below, we will apply Algorithm 2 to obtain an algorithm for this problem with a faster time of O(mlog 2 m).…”

Section: Application To Linear Cost Network Flow On Series-parallel Nmentioning

confidence: 99%

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On Computing the Nested Sums and Infimal Convolutions of Convex Piecewise-Linear Functions

Tseng

Luo

1996

Journal of Algorithms

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We consider the problem of evaluating a functional expression comprising the nested sums and infimal convolutions of convex piecewise-linear functions defined on the reals. For the special case where the nesting is serial, we give an O(NlogN) time algorithm, where N is the total nunber of breakpoints of the functions. We also prove a lower bound of fl(N log N) on the number of comparisons needed to solve this problem, thus showing that our algorithm is essentially optimal. For the general case, we give an O(Nlog 2 N) time algorithm. We apply this latter algorithm to the linear cost network flow problem on series-parallel networks to obtain an O(m log 2 m) time algorithm for this problem, where in is the number of arcs in the network. This result improves upon the previous algorithm of Bein, Brucker, and Tailir which has a time complexity of O(nm+m log m), where n is the number of nodes.KEY WORDS: Convex program, balanced binary search tree, infimal convolution, series-parallel graphs, linear cost network flow.

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Section: Introductionsupporting

confidence: 62%

Section: Application To Linear Cost Network Flow On Series-parallel Nmentioning

confidence: 99%

On Computing the Nested Sums and Infimal Convolutions of Convex Piecewise-Linear Functions

Tseng

Luo

1996

Journal of Algorithms

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“…Several researchers [2,11,21,28] have studied network flow problems on directed seriesparallel graphs, and described polynomial time dynamic programming algorithms for solving them. Since the shortest path problem is a special case of the minimum cost network flow problem, it can be solved in ᏻ(͉A͉) time, where ͉A͉ is the number of arcs, using the algorithms described therein.…”

Section: Introductionmentioning

confidence: 99%

Bicriteria product design optimization: An efficient solution procedure using AND/OR trees

Raghavan

Ball

Trichur

2002

Naval Research Logistics

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Abstract:Competitive imperatives are causing manufacturing firms to consider multiple criteria when designing products. However, current methods to deal with multiple criteria in product design are ad hoc in nature. In this paper we present a systematic procedure to efficiently solve bicriteria product design optimization problems. We first present a modeling framework, the AND/OR tree, which permits a simplified representation of product design optimization problems. We then show how product design optimization problems on AND/OR trees can be framed as network design problems on a special graph-a directed series-parallel graph. We develop an enumerative solution algorithm for the bicriteria problem that requires as a subroutine the solution of the parametric shortest path problem. Although this parametric problem is hard on general graphs, we show that it is polynomially solvable on the series-parallel graph. As a result we develop an efficient solution algorithm for the product design optimization problem that does not require the use of complex and expensive linear/integer programming solvers. As a byproduct of the solution algorithm, sensitivity analysis for product design optimization is also efficiently performed under this framework.

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“…With this technique, it can be shown, for example, that network flow problems with an underlying series-parallel directed graph can be solved by the greedy algorithm. On the other hand, Bein et al (1985) demonstrate that greedy solvability of flow problems generally does not extend beyond series-parallel networks. One can now show (Faigle and Kern (1993b) Here the pattern matrix of A is the matrix obtained when each non-zero element of A is replaced by "1".…”

Section: Greedy Linear Programsmentioning

confidence: 99%

Some recent results in the analysis of greedy algorithms for assignment problems

Faigle

1994

OR Spektrum

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Abstract.We survey some recent developments in the analysis of greedy algorithms for assignment and transportation problems. We focus on the linear programming model for matroids and linear assignment problems with Monge property, on general linear programs, probabilistic analysis for linear assignment and makespan minimization, and on-line algorithms for linear and non-linear assignment problems.

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Minimum cost flow algorithms for series-parallel networks

Cited by 48 publications

References 2 publications

On Computing the Nested Sums and Infimal Convolutions of Convex Piecewise-Linear Functions

On Computing the Nested Sums and Infimal Convolutions of Convex Piecewise-Linear Functions

Bicriteria product design optimization: An efficient solution procedure using AND/OR trees

Some recent results in the analysis of greedy algorithms for assignment problems

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