A Minimax Theorem for Directed Graphs

Lucchesi, Cláudio L.; Younger, D. H.

doi:10.1112/jlms/s2-17.3.369

Cited by 215 publications

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“…This conjecture was proved independently by at least four groups of researchers [1,2,8,9]. Interestingly, FAS is polynomial time solvable for planar digraphs [4,35] and trivially polynomial time solvable for undirected graphs.…”

Section: Feedback Arc and Vertex Set Problemsmentioning

confidence: 93%

Some Parameterized Problems On Digraphs

Gutin

Yeo

2007

The Computer Journal

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Section: Feedback Arc and Vertex Set Problemsmentioning

confidence: 93%

Some Parameterized Problems On Digraphs

Gutin

Yeo

2007

The Computer Journal

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“…For example, Menger's Theorem [7] proves that the minimum number of arcs separating node s from node t equals the maximum number of arc-disjoint dipaths from s to t. Reversing the roles of these objects gives another min-max theorem: the minimum number of arcs in a dipath from s to t equals the maximum number of arc-disjoint cuts separating s from t. Similarly, the celebrated Lucchesi-Younger Theorem [6] proves that the minimum number of arcs in a dijoin equals the maximum number of arc-disjoint dicuts. In all three cases, the min-max theorems can be extended from digraphs to weighted digraphs.…”

Section: Introductionmentioning

confidence: 99%

Packing Dicycle Covers in Planar Graphs with No K 5–e Minor

Lee

Williams

2006

LATIN 2006: Theoretical Informatics

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Abstract. We prove that the minimum weight of a dicycle is equal to the maximum number of disjoint dicycle covers, for every weighted digraph whose underlying graph is planar and does not have K5 − e as a minor (K5 − e is the complete graph on five vertices, minus one edge). Equality was previously known when forbidding K4 as a minor, while an infinite number of weighted digraphs show that planarity does not guarantee equality. The result also improves upon results known for Woodall's Conjecture and the Edmonds-Giles Conjecture for packing dijoins. Our proof uses Wagner's characterization of planar 3-connected graphs that do not have K5 − e as a minor.

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“…We call this operation flushing the cycle C. We may make the resultant flush cycle have either clockwise or anti-clockwise orientation by adjusting whether we flush on C or C. These two possibilities are shown in Figure 1.3. Similar operations have been applied in various ways previously to paths instead of cycles (see, for example, Frank (1981); Fujishige (1978); Lucchesi and Younger (1978); McWhirter and Younger (1971);Zimmermann (1982)). One key difference is that we introduce the reverse of arcs from outside the current dijoin.…”

Section: Augmentation and Optimality In The Original Topologymentioning

confidence: 99%

“…The size of a packing is |C|. Theorem 1.1 (Lucchesi and Younger (1978)) Let D be a digraph with a cost c a on each arc. Then min { a∈T c a : T is a dijoin} = max {|C| : C is a c − packing}.…”

Section: Introductionmentioning

confidence: 99%

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Visualizing, Finding and Packing Dijoins

Shepherd¹,

Vetta²

Graph Theory and Combinatorial Optimization

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We consider the problem of making a directed graph strongly connected. To achieve this, we are allowed for assorted costs to add the reverse of any arc. A successful set of arcs, called a dijoin, must intersect every directed cut. Lucchesi and Younger gave a min-max theorem for the problem of finding a minimum cost dijoin. Less understood is the extent to which dijoins pack. One difficulty is that dijoins are not as easily visualized as other combinatorial objects such as matchings, trees or flows. We give two results which act as visual certificates for dijoins. One of these, called a lobe decomposition, resembles Whitney's ear decomposition for 2-connected graphs. The decomposition leads to a natural optimality condition for dijoins. Based on this, we give a simple description of Frank's primal-dual algorithm to find a minimum dijoin. Our implementation is purely primal and only uses greedy tree growing procedures. It runtime is O(n 2 m), matching the best known, due to Gabow. We then consider the function f (k) which is the maximum value such that every weighted directed graph whose minimum weight of a directed cut is at least k, admits a weighted packing of f (k) dijoins (a weighted packing means that the number dijoins containing an arc is at most its weight). We ask whether f (k) approaches infinity. It is not yet known whether f (k 0 ) ≥ 2 for some constant k 0 . We consider a concept of skew submodular flow polyhedra and show that this dijoinpair question reduces to finding conditions on when their integer hulls are non-empty. We also show that for any k, there exists a half-integral dijoin packing of size k 2 . 2 IntroductionWe consider the basic problem of strengthening a network D = (V, A) so that it becomes strongly connected. That is, we require that there be a directed path between any pair of nodes in both directions. To achieve this goal we are allowed to add to the graph (at varying costs) the complements of some of the network arcs. Equivalently, we are searching for a collection of arcs that induce a strongly connected graph when they are either contracted or made bi-directional. It is easy to see that our problem is that of finding a dijoin, a set of arcs that intersects every directed cut.Despite its fundamental nature, the minimum cost dijoin problem is not a standard modelling tool for the combinatorial optimizer in the same way that shortest paths, matchings and network flows are. We believe that widespread adoption of these other problems stems from the fact that they can be tackled using standard concepts such as dynamic programming, greedy algorithms, shrinking, and tree growing. Consequently, simple and efficient algorithms can be implemented and also, importantly, taught. One objective of this paper, therefore, is to examine dijoins using classical combinatorial optimization techniques such as decomposition, arborescence growing and negative cycle detection. Under this framework, we present a primal version of Frank's seminal primal-dual algorithm for finding a minimum cost d...

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A Minimax Theorem for Directed Graphs

Cited by 215 publications

References 0 publications

Some Parameterized Problems On Digraphs

Some Parameterized Problems On Digraphs

Packing Dicycle Covers in Planar Graphs with No K 5–e Minor

Visualizing, Finding and Packing Dijoins

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