Visualizing, Finding and Packing Dijoins

Abstract

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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