Enumerating kk-arc-connected orientations

12 pagesWe study the problem of enumerating the $k$-arc-connected orientations of a graph $G$, i.e., generating each exactly once. A first algorithm using submodular flow optimization is easy to state, but intricate to implement. In a second approach we present a simple algorithm with delay $O(knm^2)$ and amortized time $O(m^2)$, which improves over the analysis of the submodular flow algorithm. As ingredients, we obtain enumeration algorithms for the $\alpha$-orientations of a graph $G$ in delay $O(m^2)$ and for the outdegree sequences attained by $k$-arc-connected orientations of $G$ in delay $O(knm^2)$

Finding a subdivision of a prescribed digraph of order 4

The problem of when a given digraph contains a subdivision of a fixed digraph F is considered.Bang-Jensen et al. [2] laid out foundations for approaching this problem from the algorithmic pointof view. In this paper we give further support to several open conjectures and speculations about algorithmiccomplexity of finding F-subdivisions. In particular, up to 5 exceptions, we completely classify forwhich 4-vertex digraphs F, the F-subdivision problem is polynomial-time solvable and for which it is NPcomplete.While all NP-hardness proofs are made by reduction from some version of the 2-linkage problemin digraphs, some of the polynomial-time solvable cases involve relatively complicated algorithms

Algorithms for flows and disjoint paths in planar graphs

In this dissertation we describe several algorithms for computing flows, connectivity, and disjoint paths in planar graphs. In all cases, the algorithms are either the first polynomial-time algorithms or are faster than all previously-known algorithms. First, we describe algorithms for the maximum flow problem in directed planar graphs with integer capacities on both vertices and arcs and with multiple sources and sinks. The algorithms are the first to solve the problem in near-linear time when the number of terminals is fixed and the capacities are polynomially bounded. As a byproduct, we get the first algorithm to solve the vertex-disjoint S-T paths problem in near-linear time when the number of terminals is fixed but greater than 2. We also modify our algorithms to handle real capacities in near-linear time when they are three terminals. Second, we describe algorithms to compute element-connectivity and a related structure called the reduced graph. We show that global element-connectivity in planar graphs can be found in linear time if the terminals can be covered by O(1) faces. We also show that the reduced graph can be computed in subquadratic time in planar graphs if the number of terminals is fixed. Third, we describe algorithms for solving or approximately solving the vertex-disjoint paths problem when we want to minimize the total length of the paths. For planar graphs, we describe: (1) an exact algorithm for the case of four pairs of terminals on a single face; and (2) a k-approximation algorithm for the case of k pairs of terminals on a single face. Fourth, we describe algorithms and a hardness result for the ideal orientation problem. We show that the problem is NP-hard in planar graphs. On the other hand, we show that the problem is polynomial-time solvable in planar graphs when the number of terminals is fixed, the terminals are all on the same face, and no two of the terminal pairs cross. We also describe an algorithm for serial instances of a generalization of the ideal orientation problem called the k-min-sum orientation problem