93 research outputs found
Graphs with many strong orientations
We establish mild conditions under which a possibly irregular, sparse graph
has "many" strong orientations. Given a graph on vertices, orient
each edge in either direction with probability independently. We show
that if satisfies a minimum degree condition of and has
Cheeger constant at least , then the
resulting randomly oriented directed graph is strongly connected with high
probability. This Cheeger constant bound can be replaced by an analogous
spectral condition via the Cheeger inequality. Additionally, we provide an
explicit construction to show our minimum degree condition is tight while the
Cheeger constant bound is tight up to a factor.Comment: 14 pages, 4 figures; revised version includes more background and
minor changes that better clarify the expositio
One-Way Trail Orientations
Given a graph, does there exist an orientation of the edges such that the resulting directed graph is strongly connected? Robbins\u27 theorem [Robbins, Am. Math. Monthly, 1939] asserts that such an orientation exists if and only if the graph is 2-edge connected. A natural extension of this problem is the following: Suppose that the edges of the graph are partitioned into trails. Can the trails be oriented consistently such that the resulting directed graph is strongly connected?
We show that 2-edge connectivity is again a sufficient condition and we provide a linear time algorithm for finding such an orientation.
The generalised Robbins\u27 theorem [Boesch, Am. Math. Monthly, 1980] for mixed multigraphs asserts that the undirected edges of a mixed multigraph can be oriented to make the resulting directed graph strongly connected exactly when the mixed graph is strongly connected and the underlying graph is bridgeless.
We consider the natural extension where the undirected edges of a mixed multigraph are partitioned into trails. It turns out that in this case the condition of the generalised Robbin\u27s Theorem is not sufficient. However, we show that as long as each cut either contains at least 2 undirected edges or directed edges in both directions, there exists an orientation of the trails such that the resulting directed graph is strongly connected. Moreover, if the condition is satisfied, we may start by orienting an arbitrary trail in an arbitrary direction. Using this result one obtains a very simple polynomial time algorithm for finding a strong trail orientation if it exists, both in the undirected and the mixed setting
Enumerating -arc-connected orientations
12 pagesWe study the problem of enumerating the -arc-connected orientations of a graph , 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 and amortized time , which improves over the analysis of the submodular flow algorithm. As ingredients, we obtain enumeration algorithms for the -orientations of a graph in delay and for the outdegree sequences attained by -arc-connected orientations of in delay
Approximating Minimum Cost Connectivity Orientation and Augmentation
We investigate problems addressing combined connectivity augmentation and
orientations settings. We give a polynomial-time 6-approximation algorithm for
finding a minimum cost subgraph of an undirected graph that admits an
orientation covering a nonnegative crossing -supermodular demand function,
as defined by Frank. An important example is -edge-connectivity, a
common generalization of global and rooted edge-connectivity.
Our algorithm is based on a non-standard application of the iterative
rounding method. We observe that the standard linear program with cut
constraints is not amenable and use an alternative linear program with
partition and co-partition constraints instead. The proof requires a new type
of uncrossing technique on partitions and co-partitions.
We also consider the problem setting when the cost of an edge can be
different for the two possible orientations. The problem becomes substantially
more difficult already for the simpler requirement of -edge-connectivity.
Khanna, Naor, and Shepherd showed that the integrality gap of the natural
linear program is at most when and conjectured that it is constant
for all fixed . We disprove this conjecture by showing an
integrality gap even when
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