Transiently Consistent SDN Updates: Being Greedy is Hard

The software-defined networking paradigm introduces interesting opportunities to operate networks in a more flexible, optimized, yet formally verifiable manner. Despite the logically centralized control, however, a Software-Defined Network (SDN) is still a distributed system, with inherent delays between the switches and the controller. Especially the problem of changing network configurations in a consistent manner, also known as the consistent network update problem, has received much attention over the last years. In particular, it has been shown that there exists an inherent tradeoff between update consistency and speed. This paper revisits the problem of updating an SDN in a transiently consistent, loop-free manner. First, we rigorously prove that computing a maximum (greedy) loop-free network update is generally NP-hard; this result has implications for the classic maximum acyclic subgraph problem (the dual feedback arc set problem) as well. Second, we show that for special problem instances, fast and good approximation algorithms exist

The Minimum Shared Edges Problem on Grid-like Graphs

We study the NP-hard Minimum Shared Edges (MSE) problem on graphs: decide whether it is possible to route $p$ paths from a start vertex to a target vertex in a given graph while using at most $k$ edges more than once. We show that MSE can be decided on bounded (i.e. finite) grids in linear time when both dimensions are either small or large compared to the number $p$ of paths. On the contrary, we show that MSE remains NP-hard on subgraphs of bounded grids. Finally, we study MSE from a parametrised complexity point of view. It is known that MSE is fixed-parameter tractable with respect to the number $p$ of paths. We show that, under standard complexity-theoretical assumptions, the problem parametrised by the combined parameter $k$, $p$, maximum degree, diameter, and treewidth does not admit a polynomial-size problem kernel, even when restricted to planar graphs

Transversal structures on triangulations: a combinatorial study and straight-line drawings

This article focuses on a combinatorial structure specific to triangulated plane graphs with quadrangular outer face and no separating triangle, which are called irreducible triangulations. The structure has been introduced by Xin He under the name of regular edge-labelling and consists of two bipolar orientations that are transversal. For this reason, the terminology used here is that of transversal structures. The main results obtained in the article are a bijection between irreducible triangulations and ternary trees, and a straight-line drawing algorithm for irreducible triangulations. For a random irreducible triangulation with $n$ vertices, the grid size of the drawing is asymptotically with high probability $11n/27\times 11n/27$ up to an additive error of \cO(\sqrt{n}). In contrast, the best previously known algorithm for these triangulations only guarantees a grid size $(\lceil n/2\rceil -1)\times \lfloor n/2\rfloor$.Comment: 42 pages, the second version is shorter, focusing on the bijection (with application to counting) and on the graph drawing algorithm. The title has been slightly change