Routing with Congestion in Acyclic Digraphs

We study the version of the $k$-disjoint paths problem where $k$ demand pairs $(s_1,t_1)$, $\dots$, $(s_k,t_k)$ are specified in the input and the paths in the solution are allowed to intersect, but such that no vertex is on more than $c$ paths. We show that on directed acyclic graphs the problem is solvable in time $n^{O(d)}$ if we allow congestion $k-d$ for $k$ paths. Furthermore, we show that, under a suitable complexity theoretic assumption, the problem cannot be solved in time $f(k)n^{o(d/\log d)}$ for any computable function $f$

Congestion-Free Rerouting of Flows on DAGs

Changing a given configuration in a graph into another one is known as a reconfiguration problem. Such problems have recently received much interest in the context of algorithmic graph theory. We initiate the theoretical study of the following reconfiguration problem: How to reroute k unsplittable flows of a certain demand in a capacitated network from their current paths to their respective new paths, in a congestion-free manner? This problem finds immediate applications, e.g., in traffic engineering in computer networks. We show that the problem is generally NP-hard already for k=2 flows, which motivates us to study rerouting on a most basic class of flow graphs, namely DAGs. Interestingly, we find that for general k, deciding whether an unsplittable multi-commodity flow rerouting schedule exists, is NP-hard even on DAGs. Our main contribution is a polynomial-time (fixed parameter tractable) algorithm to solve the route update problem for a bounded number of flows on DAGs. At the heart of our algorithm lies a novel decomposition of the flow network that allows us to express and resolve reconfiguration dependencies among flows

New Menger-like dualities in digraphs and applications to half-integral linkages

We present new min-max relations in digraphs between the number of paths satisfying certain conditions and the order of the corresponding cuts. We define these objects in order to capture, in the context of solving the half-integral linkage problem, the essential properties needed for reaching a large bramble of congestion two (or any other constant) from the terminal set. This strategy has been used ad-hoc in several articles, usually with lengthy technical proofs, and our objective is to abstract it to make it applicable in a simpler and unified way. We provide two proofs of the min-max relations, one consisting in applying Menger's Theorem on appropriately defined auxiliary digraphs, and an alternative simpler one using matroids, however with worse polynomial running time. As an application, we manage to simplify and improve several results of Edwards et al. [ESA 2017] and of Giannopoulou et al. [SODA 2022] about finding half-integral linkages in digraphs. Concerning the former, besides being simpler, our proof provides an almost optimal bound on the strong connectivity of a digraph for it to be half-integrally feasible under the presence of a large bramble of congestion two (or equivalently, if the directed tree-width is large, which is the hard case). Concerning the latter, our proof uses brambles as rerouting objects instead of cylindrical grids, hence yielding much better bounds and being somehow independent of a particular topology. We hope that our min-max relations will find further applications as, in our opinion, they are simple, robust, and versatile to be easily applicable to different types of routing problems in digraphs

Constant Congestion Routing of Symmetric Demands in Planar Directed Graphs

We study the problem of routing symmetric demand pairs in planar digraphs. The input consists of a directed planar graph G = (V, E) and a collection of k source-destination pairs M = {s_1t_1, ..., s_kt_k}. The goal is to maximize the number of pairs that are routed along disjoint paths. A pair s_it_i is routed in the symmetric setting if there is a directed path connecting s_i to t_i and a directed path connecting t_i to s_i. In this paper we obtain a randomized poly-logarithmic approximation with constant congestion for this problem in planar digraphs. The main technical contribution is to show that a planar digraph with directed treewidth h contains a constant congestion crossbar of size Omega(h/polylog(h))

Algorithmic Graph Theory

The main focus of this workshop was on mathematical techniques needed for the development of efficient solutions and algorithms for computationally difficult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions