In the Vertex-Disjoint-Paths-With-Congestion problem, the input consists of a digraph D, an integer c and k pairs of vertices (s_i, t_i), and the task is to find a set of paths connecting each s_i to its corresponding t_i, whereas each vertex of D appears in at most c many paths. The case where c = 1 is known to be NP-complete even if k = 2 [Fortune, Hopcroft and Wyllie, 1980] on general digraphs and is W[1]-hard with respect to k (excluding the possibility of an f(k)n^O(1)-time algorithm under standard assumptions) on acyclic digraphs [Slivkins, 2010]. The proof of [Slivkins, 2010] can also be adapted to show W[1]-hardness with respect to k for every congestion c ≥ 1. We strengthen the existing hardness result by showing that the problem remains W[1]-hard for every congestion c ≥ 1 even if: (1) the input digraph D is acyclic, (2) D does not contain an acyclic (5, 5)-grid as a butterfly minor, (3) D does not contain an acyclic tournament on 9 vertices as a butterfly minor, and (4) D has ear-anonymity at most 5. Further, we also show that the edge-congestion variant of the problem remains W[1]-hard for every congestion c ≥ 1 even if: (1) the input digraph D is acyclic, (2) D has maximum undirected degree 3, (3) D does not contain an acyclic (7, 7)-wall as a weak immersion and (4) D has ear-anonymity at most 5.

In 2015, Kawarabayashi and Kreutzer proved the directed grid theorem - the generalisation of the well-known excluded grid theorem to directed graphs - confirming a conjecture by Reed, Johnson, Robertson, Seymour, and Thomas from the mid-nineties. The theorem states the existence of a function$f$such that every digraph of directed tree-width$f(k)$contains a cylindrical grid of order$k$as a butterfly minor, but the given function grows non-elementarily with the size of the grid minor. More precisely, it contains a tower whose height depends on the size of the grid. In this paper, we present an alternative proof of the directed grid theorem which is conceptually much simpler, more modular in its composition and also improves the upper bound for the function$f$to a power tower of height 22. Our proof is inspired by the breakthrough result of Chekuri and Chuzhoy, who proved a polynomial bound for the excluded grid theorem for undirected graphs. We translate a key concept of their proof to directed graphs by introducing cycles of well-linked sets (CWS), and show that any digraph of high directed tree-width contains a large CWS, which in turn contains a large cylindrical grid, improving the result due to Kawarabayashi and Kreutzer from a non-elementary to an elementary function. An immediate application of our result is that we can improve the bound for Younger's conjecture-the directed Erdős-Pósa property-proved by Reed, Robertson, Seymour and Thomas [2] from a non-elementary to an elementary function. The same improvement applies to other types of Erdős-Pósa style problems on directed graphs. To the best of our knowledge, this is the first significant improvement on the bound for Younger's conjecture since it was proved in 1996. Since its publication in STOC 2015, the Directed Grid Theorem has found numerous applications (see for example [3]–[7]), all of which directly benefit from our main result. Finally, we believe that the theoretical tools developed in this work may find applications beyond the directed grid theorem, in a similar way as the path-of-sets-system framework due to Chekuri and Chuzhoy [8] did for undirected graphs (see for example [9]–[11]).

Abstract In Directed Feedback Arc Set (DFAS) we search for a set of at most k arcs which intersect every cycle in the input digraph. It is a well-known open problem in parameterized complexity to decide if DFAS admits a kernel of polynomial size. We consider $$\mathcal {C}$$ C -Arc Deletion Set ( $$\mathcal {C}$$ C -ADS), a variant of DFAS where we want to remove at most k arcs from the input digraph in order to turn it into a digraph of a class $$\mathcal {C}$$ C . In this work, we choose $$\mathcal {C}$$ C to be the class of funnels. Funnel-ADS is NP-hard even if the input is a DAG, but is fixed-parameter tractable with respect to k. So far no polynomial kernels for this problem were known. Our main result is a kernel for Funnel-ADS with $$\mathcal {O}(k^6)$$ O ( k 6 ) many vertices and $$\mathcal {O}(k^7)$$ O ( k 7 ) many arcs, computable in $$\mathcal {O}(nm)$$ O ( n m ) time, where n is the number of vertices and m the number of arcs in the input digraph.

In Directed Feedback Arc Set (DFAS) we search for a set of at most k arcs which intersect every cycle in the input digraph. It is a well-known open problem in parameterized complexity to decide if DFAS admits a kernel of polynomial size. We consider 𝒞-Arc Deletion Set (𝒞-ADS), a variant of DFAS where we want to remove at most k arcs from the input digraph in order to turn it into a digraph of a class 𝒞. In this work, we choose 𝒞 to be the class of funnels. Funnel-ADS is NP-hard even if the input is a DAG, but is fixed-parameter tractable with respect to k. So far no polynomial kernel for this problem was known. Our main result is a kernel for Funnel-ADS with 𝒪(k⁶) many vertices and 𝒪(k⁷) many arcs, computable in 𝒪(nm) time, where n is the number of vertices and m the number of arcs of the input digraph.

There has been intensive work on the parameterized complexity of the typically NP-hard task to edit undirected graphs into graphs fulfilling certain given vertex degree constraints. In this work, we lift the investigations to the case of directed graphs; herein, we focus on arc insertions. To this end, we develop a general two-stage framework which consists of efficiently solving a problem-specific number problem and transferring its solution to a solution for the graph problem by applying flow computations. In this way, we obtain fixed-parameter tractability and polynomial kernelizability results, with the central parameter being the maximum vertex in- or outdegree of the output digraph. Although there are certain similarities with the much better studied undirected case, the flow computation used in the directed case seems not to work for the undirected case while f -factor computations as used in the undirected case seem not to work for the directed case.