arXiv subjects

Marcelo Garlet Milani

Publications and source records attributed to Marcelo Garlet Milani.

5 recordsLinked to original sources

arXiv2026

Cycles of Well-Linked Sets I: an Elementary Bound for Directed Cycle Packing

In 1996, Reed, Robertson, Seymour and Thomas [Combinatorica 1996] proved Younger's Conjecture, which states that, for all directed graphs $D$, there exists a function $f$ such that, if $D$ does not contain $k$ disjoint cycles, then $D$ contains a feedback vertex set, i.e.~a subset of vertices whose deletion renders the graph acyclic, of size bounded by $f(k)$. However, the function obtained by Reed, Robertson, Seymour and Thomas in their paper is enormous and, in fact, not even elementary. We prove the first elementary upper bound for the function $f$ above, showing it is upper-bounded by a power tower of height 8. Our proof is inspired by the breakthrough result of Chekuri and Chuzhoy [J. ACM 2016], 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 paths of well-linked sets (PWS), and show that any digraph of large directed treewidth contains a large PWS, which in turn contains a large fence. We believe that the theoretical tools developed in this work may find applications beyond the results above, in a similar way as the path-of-sets-system framework due to Chekuri and Chuzhoy [J. ACM 2016] did for undirected graphs (see, for example, Hatzel, Komosa, Pilipczuk and Sorge [Discret. Math. Theor. Comput. Sci. 2022], Chekuri and Chuzhoy [SODA 2015] and Chuzhoy and Nimavat [arXiv 2019]). Indeed, in a follow-up paper, we apply this framework to improve the bounds of the Directed Grid Theorem.

cs.DM↗

arXiv2026

Cycles of Well-Linked Sets II: an Elementary Bound for the Directed Grid Theorem

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 that there is a function $f$ such that every digraph of directed treewidth $f(k)$ contains a cylindrical grid of order $k$ as a butterfly minor. However, the given function grows faster than any non-elementary function of the size of the grid minor. More precisely, it is larger than a power 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 composition and improves the upper bound for the function $f$ to a power tower of height $22$. A key concept of our proof is a new structure called cycles of well-linked sets (CWS). We show that any digraph of large directed treewidth contains a large CWS, which in turn contains a large cylindrical grid.

cs.DM↗

arXiv2025

Directed disjoint paths remains W[1]-hard on acyclic digraphs without large grid minors

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 $c \geq 1$. We strengthen the existing hardness result by showing that the problem remains W[1]-hard for every congestion $c \geq 1$ even if: - the input digraph $D$ is acyclic, - $D$ does not contain an acyclic $(5, 5)$-grid as a butterfly minor, - $D$ does not contain an acyclic tournament on 9 vertices as a butterfly minor, and - $D$ 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 $c \geq 1$ even if: - the input digraph $D$ is acyclic, - $D$ has maximum undirected degree 3, - $D$ does not contain an acyclic $(7, 7)$-wall as a weak immersion and - $D$ has ear-anonymity at most 5.

cs.CC↗

arXiv2024

Directed Ear Anonymity

We define and study a new structural parameter for directed graphs, which we call \emph{ear anonymity}. Our parameter aims to generalize the useful properties of \emph{funnels} to larger digraph classes. In particular, funnels are exactly the acyclic digraphs with ear anonymity one. We prove that computing the ear anonymity of a digraph is \NP/-hard and that it can be solved in $O(m(n + m))$-time on acyclic digraphs (where $n$ is the number of vertices and $m$ is the number of arcs in the input digraph). It remains open where exactly in the polynomial hierarchy the problem of computing ear anonymity lies, however for a related problem we manage to show $Σ_2^p$-completeness.

cs.DS↗

arXiv2020

A Polynomial Kernel for Funnel Arc Deletion Set

In Directed Feeback 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 $\mathcal{C}$-Arc Deletion Set ($\mathcal{C}$-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 $\mathcal{C}$. In this work, we choose $\mathcal{C}$ to be the class of funnels. Funnel-Arc Deletion Set 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-Arc Deletion Set with $\mathcal{O}(k^6)$ many vertices and $\mathcal{O}(k^7)$ many arcs, computable in $\mathcal{O}(nm)$ time, where $n$ is the number of vertices and $m$ the number of arcs in the input digraph.

cs.DS↗