Constant Approximating Disjoint Paths on Acyclic Digraphs is W[1]-hard

TL;DR

This work shows that in this setting Max Disjoint Paths is W[1]-hard to $c$-approximate for any constant $c$ for any constant $c$.

Abstract

In the Disjoint Paths problem, one is given a graph with a set of $k$ vertex pairs $(s_i,t_i)$ and the task is to connect each $s_i$ to $t_i$ with a path, so that the $k$ paths are pairwise disjoint. In the optimization variant, Max Disjoint Paths, the goal is to maximize the number of vertex pairs to be connected. We study this problem on acyclic directed graphs, where Disjoint Paths is known to be W[1]-hard when parameterized by $k$. We show that in this setting Max Disjoint Paths is W[1]-hard to $c$-approximate for any constant $c$. To the best of our knowledge, this is the first non-trivial result regarding the parameterized approximation for Max Disjoint Paths with respect to the natural parameter $k$. Our proof is based on an elementary self-reduction that is guided by a certain combinatorial object constructed by the probabilistic method.

Figures (2)

• Figure 1: An illustration for \ref{['def:game:instance']} with $k=d=3$. The boxes represent copies of an instance $I$ with $|\mathcal{T}|=3$, the large instance is $J_{3,3}(I,\beta)$ for the scheme $\beta$ listed at the bottom, and the dashed rectangle surrounds the instance $J_3 = J_{3,2}(I,\beta_3)$ where $\beta_3$ is a truncation of $\beta$ to the right subtree of $T_{3,3}$. The hollow disks represent the sinks and sources on the large instance. All the arcs are oriented upwards. The leaves of $T_{3,3}$ are numbered as $1,2,\dots,27$. For the sake of legibility, most of the arcs in the last layer are omitted and the copies of the original instance within layers $2,3$ are marked with letters. The letters are also used in the representation of the scheme $\beta$ which contains 9 bijections between sets of size 3 and 3 bijections between sets of size 9 (and one bijection for size 27, which is immaterial here). The blue lines exemplify vertex pairs which belong to the request set of the large instance; the sources (in the layer 1) indexed by $6,10,23$ are mapped to the sinks in the copy $M$ (in the layer 3). If a subset $\Gamma \subseteq [27]$ includes $6,10,23$ then it has a collision with respect to the scheme $\beta$. If we work with a no-instance then such a subset $\Gamma$ of requests cannot be served as this would require routing three of them through the copy $M$.
• Figure 2: An illustration for \ref{['lem:scheme:F']}. We consider layers $F_0, F_{\tau}, F_{2\tau},\, \dots$ The vertices from $F_0^+$ and $F_\tau^+$ are marked by black disks and their subtrees $F^{v,\tau}_{k,d}$ are depicted as gray triangles. For each vertex $v \in F^+$ we apply \ref{['lem:tree-one']} to identify a vertex $\gamma(v) \in F$: the red square inside the corresponding triangle. The root also illustrates the argument from \ref{['lem:tree-one']}. We start with a vertex $v$ satisfying $\mathsf{Frac}(v) \ge \frac{1}{2q}$ and while one of its children $v'$ has $\mathsf{Frac}(v') < \frac{1}{4q}$ we can find another child $v"$ of $v$ with $\mathsf{Frac}(v") > \mathsf{Frac}(v)$. This process terminates within $\tau$ steps.

Theorems & Definitions (17)

• Theorem 1
• Definition 2
• Lemma 3: joag1983negative
• Lemma 4
• Theorem 5: joag1983negative
• Definition 6
• Definition 7
• Definition 9
• Lemma 10
• Theorem 11