Revisiting Directed Disjoint Paths on tournaments (and relatives)

TL;DR

This work decisively patterns the complexity landscape of Directed Disjoint Paths on tournaments and related digraphs by providing a correct NP-hardness proof for k-DDP on tournaments and deriving a spectrum of congestion-based results. It introduces a congestion-aware framework that yields FPT algorithms for (k,c)-DDP on semicomplete and related digraphs when c > k/2, while also proving tight hardness results that keep the problem intractable for many parameter regimes, including on graphs with bounded directed pathwidth. A key technical contribution is the k-triple structure and a sophisticated irrelevant-vertex argument, refined to accommodate vertex-disjoint paths and congestion, plus a congestionverse construction to transfer hardness to congested variants. Collectively, these results delineate where XP- and FPT-type approaches can be expected to succeed and where underlying hardness remains, informing both theory and algorithm design for directed disjoint-path problems on dense digraphs.

Abstract

In the Directed Disjoint Paths problem ($k$-DDP), we are given a digraph $k$ pairs of terminals, and the goal is to find $k$ pairwise vertex-disjoint paths connecting each pair of terminals. Bang-Jensen and Thomassen [SIAM J. Discrete Math. 1992] claimed that $k$-DDP is NP-complete on tournaments, and this result triggered a very active line of research about the complexity of the problem on tournaments and natural superclasses. We identify a flaw in their proof, which has been acknowledged by the authors, and provide a new NP-completeness proof. From an algorithmic point of view, Fomin and Pilipczuk [J. Comb. Theory B 2019] provided an FPT algorithm for the edge-disjoint version of the problem on semicomplete digraphs, and showed that their technique cannot work for the vertex-disjoint version. We overcome this obstacle by showing that the version of $k$-DDP where we allow congestion $c$ on the vertices is FPT on semicomplete digraphs provided that $c$ is greater than $k/2$. This is based on a quite elaborate irrelevant vertex argument inspired by the edge-disjoint version, and we show that our choice of $c$ is best possible for this technique, with a counterexample with no irrelevant vertices when $c \leq k/2$. We also prove that $k$-DDP on digraphs that can be partitioned into $h$ semicomplete digraphs is $W[1]$-hard parameterized by $k+h$, which shows that the XP algorithm presented by Chudnovsky, Scott, and Seymour [J. Comb. Theory B 2019] is essentially optimal.

Figures (6)

• Figure 1: A $4$-triple $(A, B, C)$. We remark that nothing is known about the arcs inside $A$, $B$, or $C$ nor about arcs between $C$ and $A$ other than the ones in the matching.
• Figure 2: Counterexample for $n = 2$, where non-black vertices of the same color correspond to endpoints of a same request. The rectangles indicate the three sets that make up the $n$-triple.
• Figure 3: Directed butterfly gadget for variable $x_i$, occurring negated in clause $C_d$ and unnegated in clauses $C_a, C_b,$ and $C_c$. White vertices represent terminals. The given orientations of the gray arcs will be important when talking about congested versions of the problem. Vertex $\alpha_i$ is the center of $B_i$, while the two disjoint paths from the $s$ vertices to the $t$ vertices that do not use $\alpha_i$ are its wings.
• Figure 4: Queuing of the butterfly gadgets performed for the construction of $T'$. Gray arcs do not participate in the critical path of $T'$, while black arcs do. Again, within the butterfly gadgets, we add the missing arcs from top-to-bottom, right-to-left. All other arcs are also right-to-left.
• Figure 5: Queuing of the butterfly gadgets performed for the construction of $G$ with the extended critical path on the left of the figure. Each of the two smaller rectangles are cliques in the host graph. Within them, missing arcs are from top-to-bottom, right-to-left. Across the thick vertical line, the only arcs are from vertices of the form $\ell^a_i \in B_i$ to $q_a$, which exists if and only if $\ell_i \in C_a$; these are shown by the dashed arc from the first butterfly to $q_1$. The dotted arc is not considered part of the critical path. Arc $(t^*, s^*) \in E(G)$ is omitted for simplicity.

Theorems & Definitions (27)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Definition 5: $k$-triple
• Lemma 6
• Lemma 8
• Theorem 8
• Theorem 9
• Theorem 9