Forbidden Tournaments and the Orientation Completion Problem

TL;DR

The paper establishes a complete complexity dichotomy for the $\mathcal{F}$-free orientation problem on finite graphs, for any fixed finite set of tournaments $\mathcal{F}$. It reduces the problem to the $\mathcal{F}$-free orientation completion problem and analyzes it via Fraïssé limits $D_{\mathcal{F}}$ and their underlying graphs $H_{\mathcal{F}}$, linking to CSPs over a Boolean structure $\mathfrak{B}_{\mathcal{F}}$ and to $D_{\mathcal{F}}$ via primitive positive interpretations. The classification splits into two cases: (i) if $K_3$ primitively positively interprets in the corresponding Boolean/graph structures, the problem is NP-complete; (ii) otherwise, the problems are in P, governed by a minority polymorphism (and a pseudo weak near unanimity in the orientation completion setting). The results connect finite forbidden-pattern problems to MMSNP$_2$, CSP dichotomies, and infinite permutation-group methods, and yield a suite of concrete corollaries for small forbidden sets and for transitive vs nontransitive tournaments. These findings advance a unified framework for orientation problems with forbidden patterns and lay groundwork for extending dichotomies to broader MMSNP$_2$ classes.

Abstract

For a fixed finite set of finite tournaments ${\mathcal F}$, the ${\mathcal F}$-free orientation problem asks whether a given finite undirected graph $G$ has an $\mathcal F$-free orientation, i.e., whether the edges of $G$ can be oriented so that the resulting digraph does not embed any of the tournaments from ${\mathcal F}$. We prove that for every ${\mathcal F}$, this problem is in P or NP-complete. Our proof reduces the classification task to a complete complexity classification of the orientation completion problem for ${\mathcal F}$, which is the variant of the problem above where the input is a directed graph instead of an undirected graph, introduced by Bang-Jensen, Huang, and Zhu (2017). Our proof uses results from the theory of constraint satisfaction, and a result of Agarwal and Kompatscher (2018) about infinite permutation groups and transformation monoids.

Figures (5)

• Figure 1: The eight labeled tournaments on $3$ vertices. The labels correspond to the associated tuple $(x_{1,2},x_{1,3},x_{2,3})$ where $x_{i,j} = 1$ if $(i,j)\in E(G)$, and $x_{i,j} = 0$ if $(j,i)\in E(G)$ for $1\le i < j \le 3$.
• Figure 2: Three digraphs $D_1$, $D_2$, and $D_3$ where in each case, the pair $(x,y)$ forces the pair $(u,v)$ with respect to $\{\overrightarrow{C_3}\}$. In $D_1$, the cardinality of $\{x,y,u,v\}$ is $3$. In $D_2$, $|\{x,y,u,v\}| = 4$, and it is obtained from $D_1$ as in the proof of \ref{['lem:mutual-implication']}. Finally, in $D_3$, $d(a,b) \ge 2$ for any $a\in\{x,y\}$ and $b\in \{u,v\}$, and $D_3$ is obtained from $D_2$ as in the proof of \ref{['lem:mutual-implication']} for $k = 2$.
• Figure 3: The four non-isomorphic oriented tournaments on $4$ vertices
• Figure 4: An illustration of the minority operation acting on a triple $(T^1,T^2,T^3)$ of tournaments isomorphic to $TC_4$, which yields a tournament isomorphic $T_4$.
• Figure 5: A gadget for reducing not-all-equal $3$-SAT to the $\overrightarrow{C_3}$-free orientation completion. Equivalently, the interpreting formula for the primitive positive interpretation of $(\{0,1\},\{0,1\}^3\setminus\{(0,0,0),(1,1,1)\})$ in $(D_\mathcal{F}, U)$ for $\mathcal{F} = \{\overrightarrow{C_3}\}$.

Theorems & Definitions (97)

• Theorem 1: see, e.g., Hodges
• Example 2
• Example 3
• Lemma 6
• proof
• Theorem 7: BulatovFVConjectureZhukFVConjecture
• Lemma 8: see, e.g., JBKBook
• Theorem 9: see, e.g., BoKaKoRoGeiger
• Lemma 10: see, e.g., Book
• Corollary 11