Restricted CSPs and F-free Digraph Algorithmics

TL;DR

This work develops a unified framework tying restricted digraph homomorphism problems to constraint satisfaction theory, introducing Restricted CSPs (RCSPs) and exploring their finite-domain dichotomies, with a key emphasis on $ ext{RCSP}( ext{A}, ext{B})$ being in P or NP-hard depending on rpp-constructibility to $({\mathbb K}_3,{\mathbb L})$. It proves a strong NP-hardness persistence result: if CSP$(\mathbb H)$ is NP-hard, it remains NP-hard on acyclic and high-girth instances and identifies a boundary at $\vec{\mathbb P}_{N-1}$ for polynomial-time solvability of acyclic, restricted inputs. The paper further establishes a finite-domain RCSP dichotomy, ties RCSPs to the finite-domain CSP dichotomy via exponential-pp constructions, and develops a GMSNP- and MMSNP-driven tractability frontier for restricted inputs. It then applies these general results to concrete small-digraph families (three-vertex digraphs) and a family of smooth tournaments, deriving precise complexity classifications under $\vec{\mathbb P}_k$-free and $\vec{\mathbb P}_k$-subgraph-free restrictions and outlining a roadmap toward a comprehensive long-term dichotomy for restricted digraph CSPs. The results illuminate the deep connections between forbidden-substructure algorithmics, dualities, and algebraic CSP dichotomies, with broad implications for digraph homomorphism problems and their restricted variants.

Abstract

In recent years, much attention has been placed on the complexity of graph homomorphism problems when the input is restricted to ${\mathbb P}_k$-free and ${\mathbb P}_k$-subgraph-free graphs. We consider the directed version of this research line, by addressing the questions, is it true that digraph homomorphism problems CSP$({\mathbb H})$ have a P versus NP-complete dichotomy when the input is restricted to $\vec{\mathbb P}_k$-free (resp.\ $\vec{\mathbb P}_k$-subgraph-free) digraphs? Our main contribution in this direction shows that if CSP$({\mathbb H})$ is NP-complete, then there is a positive integer $N$ such that CSP$({\mathbb H})$ remains NP-hard even for $\vec{\mathbb P}_N$-subgraph-free digraphs. Moreover, it remains NP-hard for acyclic $\vec{\mathbb P}_N$-subgraph-free digraphs, and becomes polynomial-time solvable for $\vec{\mathbb P}_{N-1}$-subgraph-free acyclic digraphs. We then verify the questions above for digraphs on three vertices and a family of smooth tournaments. We prove these results by establishing a connection between $\mathbb F$-(subgraph)-free algorithmics and constraint satisfaction theory. On the way, we introduce restricted CSPs, i.e., problems of the form CSP$({\mathbb H})$ restricted to yes-instances of CSP$({\mathbb H}')$ -- these were called restricted homomorphism problems by Hell and Nešetřil. Another main result of this paper presents a P versus NP-complete dichotomy for these problems. Moreover, this complexity dichotomy is accompanied by an algebraic dichotomy in the spirit of the finite domain CSP dichotomy.

Figures (11)

• Figure 1: A depiction of the flow between main results and main questions addressed in this paper. Rectangles with rounded corners mark the main questions considered here. Dotted squares indicate the (simple) remarks connecting the ${\mathcal{F}}$-subgraph-free CSPs to the theory of digraph homomorphisms. Solid rectangles with straight corners indicate the main results of the present paper. A solid (resp. dashed) edge between a question and a result indicates that the result provides an answer (resp. a partial answer) to the corresponding question. Finally, edges between results represent that the result at the head is proved using tools introduced while proving the result at the tail of the corresponding edge.
• Figure 2: Let $\Delta$ be the primitive positive definition (of $\{E\}$ in $\{E\}$) where $\delta_E(x,y):=\exists z_1,z_2.\; E(x,z_1)\land E(z_1,z_2)\land E(z_2,y)$. On the left, we depict ${\mathbb C}_5$ and its pp power $\Pi_\Delta({\mathbb C}_5)\cong {\mathbb K}_5$ (an undirected edge $xy$ represents $(x,y)$ and $(y,x)$), and on the right, we depict ${\mathbb K}_2$ and its gadget replacement $\Gamma_\Delta({\mathbb K}_2)\cong \vec{{\mathbb C}}_6$ (dashed edges and solid edges indicate the respective edge replacements).
• Figure 3: The exponential $\vec{{\mathbb C}}_3^{{\mathbb K}_2}$ where a label $ij$ represents the function $f\colon \{1,2\}\to\{1,2,3\}$ defined by $1\mapsto i$ and $2\mapsto j$.
• Figure 4: The three digraphs on three vertices with at least two directed cycles, and whose CSP is NP-complete.
• Figure 5: A $\{E,U\}$-structure where $E$ is a binary relation represented by arcs, and $U$ is a unary relation represented by black vertices.

Theorems & Definitions (101)

• Theorem 1: Conjecture 6.1 in bangjensenDAM26 proved in BartoKozikNiven
• Theorem 2: Conjecture 2.5 in bangjensenDM138 proved in BartoKozikNiven
• Theorem 3: Theorem 3 in brewsterDAM156
• Theorem 5: Theorems 2 and 11 in FoniokNesetril
• Theorem 6: Sparse Incomparability Lemma Kun
• Corollary 7
• Remark 8
• Lemma 9
• proof
• Lemma 10: Corollary 3.5 in wonderland