Minimum Cost Homomorphisms to Locally Semicomplete and Quasi-Transitive Digraphs

For digraphs $G$ and $H$, a homomorphism of $G$ to $H$ is a mapping $f:\ V(G)\dom V(H)$ such that $uv\in A(G)$ implies $f(u)f(v)\in A(H)$. If, moreover, each vertex $u \in V(G)$ is associated with costs $c_i(u), i \in V(H)$, then the cost of a homomorphism $f$ is $\sum_{u\in V(G)}c_{f(u)}(u)$. For each fixed digraph $H$, the minimum cost homomorphism problem for $H$, denoted MinHOM($H$), can be formulated as follows: Given an input digraph $G$, together with costs $c_i(u)$, $u\in V(G)$, $i\in V(H)$, decide whether there exists a homomorphism of $G$ to $H$ and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems such as the minimum cost chromatic partition and repair analysis problems. We focus on the minimum cost homomorphism problem for locally semicomplete digraphs and quasi-transitive digraphs which are two well-known generalizations of tournaments. Using graph-theoretic characterization results for the two digraph classes, we obtain a full dichotomy classification of the complexity of minimum cost homomorphism problems for both classes.

💡 Research Summary

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The paper investigates the computational complexity of the Minimum Cost Homomorphism problem (MinHOM) for two important families of directed graphs that generalize tournaments: locally semicomplete digraphs and quasi‑transitive digraphs. A homomorphism from an input digraph G to a fixed target digraph H maps each vertex of G to a vertex of H while preserving arcs; each possible mapping u→i carries a cost c_i(u). The objective is to decide whether a homomorphism exists and, if so, to find one of minimum total cost. This framework subsumes many well‑studied optimization problems such as list homomorphism, minimum cost coloring, and network repair problems, making its complexity classification a central question in combinatorial optimization.

The authors first review known dichotomies for undirected graphs and for several classes of digraphs (semicomplete, semicomplete multipartite, reflexive digraphs). They then focus on locally semicomplete digraphs—digraphs in which the in‑neighbourhood and out‑neighbourhood of every vertex each induce a semicomplete subdigraph—and quasi‑transitive digraphs—digraphs where for any directed path x→y→z at least one of the arcs x→z or z→x is present. Both classes contain tournaments as a proper subclass, yet they exhibit richer structural behavior.

Locally Semicomplete Digraphs.
Using a theorem of Bang‑Jensen, the authors show that any connected locally semicomplete digraph H that is not strongly connected can be decomposed into a unique linear ordering of its strong components H₁,…,H_p such that each component dominates the next one and there are no backward arcs. Moreover, each strong component is itself a semicomplete digraph. This ordering yields a natural vertex ordering 1,…,|V(H)| in which all arcs go forward. The authors prove that this ordering satisfies the Min‑Max property: for any arcs i→j and s→r with i<j, s<r, the “cross” arcs i→r and s→j must also be present. A digraph admitting a Min‑Max ordering is known to make MinHOM polynomial‑time solvable. Consequently, if a locally semicomplete digraph is either acyclic (hence non‑strong) or a directed cycle →C_k (k≥2), MinHOM(H) can be solved in polynomial time by constructing the Min‑Max ordering and applying the standard algorithm.

For all other locally semicomplete digraphs, the authors prove NP‑hardness. The reduction builds on known NP‑complete problems (e.g., 3‑SAT or graph coloring) and embeds them into a locally semicomplete structure that forces any feasible homomorphism to encode a solution of the original problem. The key obstacle is that such digraphs contain a strong component with a more intricate internal structure, preventing any Min‑Max ordering.

Quasi‑Transitive Digraphs.
The analysis for quasi‑transitive digraphs is more delicate. The authors introduce the bipartite graph B(H) obtained by splitting each vertex v into a white copy v′ and a black copy v″, and turning each arc v→w into an edge v′w″. They observe that if H is quasi‑transitive and acyclic, then H is a transitive oriented graph T. For such T, the bipartite graph B(T) is reflexive and, crucially, the authors show that MinHOM(T) is polynomial‑time solvable exactly when B(T) is a proper interval bigraph. Proper interval bigraphs are characterized by the absence of induced even cycles of length ≥6 and three specific forbidden subgraphs (biclaw, binet, bitent). Moreover, a proper interval bigraph admits a bipartite Min‑Max ordering (Lemma 2.1). The authors prove that any bipartite Min‑Max ordering of B(T) can be transformed into a Min‑Max ordering of T itself, thereby guaranteeing polynomial solvability.

In addition to the acyclic case, the authors identify two small cyclic families that are tractable: the directed 2‑cycle →C₂ (and its extensions) and the directed 3‑cycle →C₃ (and any digraph obtained by replacing each vertex of →C₃ by an independent set, i.e., an “extension” of →C₃). For these, earlier results already imply polynomial‑time algorithms.

The full tractable class for quasi‑transitive digraphs is therefore: 1. H is →C₂ or an extension of →C₃; 2. H is acyclic, B(H) is a proper interval bigraph, and H does not contain any of the four forbidden induced subdigraphs O₁, O₂, O₃, O₄ shown in Figure 1.

If any of these conditions fails, the authors construct NP‑hardness reductions. The presence of any O_i subdigraph or the violation of the proper interval bigraph property enables a reduction from known hard list‑homomorphism instances, establishing that MinHOM(H) becomes NP‑complete.

Algorithmic Consequences.
For the polynomial cases, the authors outline concrete algorithms:

• Detect whether H belongs to the tractable family (strong component decomposition, cycle detection, bipartite graph construction, forbidden subgraph testing). All these steps run in polynomial time.
• If tractable, compute a Min‑Max ordering (either directly from the linear ordering of strong components for locally semicomplete digraphs, or via the bipartite Min‑Max ordering for quasi‑transitive digraphs).
• Use the Min‑Max ordering to solve MinHOM by a simple dynamic programming or linear programming formulation that respects the ordering constraints.

For the NP‑hard cases, the paper provides explicit reductions and argues that no polynomial‑time algorithm exists unless P=NP.

Conclusion and Outlook.
The paper delivers a complete dichotomy for MinHOM on locally semicomplete and quasi‑transitive digraphs, extending the landscape of homomorphism‑based optimization problems on directed graphs. The results hinge on deep structural characterizations (strong component linearity, proper interval bigraphs, forbidden subdigraphs) and on the pivotal role of Min‑Max orderings. The techniques introduced—especially the translation between bipartite Min‑Max orderings and orderings of the original digraph—suggest promising avenues for tackling broader classes such as k‑transitive digraphs, digraphs with loops, or weighted homomorphism models with non‑linear costs. The work thus represents a significant step toward a unified theory of the computational complexity of minimum cost homomorphisms in directed settings.