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

The paper investigates the Minimum Cost Homomorphism problem (MinHOM $H$) for a fixed target digraph $H$, where each vertex $u$ of an input digraph $G$ carries a cost $c_i(u)$ for being mapped to a vertex $i$ of $H$. The objective is to decide whether a homomorphism $f:V(G)\rightarrow V(H)$ exists and, if so, to find one of minimum total cost $\sum_{u\in V(G)}c_{f(u)}(u)$. This framework subsumes many classic optimization tasks such as minimum‑cost colourings, list homomorphisms, and network repair problems.

The authors focus on two well‑studied generalisations of tournaments: locally semicomplete digraphs and quasi‑transitive digraphs. A digraph is locally semicomplete if every strong component induces a complete digraph; a digraph is quasi‑transitive if $u\rightarrow v$ and $v\rightarrow w$ imply $u\rightarrow w$ or $w\rightarrow u$. Both classes enjoy rich structural characterisations (strong‑component decompositions, orderings, forbidden subdigraphs) that the paper exploits.

The main contribution is a complete dichotomy classification of the computational complexity of MinHOM $H$ for every $H$ belonging to either class. The classification splits into two mutually exclusive regimes:

1. Polynomial‑time solvable cases. When $H$ is locally semicomplete and admits a reflexive ordering (i.e., its strong components can be linearly ordered so that all arcs go forward), the problem can be reduced to a minimum‑cost flow model. Each vertex of $G$ is associated with a list of admissible images in $H$, and the cost structure becomes edge capacities and costs in a flow network. Standard min‑cost max‑flow algorithms then compute an optimal homomorphism in polynomial time. The same algorithm works when $H$ is quasi‑transitive and its underlying ordering is transitive, because the ordering guarantees that the feasibility constraints are totally unimodular.

2. NP‑hard cases. If $H$ fails the reflexive‑ordering condition—e.g., it contains a directed 3‑cycle or any of the forbidden subdigraphs identified for locally semicomplete digraphs—or if a quasi‑transitive $H$ exhibits a non‑transitive ordering pattern, the authors construct a polynomial‑time reduction from 3‑SAT or from the Minimum Cost Colouring problem. The reduction encodes variables and clauses into gadgets whose feasible mappings correspond exactly to satisfying assignments, while the cost function forces the homomorphism to respect the logical structure. Consequently, MinHOM $H$ is NP‑complete in these situations.

The paper also shows that the dichotomy is robust with respect to the nature of the costs: integer, rational, or even negative costs can be transformed without affecting the classification. Moreover, the results align with known dichotomies for the unweighted List Homomorphism problem, confirming that adding a cost dimension does not create new intermediate complexity classes for these digraph families.

Beyond the theoretical classification, the authors discuss practical implications. In network design or data‑repair scenarios, the target digraph $H$ often encodes permissible routing or consistency constraints. If $H$ falls into the polynomial regime, optimal solutions can be obtained efficiently via flow algorithms; otherwise, practitioners must resort to approximation, heuristics, or fixed‑parameter tractable methods.

The paper concludes with open directions: extending the dichotomy to broader digraph families, investigating parameterised complexity (e.g., treewidth of $G$), and developing experimental evaluations of the proposed algorithms on real‑world datasets.