[Paper Review] Minimum Cost Homomorphisms to Locally Semicomplete and Quasi-Transitive Digraphs

Arvind Gupta, Gregory Gutin+3 more|ArXiv.org|Dec 5, 2007

Advanced Graph Theory Research25 references3 citations

TL;DR

This paper establishes a complete dichotomy for the minimum cost homomorphism problem (MinHOM) on locally semicomplete and quasi-transitive digraphs. By leveraging structural graph theory of these classes, the authors prove that MinHOM(H) is polynomial-time solvable if each connected component of H is acyclic or a directed cycle (for locally semicomplete), or satisfies specific structural constraints including being an extension of C₃ or acyclic with a proper interval bigraph underlying structure and no induced Oᵢ obstructions (for quasi-transitive); otherwise, the problem is NP-hard.

ABSTRACT

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.

Motivation & Objective

• To classify the computational complexity of the minimum cost homomorphism problem (MinHOM) for locally semicomplete and quasi-transitive digraphs.
• To extend existing dichotomy results from semicomplete and semicomplete multipartite digraphs to broader classes of digraphs generalizing tournaments.
• To identify precise structural conditions on H that determine whether MinHOM(H) is polynomial-time solvable or NP-hard.
• To use graph-theoretic characterizations of locally semicomplete and quasi-transitive digraphs to derive complexity dichotomies.

Proposed method

• Applying structural graph theory to characterize locally semicomplete and quasi-transitive digraphs, particularly focusing on cycles, connectivity, and adjacency patterns.
• Using forbidden subgraph characterizations, including the Oᵢ obstructions (i=1,2,3,4), to define necessary and sufficient conditions for tractability.
• Reducing the problem to known NP-hard cases via induced subdigraph isomorphisms (e.g., H₁, H₂) to prove NP-hardness.
• Leveraging quasi-transitivity and local semicompleteness to constrain vertex adjacencies and derive contradictions in non-trivial cases.
• Proving dichotomies by analyzing component structure: acyclic, directed cycles, or extensions of C₃ with specific underlying bigraphs.
• Using lemmas and case analysis to show that any deviation from the tractable classes leads to NP-hardness through reduction to known hard instances.

Experimental results

Research questions

• RQ1 For which locally semicomplete digraphs H is the minimum cost homomorphism problem polynomial-time solvable?
• RQ2 What structural properties of quasi-transitive digraphs H determine the tractability of MinHOM(H)?
• RQ3 Are there forbidden subgraphs that characterize NP-hard instances of MinHOM(H) in these digraph classes?
• RQ4 Can the complexity of MinHOM(H) be fully classified for locally semicomplete and quasi-transitive digraphs using structural graph theory?

Key findings

• MinHOM(H) is polynomial-time solvable for locally semicomplete digraphs H if and only if every connected component is acyclic or a directed cycle Cₖ for k ≥ 2.
• For quasi-transitive digraphs, MinHOM(H) is polynomial-time solvable if each connected component H′ is either C₂, an extension of C₃, or acyclic, with B(H′) being a proper interval bigraph and H′ containing no induced Oᵢ for i=1,2,3,4.
• Any deviation from these structural conditions in either class results in NP-hardness of MinHOM(H), as shown via reductions to known hard subdigraphs H₁ and H₂.
• The Oᵢ obstructions (i=1,2,3,4) are critical forbidden subgraphs in the quasi-transitive case, and their presence implies NP-hardness.
• The proof relies on the fact that quasi-transitivity and local semicompleteness constrain vertex adjacencies so strictly that only specific configurations allow tractable solutions.