Minimum Cost and List Homomorphisms to Semicomplete Digraphs

TL;DR

This paper classifies the computational complexity of list homomorphism and minimum cost homomorphism problems for semicomplete digraphs, showing polynomial solvability or NP-completeness depending on the structure of H.

Contribution

It provides a dichotomy classification for both problems when H is a semicomplete digraph, extending previous results and identifying new complexity boundaries.

Findings

List homomorphism problem is polynomial if H has at most one cycle; otherwise NP-complete.

MinHOMP(H) is polynomial if H is acyclic or a 2- or 3-cycle; otherwise NP-hard.

Dichotomy results align with known classifications for related homomorphism problems.

Abstract

The following optimization problem was introduced in \cite{gutinDAM}, where it was motivated by a real-world problem in defence logistics. Suppose we are given a pair of digraphs $D,H$ and a positive cost $c_i(u)$ for each $u\in V(D)$ and $i\in V(H)$. The cost of a homomorphism $f$ of $D$ to $H$ is $\sum_{u\in V(D)}c_{f(u)}(u)$. For a fixed digraph $H$, the minimum cost homomorphism problem for $H$, MinHOMP($H$), is stated as follows: For an input digraph $D$ and costs $c_i(u)$ for each $u\in V(D)$ and $i\in V(H)$, verify whether there is a homomorphism of $D$ to $H$ and, if it exists, find such a homomorphism of minimum cost. We obtain dichotomy classifications of the computational complexity of the list homomorphism problem and MinHOMP($H$), when $H$ is a semicomplete digraph (a digraph in which every two vertices have at least one arc between them). Our dichotomy for the list homomorphism problem coincides with the one obtained by Bang-Jensen, Hell and MacGillivray in 1988 for the homomorphism problem when $H$ is a semicomplete digraph: both problems are polynomial solvable if $H$ has at most one cycle; otherwise, both problems are NP-complete. The dichotomy for \MiP is different: the problem is polynomial time solvable if $H$ is acyclic or $H$ is a cycle of length 2 or 3; otherwise, the problem is NP-hard.