The complexity of colouring by semicomplete digraphs
SIAM Journal on Discrete Mathematics
On the complexity of H-coloring
Journal of Combinatorial Theory Series B
Discrete Applied Mathematics
A coloring problem for weighted graphs
Information Processing Letters
A nice class for the vertex packing problem
GO-II Meeting Proceedings of the second international colloquium on Graphs and optimization
Maximum weighted independent sets on transitive graphs and applications
Integration, the VLSI Journal
Augmenting graphs for independent sets
Discrete Applied Mathematics - The fourth international colloquium on graphs and optimisation (GO-IV)
Digraphs: Theory, Algorithms and Applications
Digraphs: Theory, Algorithms and Applications
A maximal tractable class of soft constraints
Journal of Artificial Intelligence Research
Communication: Level of repair analysis and minimum cost homomorphisms of graphs
Discrete Applied Mathematics
A dichotomy for minimum cost graph homomorphisms
European Journal of Combinatorics
Communication: Minimum cost homomorphisms to semicomplete multipartite digraphs
Discrete Applied Mathematics
Minimum Cost Homomorphism Dichotomy for Locally In-Semicomplete Digraphs
COCOA 2008 Proceedings of the 2nd international conference on Combinatorial Optimization and Applications
Introduction to the Maximum Solution Problem
Complexity of Constraints
New Plain-Exponential Time Classes for Graph Homomorphism
CSR '09 Proceedings of the Fourth International Computer Science Symposium in Russia on Computer Science - Theory and Applications
The complexity of soft constraint satisfaction
Artificial Intelligence
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For digraphs D and H, a mapping f : V(D) → V(H) is a homomorphism of D to H if uv ∈ A(D) implies f(u) f(v) ∈ A(H). Let H be a fixed directed or undirected graph. The homomorphism problem for H asks whether a directed or undirected input graph D admits a homomorphism to H. The list homomorphism problem for H is a generalization of the homomorphism problem for H, where every vertex x ∈ V(D) is assigned a set Lx of possible colors (vertices of H).The following optimization version of these decision problems generalizes the list homomorphism problem and was introduced in Gutin et al. [Level of repair analysis and minimum cost homomorphisms of graphs, Discrete Appl. Math., to appear], 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 integral cost ci(u) for each u ∈ V(D) and i ∈ V(H). The cost of a homomorphism f of D to H is Σu ∈ V(D) cf(u)(u). For a fixed digraph H, the minimum cost homomorphism problem for H is stated as follows: for an input digraph D and costs ci(u) for each u ∈ V(D) and i ∈ V(H), verify whether there is a homomorphism of D to H and, if one exists, find such a homomorphism of minimum cost.We obtain dichotomy classifications of the computational complexity of the list homomorphism and minimum cost homomorphism problems, when H is a semicomplete digraph (digraph in which there is at least one arc between any two vertices). 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 the minimum cost homomorphism problem 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.