Minimum cost homomorphisms to digraphs

For digraphs $D$ and $H$, a homomorphism of $D$ to $H$ is a mapping $f:\ V(D)\dom V(H)$ such that $uv\in A(D)$ implies $f(u)f(v)\in A(H)$. Suppose $D$ and $H$ are two digraphs, and $c_i(u)$, $u\in V(D)$, $i\in V(H)$, are nonnegative integer costs. The cost of the homomorphism $f$ of $D$ to $H$ is $\sum_{u\inV(D)}c_{f(u)}(u)$. The minimum cost homomorphism for a fixed digraph $H$, denoted by MinHOM($H$), asks whether or not an input digraph $D$, with nonnegative integer costs $c_i(u)$, $u\in V(D)$, $i\in V(H)$, admits a homomorphism $f$ to $H$ and if it admits one, find a homomorphism of minimum cost. Our interest is in proving a dichotomy for minimum cost homomorphism problem: we would like to prove that for each digraph $H$, MinHOM($H$) is polynomial-time solvable, or NP-hard. Gutin, Rafiey, and Yeo conjectured that such a classification exists: MinHOM($H$) is polynomial time solvable if $H$ admits a $k$-Min-Max ordering for some $k \geq 1$, and it is NP-hard otherwise. For undirected graphs, the complexity of the problem is well understood; for digraphs, the situation appears to be more complex, and only partial results are known. In this thesis, we seek to verify this conjecture for ``large'' classes of digraphs including reflexive digraphs, locally in-semicomplete digraphs, as well as some classes of particular interest such as quasi-transitive digraphs. For all classes, we exhibit a forbidden induced subgraph characterization of digraphs with $k$-Min-Max ordering; our characterizations imply a polynomial time test for the existence of a $k$-Min-Max ordering. Given these characterizations, we show that for a digraph $H$ which does not admit a $k$-Min-Max ordering, the minimum cost homomorphism problem is NP-hard. This leads us to a full dichotomy classification of the complexity of minimum cost homomorphism problems for the aforementioned classes of digraphs.