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

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 $uvin A(D)$ implies $f(u)f(v)in A(H)$. Suppose $D$ and $H$ are two digraphs, and $c_i(u)$, $uin V(D)$, $iin V(H)$, are nonnegative integer costs. The cost of the homomorphism $f$ of $D$ to $H$ is $sum_{uinV(D)}c_{f(u)}(u)$. 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 $c_i(u)$, $uin V(D)$, $iin V(H)$, 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\u27\u27 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

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum