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.
The minimum cost homomorphism problem was introduced in [17], where it was motivated by a real-world problem in defense logistics. In general, the problem appears to offer a natural and practical way to model many optimization problems. Special cases include the homomorphism problem, the list homomorphism problem [19,21] and the optimum cost chromatic partition problem [18,24,25] (which itself has a number of well-studied special cases and applications [27,28]).
For digraphs G and H, a mapping f : V (G)→V (H) is a homomorphism of G to H if f (u)f (v) is an arc of H whenever uv is an arc of G. In the homomorphism problem, given a graph H, for an input graph G we wish to decide whether there is a homomorphism of G to H. In the list homomorphism problem, our input apart from G consists of sets L(u), u ∈ V (G), of vertices of H, and we wish to decide whether there is a homomorphism f of G to H such that f (u) ∈ L(u) for each u ∈ V (G). In the minimum cost homomorphism problem we fix H as before, our inputs are a graph G and costs c i (u), u ∈ V (G), i ∈ V (H) of mapping u to i, and we wish to check whether there exists a homomorphism of G to H and if it does exist, we wish to obtain one of minimum cost, where the cost of a homomorphism f is u∈V (G) c f (u) (u). The homomorphism, list homomorphism, and minimum cost homomorphism problems are denoted by HOM(H), ListHOM(H) and MinHOM(H), respectively. If the graph H is symmetric (each uv ∈ A(H) implies vu ∈ A(H)), we may view H as an undirected graph. This way, we may view the problem MinHOM(H) as also a problem for undirected graphs. For further terminology and notation see the next section, where we define several terms used in the rest of this section.
Our interest is in obtaining dichotomies: given a problem such as HOM(H), we would like to find a class of digraphs H such that if H ∈ H, then the problem is polynomial-time solvable and if H / ∈ H, then the problem is NP-complete. For instance, in the case of undirected graphs it is well-known that HOM(H) is polynomial-time solvable when H is bipartite or has a loop, and NP-complete otherwise [20].
For undirected graphs H, a dichotomy classification for the problem MinHOM(H) has been provided in [11]. (For ListHOM(H), consult [6].) Since [11] interest has shifted to directed graphs. The first studies [14,15,16] focused on loopless digraphs and dichotomies have been obtained for semicomplete digraphs and semicomplete multipartite digraphs (we define these and other classes of digraphs in the next section). More recently, [13] initiated the study of digraphs with loops allowed; and, in particular, of reflexive digraphs, where each vertex has a loop. While [12] gave a dichotomy for semicomplete digraphs with possible loops, [10] obtained a dichotomy for all reflexive digraphs. (Partial results on ListHOM(H) for digraphs can be found in [3,5,7,8,9,23,29].)
Along with semicomplete digraphs and semicomplete multipartite digraphs, locally semicomplete digraphs and quasi-transitive digraphs are the most studied families of generalizations of tournaments [1]. Thus, it is a natural problem to obtain dichotomies for locally semicomplete digraphs and quasi-transitive digraphs and we solve this problem in the present paper. Like with semicomplete digraphs and semicomplete multipartite digraphs, structural properties of locally semicomplete digraphs and quasi-transitive digraphs play key role in proving the dichotomies. Unlike for semicomplete digraphs and semicomplete multipartite digraphs, we also use structural properties of a family of undirected graphs. We hope that the study of well-known classes of digraphs will eventually allow us to conjecture and prove a full dichotomy for loopless digraphs.
In this paper we prove the following two dichotomies: 1). Otherwise, MinHom(H) is NP-hard.
In fact, it is easy to see that it suffices to prove Theorems 1.1 and 1.2 only for connected digraphs H; for a short proof, see [11]. The rest of the paper is devoted to proving the two theorems for the case of connected H. In Section 2 we provide further terminology and notation and formulate a characterization of proper interval bigraphs that we use later. In Section 3 we prove the polynomial-time solvability parts of the two theorems. While the proof of the polynomial-time solvability part of Theorem 1.1 is relatively easy, this part of Theorem 1.2 is quite technical and lengthy. In Section 3 we prove the NP-completeness parts of the two theorems. There we use several known results and prove some new ones.
In our terminology and notation, we follow [1]. From now on, all digraphs are loopless and do not have parallel arcs. A digraph D is semicomplete if, for each pair x, y of distinct vertices either x dominates y or y dominates x or both. A digraph D obtained from a complete k-partite (undirected) graph G by replacing every edge xy of G with arc xy, arc yx, or both, is called a semicomplete k-partite digraph (or, semicomplete mul
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