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 graph input digraph 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 was introduced in [16], 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 ci(u) for each u ∈ V (D) and i ∈ V (H). The cost of a homomorphism f of D to H is � u∈V (D) cf(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 ci(u) for each u ∈ V (D) and i ∈ V (H), verify whether there is a homomorphism of D to H and, if it exists, find

Abstract. 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). If, moreover, each vertex u ∈ V (D) is associated with costs ci(u), i ∈ V (H), then the cost of the homomorphism f is � u∈V (D) c f(u)(u). For each fixed digraph H, we have the minimum cost homomorphism problem for H. The problem is to decide, for an input graph D with costs ci(u), u ∈ V (D), i ∈ V (H), whether there exists a homomorphism of D 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. We describe a dichotomy of the minimum cost homomorphism problem for semicomplete bipartite digraphs H. This solves an open problem from an earlier paper. To obtain the dichotomy of this paper, we introduce and study a new notion, a k-Min-Max ordering of digraphs. Key words. homomorphisms, minimum cost homomorphisms, semicomplete bipartite digraphs

For graphs G and H, a mapping f: V (G)→V (H) is a homomorphism of G to H if uv ∈ E(G) implies f(u)f(v) ∈ E(H). If, moreover, each vertex u ∈ V (G) is associated with costs ci(u), i ∈ V (H), then the cost of the homomorphism f is � u∈V (G) cf(u)(u). For each fixed graph H, we have the minimum cost homomorphism problem, written as MinHOM(H). The problem is to decide, for an input graph G with costs ci(u), u ∈ V (G), i ∈ V (H), 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. We prove a dichotomy of the minimum cost homomorphism problems for graphs H, with loops allowed. When each connected component of H is either a reflexive proper interval graph or an irreflexive proper interval bigraph, the problem MinHOM(H) is polynomial time solvable. In all other cases the problem MinHOM(H) is NP-hard. This solves an open problem from an earlier paper. 1

For digraphs G and H, a homomorphism of G to H is a mapping f: V (G)→V (H) such that uv ∈ A(G) implies f(u)f(v) ∈ A(H). If moreover each vertex u ∈ V (G) is associated with costs ci(u), i ∈ V (H), then the cost of a homomorphism f is u∈V (G) c f(u)(u). For each fixed digraph H, the minimum cost homomorphism problem for H, denoted MinHOM(H), is the following problem. Given an input digraph G, together with costs ci(u), u ∈ V (G), i ∈ V (H), and an integer k, decide if G admits a homomorphism to H of cost not exceeding k. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems such as chromatic partition optimization and applied problems in repair analysis. For undirected graphs the complexity of the problem, as a function of the parameter H, is well understood; for digraphs, the situation appears to be more complex, and only partial results are known. We focus on the minimum cost homomorphism problem for reflexive digraphs H (every vertex of H has a loop). It is known that the problem MinHOM(H) is polynomial time solvable if the digraph H has a Min-Max ordering, i.e., if its vertices can be linearly ordered by < so that i < j, s < r and ir, js ∈ A(H) imply that is ∈ A(H) and jr ∈ A(H). We give a forbidden induced subgraph characterization of reflexive digraphs with a Min-Max ordering; our characterization implies a polynomial time test for the existence of a Min-Max ordering. Using this characterization, we show that for a reflexive digraph H which does not admit a Min-Max ordering, the minimum cost homomorphism problem is NP-complete. Thus we obtain a full dichotomy classification of the complexity of minimum cost homomorphism problems for reflexive digraphs. 1

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). For a fixed digraph H, the homomorphism problem is to decide whether an input digraph D admits a homomorphism to H or not, and is denoted as HOM(H). An optimization version of the homomorphism problem was motivated by a realworld problem in defence logistics and was introduced in [13]. If each vertex u ∈ V (D) is associated with costs ci(u), i ∈ V (H), then the cost of the homomorphism f is u∈V (D) c f(u)(u). For each fixed digraph H, we have the minimum cost homomorphism problem for H and denote it as MinHOM(H). The problem is to decide, for an input graph D with costs ci(u), u ∈ V (D), i ∈ V (H), whether there exists a homomorphism of D to H and, if one exists, to find one of minimum cost. Although a complete dichotomy classification of the complexity of MinHOM(H) for a digraph H remains an unsolved problem, complete dichotomy classifications for MinHOM(H) were proved when H is a semicomplete digraph [10], and a semicomplete multipartite digraph [12, 11]. In these studies, it is assumed that the digraph H is loopless. In this paper, we present a full dichotomy classification for semicomplete digraphs with possible loops, which solves a problem in [9].

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). If, moreover, each vertex u ∈ V (D) is associated with costs ci(u), i ∈ V (H), then the cost of the homomorphism f is u∈V (D) c f(u)(u). For each fixed digraph H, we have the minimum cost homomorphism problem for H (abbreviated MinHOM(H)). The problem is to decide, for an input graph D with costs ci(u), u ∈ V (D), i ∈ V (H), whether there exists a homomorphism of D to H and, if one exists, to find one of minimum cost. We obtain a dichotomy classification for the time complexity of MinHOM(H) when H is an oriented cycle. We conjecture a dichotomy classification for all digraphs with possible loops. 1

Abstract. We study the complexity of the problem MAX SOL which is a natural optimisation version of the graph homomorphism problem. Given a fixed target graph H with V (H) ⊆ N, and a weight function w: V (G) → Q +, an instance of the problem is a graph G and the goal is to find a homomorphism f: G → H which maximises P v∈G f(v) · w(v). MAX SOL can be seen as a restriction of the MIN HOM-problem [Gutin et al., Disc. App. Math., 154 (2006), pp. 881-889] and as a natural generalisation of MAX ONES to larger domains. We present new tools with which we classify the complexity of MAX SOL for irreflexive graphs with degree less than or equal to 2 as well as for small graphs (|V (H) | ≤ 4). We also study an extension of MAX SOL where value lists and arbitrary weights are allowed; somewhat surprisingly, this problem is polynomial-time equivalent to MIN HOM.

For digraphs G and H, a homomorphism of G to H is a mapping f: V (G)→V (H) such that uv ∈ A(G) implies f(u)f(v) ∈ A(H). If, moreover, each vertex u ∈ V (G) is associated with costs ci(u), i ∈ V (H), then the cost of a homomorphism f is u∈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 ci(u), u ∈ V (G), i ∈ 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.

Constraint satisfaction problems have enjoyed much attention since the early seventies, and in the last decade have become also a focus of attention amongst theoreticians. Graph colourings are a special class of constraint satisfaction problems; they offer a microcosm of many of the considerations that occur in constraint satisfaction. From the point of view of theory, they are well known to exhibit a dichotomy of complexity- the k-colouring problem is polynomial time solvable when k ≤ 2, and NP-complete when k ≥ 3. Similar dichotomy has been proved for the class of graph homomorphism problems, which are intermediate problems between graph colouring and constraint satisfaction

Abstract. We introduce the maximum graph homomorphism (MGH) problem: given a graph G, and a target graph H, find a mapping ϕ: VG ↦ → VH that maximizes the number of edges of G that are mapped to edges of H. This problem encodes various fundamental NP-hard problems including Maxcut and Max-k-cut. We also consider the multiway uncut problem. We are given a graph G and a set of terminals T ⊆ VG. We want to partition VG into |T | parts, each containing exactly one terminal, so as to maximize the number of edges in EG having both endpoints in the same part. Multiway uncut can be viewed as a special case of prelabeled MGH where one is also given a prelabeling ϕ ′ : U ↦ → VH, U ⊆ VG, and the output has to be an extension of ϕ ′. Both MGH and multiway uncut have a trivial 0.5-approximation algorithm. We present a 0.8535-approximation algorithm for multiway uncut based on a natural linear programming relaxation. This relaxation has an integrality gap of 6 � 0.8571, showing that our guarantee is almost 7 tight. For maximum graph homomorphism, we show that a � � 1 + ε0-2 approximation algorithm, for any constant ε0> 0, implies an algorithm for distinguishing between certain average-case instances of the subgraph isomorphism problem that appear to be hard. Complementing this, we give a � 1 1 + Ω(