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A dichotomy for minimum cost graph homomorphisms
 European J. Combin
, 2007
"... 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 h ..."
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Cited by 19 (5 self)
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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 NPhard. This solves an open problem from an earlier paper. 1
Minimum Cost Homomorphisms to reflexive digraphs
 8th Latin American Theoretical Informatics (LATIN), Rio de Janeiro, Brazil
"... 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 hom ..."
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Cited by 13 (7 self)
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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 MinMax 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 MinMax ordering; our characterization implies a polynomial time test for the existence of a MinMax ordering. Using this characterization, we show that for a reflexive digraph H which does not admit a MinMax ordering, the minimum cost homomorphism problem is NPcomplete. Thus we obtain a full dichotomy classification of the complexity of minimum cost homomorphism problems for reflexive digraphs. 1
Minimum Cost “Homomorphisms to proper interval graphs and bigraphs
"... 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 h ..."
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Cited by 6 (4 self)
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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 describe 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 NPhard. This solves an open problem from an earlier paper. Along the way, we prove a new characterization of the class of proper interval bigraphs. 1
Complexity of the Minimum Cost Homomorphism Problem for Semicomplete Digraphs with Possible Loops
"... 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 homomo ..."
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Cited by 6 (1 self)
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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). 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 [12]. 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 [9], and a semicomplete multipartite digraph [11, 10]. 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 [8].
Minimum Cost Homomorphism Dichotomy for Oriented Cycles
 Proc. AAIM 2008, Lecture Notes Comput. Sci
"... 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 cos ..."
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Cited by 5 (1 self)
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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). 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
Minimum Cost Homomorphisms to Locally Semicomplete Quasitransitive Digraphs
, 2007
"... 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 h ..."
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Cited by 2 (1 self)
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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 quasitransitive digraphs which are two wellknown generalizations of tournaments. Using graphtheoretic 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 Oriented Cycles with Some Loops
"... For digraphs D and H, a homomorphism of D to H is a mapping f: V (D)→V (H) such that uv ∈ A(D) implies f(u)f(v) ∈ A(H). Suppose D and H are two digraphs, and ci(u), u ∈ V (D), i ∈ V (H), are nonnegative real costs. The cost of the homomorphism f of D to H is ∑ u∈V (D) cf(u)(u). The minimum cost hom ..."
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Cited by 1 (1 self)
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For digraphs D and H, a homomorphism of D to H is a mapping f: V (D)→V (H) such that uv ∈ A(D) implies f(u)f(v) ∈ A(H). Suppose D and H are two digraphs, and ci(u), u ∈ V (D), i ∈ V (H), are nonnegative real costs. The cost of the homomorphism f of D to H is ∑ u∈V (D) cf(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 real costs ci(u), u ∈ V (D), i ∈ V (H), admits a homomorphism f to H and if it admits one, find a homomorphism of minimum cost. The minimum cost homomorphism problem seems to offer a natural and practical way to model many optimization problems such as list homomorphism problems, retraction and precolouring extension problems, chromatic partition optimization, and applied problems in repair analysis. 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 polynomialtime solvable, or NPhard. We say that H is a digraph with some loops, if H has at least one loop. For reflexive digraphs H (every vertex has a loop) the complexity of MinHOM(H) is well understood. In this paper, we obtain a full dichotomy for MinHOM(H) when H is an oriented cycle with some loops. Furthermore, we show that this dichotomy is a remarkable progress toward a dichotomy for oriented graphs with some loops.
Minimum Cost Homomorphisms to Digraphs
, 2009
"... For digraphs D and H, a homomorphism of D to H is a mapping f: V (D)→V (H) such that uv ∈ A(D) implies f(u)f(v) ∈ A(H). Suppose D and H are two digraphs, and ci(u), u ∈ V (D), i ∈ V (H), are nonnegative integer costs. The cost of the homomorphism f of D to H is ∑ u∈V (D) c f(u)(u). The minimum cost ..."
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For digraphs D and H, a homomorphism of D to H is a mapping f: V (D)→V (H) such that uv ∈ A(D) implies f(u)f(v) ∈ A(H). Suppose D and H are two digraphs, and ci(u), u ∈ V (D), i ∈ V (H), are nonnegative integer costs. The cost of the homomorphism f of D to H is ∑ u∈V (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 ci(u), u ∈ V (D), i ∈ 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 polynomialtime solvable, or NPhard. Gutin, Rafiey, and Yeo conjectured that such a classification exists: MinHOM(H) is polynomial time solvable if H admits a kMinMax ordering for some k ≥ 1, and it is NPhard 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,