Minimum Cost Homomorphisms with Constrained Costs

Theory Comput Syst, 2007

Graph homomorphism, also called H-coloring, is a natural generalization of graph coloring: There is a homomorphism from a graph G to a complete graph on k vertices if and only if G is k-colorable. During the recent years the topic of exact (exponential-time) algorithms for NP-hard problems in general, and for graph coloring in particular, has led to extensive research. Consequently, it is natural to ask how the techniques developed for exact graph coloring algorithms can be extended to graph homomorphisms. By the celebrated result of Hell and Nešetřil, for each fixed simple graph H, deciding whether a given simple graph G has a homomorphism to H is polynomial-time solvable if H is a bipartite graph, and NP-complete otherwise. The case where H is a cycle of length 5 is the first NP-hard case different from graph coloring. We show that, for a given graph G on n vertices and an odd integer k ≥ 5, whether G is homomorphic to a cycle of length k can be decided in time min{ n n/k , 2 n/2 } · n O(1) . We extend the results obtained for cycles, which are graphs of treewidth two, to graphs of bounded treewidth as follows: If H is of treewidth at most t, then whether G is homomorphic to H can be decided in time (2t + 1) n · n O(1) .

Surveys in Combinatorics 2003

Homomorphisms are a useful model for a wide variety of combinatorial problems dealing with mappings and assignments, typified by scheduling and channel assignment problems. Homomorphism problems generalize graph colourings, and are in turn generalized by constraint satisfaction problems; they represent a robust intermediate class of problems -with greater modeling power than graph colourings, yet simpler and more manageable than general constraint satisfaction problems. We will discuss various homomorphism problems from a computational perspective. One variant, with natural applications, gives each vertex a list of allowed images. Such list homomorphisms generalize list colourings, precolouring extensions, and graph retractions. Many algorithms for finding homomorphisms adapt well to finding list homomorphisms. Semihomomorphisms are another variant; they generalize the kinds of partitions that homomorphisms induce, to allow both homomorphism type constraints, and constraints that correspond to homomorphisms of the complementary graphs. Surprisingly, semi-homomorphism partition problems cover a great variety of concepts arising in the study of perfect graphs. We illustrate some of the ideas leading to efficient algorithms for all these problems.

SIAM Journal on Discrete Mathematics, 2012

The minimum cost homomorphism problem has arisen as a natural and useful optimization problem in the study of graph (and digraph) coloring and homomorphisms: it unifies a number of other well studied optimization problems. It was shown by Gutin, Rafiey, and Yeo that the minimum cost problem for homomorphisms to a digraph H that admits a so-called extended Min-Max ordering is polynomial time solvable, and these authors conjectured that for all other digraphs H the problem is NP-complete. In a companion paper, we gave a forbidden structure characterization of digraphs that admit extended Min-Max orderings. In this paper, we apply this characterization to prove Gutin's conjecture.

Discrete Applied Mathematics, 2006

Let H be a fixed directed or undirected graph. The homomorphism problem for H asks whether a directed or undirected input graph 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 L x of possible colors (vertices of H).

SIAM Journal on Discrete Mathematics, 2008

For digraphs D and H, a mapping f :

Discrete Applied Mathematics, 2010

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 c i (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 c i (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].

Discrete Applied Mathematics, 2013

For a fixed target graph H, the minimum cost homomorphism problem, MinHOM(H), asks, for a given graph G with integer costs c i (u), u ∈ V (G), i ∈ V (H), and an integer k, whether or not there exists a homomorphism of G to H of cost not exceeding k. When the target graph H is a bipartite graph a dichotomy classification is known: MinHOM(H) is solvable in polynomial time if and only if H does not contain bipartite claws, nets, tents and any induced cycles C 2k for k ≥ 3 as an induced subgraph. In this paper, we start studying the approximability of MinHOM(H) when H is bipartite. First we note that if H has as an induced subgraph C 2k for k ≥ 3, then there is no approximation algorithm. Then we suggest an integer linear program formulation for MinHOM(H) and show that the integrality gap can be made arbitrarily large if H is a bipartite claw. Finally, we obtain a 2-approximation algorithm when H is a subclass of doubly convex bipartite graphs that has as special case bipartite nets and tents.

Discrete Applied Mathematics, 2008

. For a fixed directed or undirected graph H and an input graph D, the problem of verifying whether there exists a homomorphism of D to H has been studied in a large number of papers. We study an optimization version of this decision problem. Our optimization problem is motivated by a realworld problem in defence logistics and was introduced very recently by the authors and M. Tso.

arXiv (Cornell University), 2007

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 c i (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 c i (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]. 1