Minimum Cost Homomorphism Dichotomy for Oriented Cycles.

Graphs and Combinatorics (Impact Factor: 0.35). 01/2009; 25:521-531. DOI: 10.1007/s00373-009-0853-9
Source: DBLP

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 c

(u),i ∈ V(H), then the cost of the homomorphism f is ∑
u ∈V(D)

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 c

(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.

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    ABSTRACT: In a constraint satisfaction problem (CSP), the aim is to find an assignment of values to a given set of variables, subject to specified constraints. The CSP is known to be NP-complete in general. However, certain restrictions on the form of the allowed constraints can lead to problems solvable in polynomial time. Such restrictions are usually imposed by specifying a constraint language. The principal research direction aims to distinguish those constraint languages, which give rise to tractable CSPs from those which do not. We achieve this goal for the widely used variant of the CSP, in which the set of values for each individual variable can be restricted arbitrarily. Restrictions of this type can be expressed by including in a constraint language all possible unary constraints. Constraint languages containing all unary constraints will be called conservative. We completely characterize conservative constraint languages that give rise to CSP classes solvable in polynomial time. In particular, this result allows us to obtain a complete description of those (directed) graphs H for which the List H-Coloring problem is polynomial time solvable.
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  • SIAM J. Discrete Math. 01/2001; 14:471-480.
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    ABSTRACT: For graphs $G$ and $H$, a mapping $f: V(G)\dom V(H)$ is a homomorphism of $G$ to $H$ if $uv\in E(G)$ implies $f(u)f(v)\in E(H).$ If, moreover, each vertex $u \in V(G)$ is associated with costs $c_i(u), i \in V(H)$, then the cost of the homomorphism $f$ is $\sum_{u\in V(G)}c_{f(u)}(u)$. For each fixed graph $H$, we have the {\em minimum cost homomorphism problem}, written as MinHOM($H)$. The problem is to decide, for an input graph $G$ with costs $c_i(u),$ $u \in V(G), i\in V(H)$, 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 NP-hard. This solves an open problem from an earlier paper. Along the way, we prove a new characterization of the class of proper interval bigraphs.
    CoRR. 01/2006; abs/cs/0602038.

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