Minimum Cost Homomorphisms to Reflexive Digraphs

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$H, the {\em minimum cost homomorphism problem} for H$, denoted MinHOM($H$), is the following problem. Given an input digraph$G$, together with costs$c_i(u)$,$u\in V(G)$,$i\in V(H)$, and an integer$k$, decide if$G$admits a homomorphism to$H$of cost not exceeding$k. We focus on the minimum cost homomorphism problem for {\em reflexive} digraphs H$(every vertex of$H$has a loop). It is known that the problem MinHOM($H$) is polynomial time solvable if the digraph$H has a {\em Min-Max ordering}, i.e., if its vertices can be linearly ordered by <$so that$i<j, s<r$and$ir, js \in A(H)$imply that$is \in A(H)$and$jr \in A(H)$. 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

Minimum Cost Homomorphisms to Locally Semicomplete and Quasi-Transitive Digraphs

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

A connection between circular colorings and periodic schedules

AbstractWe show that there is a curious connection between circular colorings of edge-weighted digraphs and periodic schedules of timed marked graphs. Circular coloring of an edge-weighted digraph was introduced by Mohar [B. Mohar, Circular colorings of edge-weighted graphs, J. Graph Theory 43 (2003) 107–116]. This kind of coloring is a very natural generalization of several well-known graph coloring problems including the usual circular coloring [X. Zhu, Circular chromatic number: A survey, Discrete Math. 229 (2001) 371–410] and the circular coloring of vertex-weighted graphs [W. Deuber, X. Zhu, Circular coloring of weighted graphs, J. Graph Theory 23 (1996) 365–376]. Timed marked graphs G→ [R.M. Karp, R.E. Miller, Properties of a model for parallel computations: Determinancy, termination, queuing, SIAM J. Appl. Math. 14 (1966) 1390–1411] are used, in computer science, to model the data movement in parallel computations, where a vertex represents a task, an arc uv with weight cuv represents a data channel with communication cost, and tokens on arc uv represent the input data of task vertex v. Dynamically, if vertex u operates at time t, then u removes one token from each of its in-arc; if uv is an out-arc of u, then at time t+cuv vertex u places one token on arc uv. Computer scientists are interested in designing, for each vertex u, a sequence of time instants {fu(1),fu(2),fu(3),…} such that vertex u starts its kth operation at time fu(k) and each in-arc of u contains at least one token at that time. The set of functions {fu:u∈V(G→)} is called a schedule of G→. Computer scientists are particularly interested in periodic schedules. Given a timed marked graph G→, they ask if there exist a period p>0 and real numbers xu such that G→ has a periodic schedule of the form fu(k)=xu+p(k−1) for each vertex u and any positive integer k. In this note we demonstrate an unexpected connection between circular colorings and periodic schedules. The aim of this note is to provide a possibility of translating problems and methods from one area of graph coloring to another area of computer science