Interval-Like Graphs and Digraphs

We unify several seemingly different graph and digraph classes under one umbrella. These classes are all, broadly speaking, different generalizations of interval graphs, and include, in addition to interval graphs, adjusted interval digraphs, threshold graphs, complements of threshold tolerance graphs (known as `co-TT\u27 graphs), bipartite interval containment graphs, bipartite co-circular arc graphs, and two-directional orthogonal ray graphs. (The last three classes coincide, but have been investigated in different contexts.) This common view is made possible by introducing reflexive relationships (loops) into the analysis. We also show that all the above classes are united by a common ordering characterization, the existence of a min ordering. We propose a common generalization of all these graph and digraph classes, namely signed-interval digraphs, and show that they are precisely the digraphs that are characterized by the existence of a min ordering. We also offer an alternative geometric characterization of these digraphs. For most of the above graph and digraph classes, we show that they are exactly those signed-interval digraphs that satisfy a suitable natural restriction on the digraph, like having a loop on every vertex, or having a symmetric edge-set, or being bipartite. For instance, co-TT graphs are precisely those signed-interval digraphs that have each edge symmetric. We also offer some discussion of future work on recognition algorithms and characterizations

Toward a Dichotomy for Approximation of H-Coloring

Given two (di)graphs G, H and a cost function c:V(G) x V(H) -> Q_{>= 0} cup {+infty}, in the minimum cost homomorphism problem, MinHOM(H), we are interested in finding a homomorphism f:V(G)-> V(H) (a.k.a H-coloring) that minimizes sum limits_{v in V(G)}c(v,f(v)). The complexity of exact minimization of this problem is well understood [Pavol Hell and Arash Rafiey, 2012], and the class of digraphs H, for which the MinHOM(H) is polynomial time solvable is a small subset of all digraphs. In this paper, we consider the approximation of MinHOM within a constant factor. In terms of digraphs, MinHOM(H) is not approximable if H contains a digraph asteroidal triple (DAT). We take a major step toward a dichotomy classification of approximable cases. We give a dichotomy classification for approximating the MinHOM(H) when H is a graph (i.e. symmetric digraph). For digraphs, we provide constant factor approximation algorithms for two important classes of digraphs, namely bi-arc digraphs (digraphs with a conservative semi-lattice polymorphism or min-ordering), and k-arc digraphs (digraphs with an extended min-ordering). Specifically, we show that: - Dichotomy for Graphs: MinHOM(H) has a 2|V(H)|-approximation algorithm if graph H admits a conservative majority polymorphims (i.e. H is a bi-arc graph), otherwise, it is inapproximable; - MinHOM(H) has a |V(H)|^2-approximation algorithm if H is a bi-arc digraph; - MinHOM(H) has a |V(H)|^2-approximation algorithm if H is a k-arc digraph. In conclusion, we show the importance of these results and provide insights for achieving a dichotomy classification of approximable cases. Our constant factors depend on the size of H. However, the implementation of our algorithms provides a much better approximation ratio. It leaves open to investigate a classification of digraphs H, where MinHOM(H) admits a constant factor approximation algorithm that is independent of |V(H)|