Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Toward a Dichotomy for Approximation of $H$-coloring   Given two (di)graphs G, H and a cost function $c:V(G)\times V(H) \to
\mathbb{Q}_{\geq 0}\cup\{+\infty\}$, in the minimum cost homomorphism problem,
MinHOM(H), goal is finding a homomorphism $f:V(G)\to V(H)$ (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 [34], 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. For 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. 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:
  1. \textbf{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;
  2. MinHOM(H) has a $|V(H)|^2$-approximation algorithm if H is a bi-arc
digraph;
  3. 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 H.

33
The categories of this paper are: plasm-ph
17
The categories of this paper are: q-fin
18
The categories of this paper are: econ
7
The categories of this paper are: hep-th
11
The categories of this paper are: hep-lat
30
The categories of this paper are: q-alg
14
The categories of this paper are: math-ph
29
The categories of this paper are: funct-an
31
The categories of this paper are: ao-sci
2
The categories of this paper are: math
25
The categories of this paper are: patt-sol
36
The categories of this paper are: cmp-lg
6
The categories of this paper are: astro-ph
32
The categories of this paper are: atom-ph
4
The categories of this paper are: cond-mat
37
The categories of this paper are: supr-con