Abstract
Graph Theory We introduce a binary parameter on optimisation problems called separation. The parameter is used to relate the approximation ratios of different optimisation problems; in other words, we can convert approximability (and non-approximability) result for one problem into (non)-approximability results for other problems. Our main application is the problem (weighted) maximum H-colourable subgraph (Max H-Col), which is a restriction of the general maximum constraint satisfaction problem (Max CSP) to a single, binary, and symmetric relation. Using known approximation ratios for Max k-cut, we obtain general asymptotic approximability results for Max H-Col for an arbitrary graph H. For several classes of graphs, we provide near-optimal results under the unique games conjecture. We also investigate separation as a graph parameter. In this vein, we study its properties on circular complete graphs. Furthermore, we establish a close connection to work by Šámal on cubical colourings of graphs. This connection shows that our parameter is closely related to a special type of chromatic number. We believe that this insight may turn out to be crucial for understanding the behaviour of the parameter, and in the longer term, for understanding the approximability of optimisation problems such as Max H-Col.
Highlights
In this article we study an approximation-preserving reducibility called continuous reduction (Simon, 1989)
We introduce a binary parameter on optimisation problems which measures the degradation in the approximation guarantee of a given continuous reduction
These two topics relate to the application of our approach to the approximability of the problem MAX H -COL (and more generally to maximum constraint satisfaction problem (MAX CSP)(Γ)), and to the computation and interpretation of the separation parameter
Summary
In this article we study an approximation-preserving reducibility called continuous reduction (Simon, 1989). We introduce a binary parameter on optimisation problems which measures the degradation in the approximation guarantee of a given continuous reduction. We call this parameter the separation of the two problems. The separation parameter is used to study a concrete family of optimisation problems called the maximum H-colourable subgraph problems, or MAX H -COL for short. This family includes the problems MAX k-CUT for which good approximation ratios are known (Frieze and Jerrum, 1997). Among the most striking results is the connection between separation and a (generalisation of) cubical colourings and fractional covering by cuts previously studied by Šámal (2005, 2006, 2012)
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