An approximability-related parameter on graphs―-properties and applicationsReport as inadecuate

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1 Department of Computer and Information Science - Linköping University 2 LIGM - Laboratoire d-Informatique Gaspard-Monge

Abstract : 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.

Keywords : graph H-colouring approximation graph homomorphism circular colouring combinatorial optimisation graph theory

Author: Robert Engström - Tommy Färnqvist - Peter Jonsson - Johan Thapper -



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