Finding Diverse Solutions Parameterized by Cliquewidth

Finding a few solutions for a given problem that are diverse, as opposed to finding a single best solution to solve the problem, has recently become a notable topic in theoretical computer science. It is important to mention that finding a single solution is computationally hard in most interesting cases. To overcome this issue, we analyze the problems under a finer scope of parameterized complexity, which often allows for tractable algorithms. That, in turn, enables us to find a diverse set of solutions. Recently, Baste, Fellows, Jaffke, Masařík, Oliveira, Philip, and Rosamond showed that under a standard structural parameterization by treewidth, one can find a set of diverse solutions for many problems with only a very small additional cost [Artificial Intelligence 2022]. In this paper, we investigate a much stronger graph parameter, the cliquewidth, which can additionally describe some dense graph classes. Broadly speaking, it describes graphs that can be recursively constructed by a few operations defined on graphs whose vertices are divided into a bounded number of groups, while each such group behaves uniformly with respect to any operation. We show that for any vertex problem, if we are given a dynamic program solving that problem on cliquewidth decomposition, we can modify it to produce a few solutions that are as diverse as possible with as little overhead as in the above-mentioned treewidth paper. As a consequence, we prove that a diverse version of any MSO1 expressible problem can be solved in linear FPT time parameterized by the cliquewidth, the number of sought solutions, and the number of quantifiers in the formula, which was a natural missing piece in the complexity landscape of structural graph parameters and logic for the diverse problems. We prove our results, allowing for a more general natural collection of diversity functions compared to only two mostly studied diversity functions previously. That might be of independent interest as a larger pool of different diversity functions can highlight various aspects of different solutions to a problem.