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Most of my research so far is within the field of parameterized complexity. Mainly, the field studies NP-hard problems and seeks to understand the influence of problem parameters on the problem complexity; in particular, one seeks algorithms that are fast when certain parameters are small. As an example, many optimization problems are expected (under the so-called exponential-time hypothesis) to take time O(a^n) to exactly solve (worst-case) inputs of size n for some a > 1. However, if the task is only to find a solution of size at most k (or find that none exists) then this can often be done in time, e.g., O(b^k n^c), that is, with only polynomial dependence on the input size. The same can of course be asked when k instead stands for some structural parameter as for example the treewidth of input graphs. The field also has various techniques for getting lower bounds on running times or ruling out time O(f(k) n^c) for all functions f (modulo complexity assumptions). From such considerations we get two things: First, we may find algorithms that are sufficiently fast at solving our problem for a range of the parameter that actually occurs in practice. Second, we get a better understanding of what makes a problem hard and we may be able to deduce a different, hopefully more tractable, problem formulation. Multivariate analysis of computational problems is a very general perspective (predating parameterized complexity) and can also be used to talk about approximation schemes, adaptive algorithms, or data structures.
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CoRR (2023): 35:1-35:15
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arXiv (Cornell University) (2023): 102-119
ACM Transactions on Algorithms (2023)
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