Parameterized Algorithms for Zero Extension and Metric Labelling Problems
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We consider the problems ZERO EXTENSION and METRIC LABELLING under the paradigm of parameterized complexity. These are natural, well-studied problems with important applications, but have previously not received much attention from parameterized complexity. Depending on the chosen cost function μ, we find that different algorithmic approaches can be applied to design FPT-algorithms: for arbitrary μ we parameterized by the number of edges that cross the cut (not the cost) and show how to solve ZERO EXTENSION in time O(|D|^O(k^2) n^4 n) using randomized contractions. We improve this running time with respect to both parameter and input size to O(|D|^O(k) m) in the case where μ is a metric. We further show that the problem admits a polynomial sparsifier, that is, a kernel of size O(k^|D|+1) that is independent of the metric μ. With the stronger condition that μ is described by the distances of leaves in a tree, we parameterize by a gap parameter (q - p) between the cost of a true solution q and a `discrete relaxation' p and achieve a running time of O(|D|^q-p |T|m + |T|ϕ(n,m)) where T is the size of the tree over which μ is defined and ϕ(n,m) is the running time of a max-flow computation. We achieve a similar running for the more general METRIC LABELLING, while also allowing μ to be the distance metric between an arbitrary subset of nodes in a tree using tools from the theory of VCSPs. We expect the methods used in the latter result to have further applications.
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