4 research outputs found
Preprocessing under uncertainty
In this work we study preprocessing for tractable problems when part of the
input is unknown or uncertain. This comes up naturally if, e.g., the load of
some machines or the congestion of some roads is not known far enough in
advance, or if we have to regularly solve a problem over instances that are
largely similar, e.g., daily airport scheduling with few charter flights.
Unlike robust optimization, which also studies settings like this, our goal
lies not in computing solutions that are (approximately) good for every
instantiation. Rather, we seek to preprocess the known parts of the input, to
speed up finding an optimal solution once the missing data is known.
We present efficient algorithms that given an instance with partially
uncertain input generate an instance of size polynomial in the amount of
uncertain data that is equivalent for every instantiation of the unknown part.
Concretely, we obtain such algorithms for Minimum Spanning Tree, Minimum Weight
Matroid Basis, and Maximum Cardinality Bipartite Maxing, where respectively the
weight of edges, weight of elements, and the availability of vertices is
unknown for part of the input. Furthermore, we show that there are tractable
problems, such as Small Connected Vertex Cover, for which one cannot hope to
obtain similar results.Comment: 18 page
Preprocessing Under Uncertainty: Matroid Intersection
We continue the study of preprocessing under uncertainty that was initiated independently by Assadi et al. (FSTTCS 2015) and Fafianie et al. (STACS 2016). Here, we are given an instance of a tractable problem with a large static/known part and a small part that is dynamic/uncertain, and ask if there is an efficient algorithm that computes an instance of size polynomial in the uncertain part of the input, from which we can extract an optimal solution to the original instance for all (usually exponentially many) instantiations of the uncertain part.
In the present work, we focus on the Matroid Intersection problem. Amongst others we present a positive preprocessing result for the important case of finding a largest common independent set in two linear matroids. Motivated by an application for intersecting two gammoids we also revisit Maximum Flow. There we tighten a lower bound of Assadi et al. and give an alternative positive result for the case of low uncertain capacity that yields a Maximum Flow instance as output rather than a matrix