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Max $s,t$-Flow Oracles and Negative Cycle Detection in Planar Digraphs by Karczmarz, Adam is a scholarly article available to read on EtoBox.

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We study the maximum $s,t$-flow oracle problem on planar directed graphs where the goal is to design a data structure answering max $s,t$-flow value (or equivalently, min $s,t$-cut value) queries for arbitrary source-target pairs $(s,t)$. For the case of polynomially bounded integer edge capacities, we describe an exact max $s,t$-flow oracle with truly subquadratic space and preprocessing, and sublinear query time. Moreover, if $(1-\epsilon)$-approximate answers are acceptable, we obtain a static oracle with near-linear preprocessing and $\tilde{O}(n^{3/4})$ query time and a dynamic oracle supporting edge capacity updates and queries in $\tilde{O}(n^{6/7})$ worst-case time. To the best of our knowledge, for directed planar graphs, no (approximate) max $s,t$-flow oracles have been described even in the unweighted case, and only trivial tradeoffs involving either no preprocessing or precomputing all the $n^2$ possible answers have been known. One key technical tool we develop on the way is a sublinear (in the number of edges) algorithm for finding a negative cycle in so-called dense distance graphs. By plugging it in earlier frameworks, we obtain improved bounds for other fundamental

Author

Karczmarz, Adam

Published

2023

Language

EN