We study the problem of finding the cycle of minimum cost-to-time ratio in a directed graph with n nodes and m edges. This problem has a long history in combinatorial optimization and has recently seen interesting applications in the context of quantitative verification. We focus on strongly polynomial algorithms to cover the use-case where the weights are relatively large compared to the size of the graph. Our main result is an algorithm with running time ~O(m^{3/4} n^{3/2}), which gives the first improvement over Megiddo's ~O(n^3) algorithm [JACM'83] for sparse graphs (We use the notation ~O(.) to hide factors that are polylogarithmic in n.) We further demonstrate how to obtain both an algorithm with running time n^3/2^{Omega(sqrt(log n)} on general graphs and an algorithm with running time ~O(n) on constant treewidth graphs. To obtain our main result, we develop a parallel algorithm for negative cycle detection and single-source shortest paths that might be of independent interest.
@InProceedings{bringmann_et_al:LIPIcs.ICALP.2017.124, author = {Bringmann, Karl and Dueholm Hansen, Thomas and Krinninger, Sebastian}, title = {{Improved Algorithms for Computing the Cycle of Minimum Cost-to-Time Ratio in Directed Graphs}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {124:1--124:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.124}, URN = {urn:nbn:de:0030-drops-74398}, doi = {10.4230/LIPIcs.ICALP.2017.124}, annote = {Keywords: quantitative verification and synthesis, parametric search, shortest paths, negative cycle detection} }
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