article published yes Krishnendu Chatterjee author 2E5DCA20-F248-11E8-B48F-1D18A9856A870000-0002-4561-241X Monika H Henzinger author 540c9bbd-f2de-11ec-812d-d04a5be856300000-0002-5008-6530 Sebastian Krinninger author Veronika Loitzenbauer author Michael Raskin author KrCh department Modern Graph Algorithmic Techniques in Formal Verification project Game Theory project Quantitative Graph Games: Theory and Applications project Microsoft Research Faculty Fellowship project We consider directed graphs where each edge is labeled with an integer weight and study the fundamental algorithmic question of computing the value of a cycle with minimum mean weight. Our contributions are twofold: (1) First we show that the algorithmic question is reducible to the problem of a logarithmic number of min-plus matrix multiplications of n×n-matrices, where n is the number of vertices of the graph. (2) Second, when the weights are nonnegative, we present the first (1+ε)-approximation algorithm for the problem and the running time of our algorithm is Õ(nωlog3(nW/ε)/ε),1 where O(nω) is the time required for the classic n×n-matrix multiplication and W is the maximum value of the weights. With an additional O(log(nW/ε)) factor in space a cycle with approximately optimal weight can be computed within the same time bound. Elsevier2014 eng 1307.4473 00034069400000810.1016/j.tcs.2014.06.031 547C104 - 116 K. Chatterjee, M. Henzinger, S. Krinninger, V. Loitzenbauer, and M. Raskin, “Approximating the minimum cycle mean,” <i>Theoretical Computer Science</i>, vol. 547, no. C. Elsevier, pp. 104–116, 2014. Chatterjee K, Henzinger M, Krinninger S, Loitzenbauer V, Raskin M. Approximating the minimum cycle mean. <i>Theoretical Computer Science</i>. 2014;547(C):104-116. doi:<a href="https://doi.org/10.1016/j.tcs.2014.06.031">10.1016/j.tcs.2014.06.031</a> Chatterjee K, Henzinger M, Krinninger S, Loitzenbauer V, Raskin M. 2014. Approximating the minimum cycle mean. Theoretical Computer Science. 547(C), 104–116. K. Chatterjee, M. Henzinger, S. Krinninger, V. Loitzenbauer, M. Raskin, Theoretical Computer Science 547 (2014) 104–116. Chatterjee, Krishnendu, Monika Henzinger, Sebastian Krinninger, Veronika Loitzenbauer, and Michael Raskin. “Approximating the Minimum Cycle Mean.” <i>Theoretical Computer Science</i>. Elsevier, 2014. <a href="https://doi.org/10.1016/j.tcs.2014.06.031">https://doi.org/10.1016/j.tcs.2014.06.031</a>. Chatterjee, Krishnendu, et al. “Approximating the Minimum Cycle Mean.” <i>Theoretical Computer Science</i>, vol. 547, no. C, Elsevier, 2014, pp. 104–16, doi:<a href="https://doi.org/10.1016/j.tcs.2014.06.031">10.1016/j.tcs.2014.06.031</a>. Chatterjee, K., Henzinger, M., Krinninger, S., Loitzenbauer, V., &#38; Raskin, M. (2014). Approximating the minimum cycle mean. <i>Theoretical Computer Science</i>. Elsevier. <a href="https://doi.org/10.1016/j.tcs.2014.06.031">https://doi.org/10.1016/j.tcs.2014.06.031</a> 13752018-12-11T11:51:40Z2025-09-29T13:17:21Z