%0 Journal Article %1 balliu2019lower %A Balliu, Alkida %A Brandt, Sebastian %A Hirvonen, Juho %A Olivetti, Dennis %A Rabie, Mikaël %A Suomela, Jukka %D 2019 %K bounds mathematics theory %T Lower bounds for maximal matchings and maximal independent sets %U http://arxiv.org/abs/1901.02441 %X There are distributed graph algorithms for finding maximal matchings and maximal independent sets in $O(\Delta + łog^* n)$ communication rounds; here $n$ is the number of nodes and $\Delta$ is the maximum degree. The lower bound by Linial (1987, 1992) shows that the dependency on $n$ is optimal: these problems cannot be solved in $o(łog^* n)$ rounds even if $\Delta = 2$. However, the dependency on $\Delta$ is a long-standing open question, and there is currently an exponential gap between the upper and lower bounds. We prove that the upper bounds are tight. We show that maximal matchings and maximal independent sets cannot be found in $o(\Delta + n / łog n)$ rounds with any randomized algorithm in the LOCAL model of distributed computing. As a corollary, it follows that there is no deterministic algorithm for maximal matchings or maximal independent sets that runs in $o(\Delta + n / n)$ rounds; this is an improvement over prior lower bounds also as a function of $n$.

@article{balliu2019lower, abstract = {There are distributed graph algorithms for finding maximal matchings and maximal independent sets in $O(\Delta + \log^* n)$ communication rounds; here $n$ is the number of nodes and $\Delta$ is the maximum degree. The lower bound by Linial (1987, 1992) shows that the dependency on $n$ is optimal: these problems cannot be solved in $o(\log^* n)$ rounds even if $\Delta = 2$. However, the dependency on $\Delta$ is a long-standing open question, and there is currently an exponential gap between the upper and lower bounds. We prove that the upper bounds are tight. We show that maximal matchings and maximal independent sets cannot be found in $o(\Delta + \log \log n / \log \log \log n)$ rounds with any randomized algorithm in the LOCAL model of distributed computing. As a corollary, it follows that there is no deterministic algorithm for maximal matchings or maximal independent sets that runs in $o(\Delta + \log n / \log \log n)$ rounds; this is an improvement over prior lower bounds also as a function of $n$.}, added-at = {2019-09-09T22:02:25.000+0200}, author = {Balliu, Alkida and Brandt, Sebastian and Hirvonen, Juho and Olivetti, Dennis and Rabie, Mikaël and Suomela, Jukka}, biburl = {https://www.bibsonomy.org/bibtex/2d22fe55726c5d42ff02d4103706add52/kirk86}, description = {[1901.02441] Lower bounds for maximal matchings and maximal independent sets}, interhash = {2ce42b2c3d0e54528658a8ce7657a875}, intrahash = {d22fe55726c5d42ff02d4103706add52}, keywords = {bounds mathematics theory}, note = {cite arxiv:1901.02441}, timestamp = {2019-09-09T22:02:25.000+0200}, title = {Lower bounds for maximal matchings and maximal independent sets}, url = {http://arxiv.org/abs/1901.02441}, year = 2019 }