In this paper, we prove a super-cubic lower bound on the size of a communication protocol for generalized Karchmer-Wigderson game for an explicit function f: {0,1}ⁿ → {0,1}^{log n}. Lower bounds for original Karchmer-Wigderson games correspond to De Morgan formula lower bounds, thus the best known size lower bound is cubic. The generalized Karchmer-Wigderson games are similar to the original ones, so we hope that our approach can provide an insight for proving better lower bounds on the original Karchmer-Wigderson games, and hence for proving new lower bounds on De Morgan formula size. To achieve super-cubic lower bound we adapt several techniques used in formula complexity to communication protocols, prove communication complexity lower bound for a composition of several functions with a multiplexer relation, and use a technique from [Ivan Mihajlin and Alexander Smal, 2021] to extract the "hardest" function from it. As a result, in this setting we are able to show that there is a relatively small set of functions such that at least one of them does not have a small protocol. The resulting lower bound of Ω̃(n^3.156) is significantly better than the bound obtained from the counting argument.
@InProceedings{ignatiev_et_al:LIPIcs.ISAAC.2022.66, author = {Ignatiev, Artur and Mihajlin, Ivan and Smal, Alexander}, title = {{Super-Cubic Lower Bound for Generalized Karchmer-Wigderson Games}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {66:1--66:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.66}, URN = {urn:nbn:de:0030-drops-173510}, doi = {10.4230/LIPIcs.ISAAC.2022.66}, annote = {Keywords: communication complexity, circuit complexity, Karchmer-Wigderson games} }
Feedback for Dagstuhl Publishing