Lower Bounds for $k$-Set Agreement in Fault-Prone Networks

We develop a new lower bound for k-set agreement in synchronous message-passing systems connected by an arbitrary directed communication network, where up to t processes may crash. Our result thus generalizes the t/k+1 lower bound for complete networks in the t-resilient model by Chaudhuri, Herlihy, Lynch, and Tuttle [JACM'00]. Moreover, it generalizes two lower bounds for oblivious algorithms in synchronous systems connected by an arbitrary undirected communication network known to the processes, namely, the domination number-based lower bound by Castaneda, Fraigniaud, Paz, Rajsbaum, Roy, and Travers [TCS'21] for failure-free processes, and the radius-based lower bound in the t-resilient model by Fraigniaud, Nguyen, and Paz [STACS'24]. Our topological proof non-trivially generalizes and extends the connectivity-based approach for the complete network, as presented in the book by Herlihy, Kozlov, and Rajsbaum (2013). It is based on a sequence of shellable carrier maps that, starting from a shellable input complex, determine the evolution of the protocol complex: During the first t/k rounds, carrier maps that crash exactly k processes per round are used, ensuring high connectivity of their images. A Sperner's lemma style argument is used to prove that k-set agreement is still impossible by that round. From round t/k+1 up to our lower bound, we employ a novel carrier map that maintains high connectivity. Our proof also provides a strikingly simple lower bound for k-set agreement in synchronous systems with an arbitrary communication network with initial crashes. We express the resulting additional agreement overhead via an appropriately defined radius of the communication graphs. Finally, we prove that the usual input pseudosphere complex for k-set agreement can be replaced by an exponentially smaller input complex based on Kuhn triangulations, which we prove to be also shellable.