Parallel Consensus is Harder than Set Agreement in Message Passing

In the traditional consensus task, processes are required to agree on a common value chosen among the initial values of the participating processes. It is well known that consensus cannot be solved in crash-prone, asynchronous distributed systems. Two generalizations of the consensus tasks have been introduced: k-set agreement and k-parallel consensus. The k-set agreement task has the same requirements as consensus except that processes are allowed to decide up to k distinct values. In the k-parallel consensus task, each process participates simultaneously in k instances of consensus and is required to decide in at least one of them; any two processes deciding in the same instance must decide the same value. It is known that both tasks are equivalent in the wait-free shared memory model. Perhaps surprisingly, this paper shows that this is no longer the case in the n-process asynchronous message passing model with at most t process crashes. Specifically, the paper establishes that for parameters t, n, k such that t > n+ k-2/2, k-parallel consensus is strictly harder than k-set agreement. The proof compares the information on failures necessary to solve each task in the failure detector framework and relies on a result in topological combinatorics, namely, the chromatic number of Kneser graphs. The paper also introduces the new failure detector class V Sigma k, which is a generalization of the quorum failures detector class Sigma suited to k-parallel consensus.