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Abstract.A system with limited-scope failure detectors ensures that there are q subsets Xi of xi processes, 0 ≤ i ≤ q − 1, such that some correct process in Xi is never suspected by any process in Xi. Let x be the sum of xi and X be the union of Xi. The failure detector class Sx,q satisfies this property all the time, while Sx,q satisfies it eventually. This paper gives the first tight bounds for the k-set agreement task in asynchronous message-passing models augmented with failure detectors from either the Sx,q or Sx,q classes. For Sx,q, we show that any k-set agreement protocol that tolerates f failures must satisfy f < k + x − q. This result establishes for the first time that the protocol of Mostéfaoui and Raynal for the Sx = Sx,1 failure detector is optimal. For Sx,q, our lower bound is f < min(, k+x−q). We give a novel protocol that matches our lower bound, disproving a conjecture of Mostéfaoui and Raynal for the Sx = Sx,1 failure detector. Our lower bounds exploit techniques borrowed from Combinatorial Topology, demonstrating for the first time that this approach is applicable to models that encompass failure detectors.

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