Algorithms to solve fault-tolerant consensus in asynchronous systems often rely on primitives such as crusader agreement, adopt-commit, and graded broadcast, which provide weaker agreement properties than consensus. Although these primitives have a similar flavor, they have been defined and implemented separately in ad hoc ways. We propose a new problem called connected consensus that has as special cases crusader agreement, adopt-commit, and graded broadcast, and generalizes them to handle multi-valued inputs. The generalization is accomplished by relating the problem to approximate agreement on graphs. We present three algorithms for multi-valued connected consensus in asynchronous message-passing systems, one tolerating crash failures and two tolerating malicious (unauthenticated Byzantine) failures. We extend the definition of binding, a desirable property recently identified as supporting binary consensus algorithms that are correct against adaptive adversaries, to the multi-valued input case and show that all our algorithms satisfy the property. Our crash-resilient algorithm has failure-resilience and time complexity that we show are optimal. When restricted to the case of binary inputs, the algorithm has improved time complexity over prior algorithms. Our two algorithms for malicious failures trade off failure resilience and time complexity. The first algorithm has time complexity that we prove is optimal but worse failure-resilience, while the second has failure-resilience that we prove is optimal but worse time complexity. When restricted to the case of binary inputs, the time complexity (as well as resilience) of the second algorithm matches that of prior algorithms.

翻译：在异步系统中解决容错共识的算法往往依赖于十字军协议、采纳-提交和分级广播等原语，这些原语提供的共识性质弱于完整共识。尽管这些原语具有相似特性，但它们通常以特定方式分别定义和实现。我们提出一类称为"连通共识"的新问题，其特例涵盖十字军协议、采纳-提交和分级广播，并将其推广至处理多值输入。该推广通过将问题与图上的近似共识建立关联来实现。我们针对异步消息传递系统提出三种多值连通共识算法：一种容忍崩溃故障，两种容忍恶意（未认证拜占庭）故障。我们将最近被识别为支持针对自适应敌手正确的二进制共识算法的理想性质"绑定"扩展至多值输入情形，并证明所有算法均满足该性质。我们的崩溃容错算法具有最优的故障容错性和时间复杂度。当限制为二进制输入时，该算法的时间复杂度优于已有算法。两种针对恶意故障的算法在故障容错性和时间复杂度间进行权衡：第一种算法的时间复杂度经证明最优但故障容错性较差，第二种算法的故障容错性经证明最优但时间复杂度较高。当限制为二进制输入时，第二种算法的时间复杂度（及容错性）与已有算法相当。