In this paper, we investigate refutability in Difference Constraint Systems (DCS) under the Unit Read-Once Refutation (UROR) system. A difference constraint is a linear relationship of the form: \(x_{i}-x_{j} \le b_{ij}\) and a DCS is a conjunction of such constraints. In the UROR refutation system, each constraint can be used by at most one inference. Additionally, each inference has to use at least one one-variable (absolute) constraint. Note that an unsatisfiable difference constraint system may not have a UROR. Thus, the UROR refutation system is incomplete for DCSs. The UROR refutation system is useful for proving that the infeasibility of a DCS is caused by the current variable domains. These domains are determined by the absolute constraints in the system. Thus, the UROR refutations of a DCS depend on these variable domains. This is in contrast to unrestricted refutations which do not need to depend on these domain constraints. Investigating weak (incomplete) refutation systems leads to a better understanding of the inference rules required for establishing contradictions in the given constraint system. Thus, this study is well-motivated. Likewise, difference constraint systems arise in a number of application domains such as program verification and scheduling. It follows that efficient refutation systems are of paramount interest. In this paper, we show that problem of checking if a DCS has a unit read-once refutation is NP-complete. Additionally, we provide parameterized and exact exponential algorithms for solving this problem. Finally, we show that the problem of finding the length of the shortest unit read-once refutation is NPO PB-complete.

在本文中，我们研究了单元一次性反驳（UROR）系统下差异约束系统（DCS）的可反驳性。差异约束是以下形式的线性关系：\(x_{i}-x_{j} \le b_{ij}\)DCS 就是这些约束的结合。在UROR反驳系统中，每个约束最多可以被一个推论使用。此外，每个推论必须使用至少一个单变量（绝对）约束。请注意，不可满足的差异约束系统可能没有 UROR。因此，UROR 反驳系统对于 DCS 来说是不完整的。UROR 反驳系统可用于证明 DCS 的不可行性是由当前变量域引起的。这些域是由系统中的绝对约束确定的。因此，DCS 的 UROR 反驳取决于这些可变域。这与不需要依赖于这些领域约束的无限制反驳形成对比。调查薄弱环节（不完全）反驳系统可以更好地理解在给定约束系统中建立矛盾所需的推理规则。因此，这项研究是有充分动机的。同样，差异约束系统出现在许多应用领域，例如程序验证和调度。由此可见，有效的反驳系统至关重要。在本文中，我们证明了检查 DCS 是否具有单元一次读取反驳的问题是NP 完全的。此外，我们还提供参数化和精确的指数算法来解决这个问题。最后，我们证明寻找最短单元一次读取反驳的长度问题是NPO PB-complete 的。