1- VASS的可覆盖性及其不等式检验
Coverability in 1-VASS with Disequality Tests
摘要：
研究了一类带权图的可达性问题，其中路的累计权是有约束的。我们所研究的问题可以等价地表示为状态向量加法系统模型（VASS）。我们考虑一个版本的顶点到顶点的可达性问题，其中路径的累积权重总是需要是非负的。这等价于一维VASS的所谓控制状态可达性问题（也称为可覆盖性问题）。我们证明了这个问题存在于NC中：这类问题可以在多对数并行时间中求解。在我们的主要结果中，我们将问题推广到允许边上的不等约束（即，如果累积权重等于特定值，则允许禁用边缘）。我们表明，在这种情况下，顶点到顶点的可达性问题是可解的多项式时间，即使最短路径可能有指数长度。在VASS的语言中，这意味着控制状态的可达性是在多项式时间的一维VASS与不等性测试。
英文摘要：
We study a class of reachability problems in weighted graphs with constraints on the accumulated weight of paths. The problems we study can equivalently be formulated in the model of vector addition systems with states (VASS). We consider a version of the vertex-to-vertex reachability problem in which the accumulated weight of a path is required always to be non-negative. This is equivalent to the so-called control-state reachability problem (also called the coverability problem) for 1-dimensional VASS. We show that this problem lies in NC: the class of problems solvable in polylogarithmic parallel time. In our main result we generalise the problem to allow disequality constraints on edges (i.e., we allow edges to be disabled if the accumulated weight is equal to a specific value). We show that in this case the vertex-to-vertex reachability problem is solvable in polynomial time even though a shortest path may have exponential length. In the language of VASS this means that control-state reachability is in polynomial time for 1-dimensional VASS with disequality tests.
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