We consider the core algorithmic problems related to verification of systems with respect to three classical quantitative properties, namely, the mean-payoff, the ratio, and the minimum initial credit for energy property. The algorithmic problem given a graph and a quantitative property asks to compute the optimal value (the infimum value over all traces) from every node of the graph. We consider graphs with bounded treewidth—a class that contains the control flow graphs of most programs. Let n denote the number of nodes of a graph, m the number of edges (for bounded treewidth \(m=O(n)\)) and W the largest absolute value of the weights. Our main theoretical results are as follows. First, for the minimum initial credit problem we show that (1) for general graphs the problem can be solved in \(O(n^2\cdot m)\) time and the associated decision problem in \(O(n\cdot m)\) time, improving the previous known \(O(n^3\cdot m\cdot \log (n\cdot W))\) and \(O(n^2 \cdot m)\) bounds, respectively; and (2) for bounded treewidth graphs we present an algorithm that requires \(O(n\cdot \log n)\) time. Second, for bounded treewidth graphs we present an algorithm that approximates the mean-payoff value within a factor of \(1+\epsilon \) in time \(O(n \cdot \log (n/\epsilon ))\) as compared to the classical exact algorithms on general graphs that require quadratic time. Third, for the ratio property we present an algorithm that for bounded treewidth graphs works in time \(O(n \cdot \log (|a\cdot b|))=O(n\cdot \log (n\cdot W))\), when the output is \(\frac{a}{b}\), as compared to the previously best known algorithm on general graphs with running time \(O(n^2 \cdot \log (n\cdot W))\). We have implemented some of our algorithms and show that they present a significant speedup on standard benchmarks.

我们考虑了与系统验证相关的核心算法问题，涉及三个经典的定量属性，即均值，比率和能量属性的最小初始信用。给定图形和定量属性的算法问题要求从图形的每个节点计算最佳值（所有迹线的最小值）。我们考虑具有有限树宽的图-包含大多数程序的控制流程图的类。令n表示图的节点数，m表示边数（对于有界树宽\（m = O（n）\））和W重量的最大绝对值。我们的主要理论结果如下。首先，对于最小初始信用问题，我们表明（1）对于一般图形，该问题可以在\（O（n ^ 2 \ cdot m）\）时间内解决，而相关的决策问题可以在\（O（n \ cdot m）\ m）\）时间，分别改善了先前已知的\（O（n ^ 3 \ cdot m \ cdot \ log（n \ cdot W））\）和\（O（n ^ 2 \ cdot m）\）范围; （2）对于有界树宽图，我们提出了一种算法，该算法需要\（O（n \ cdot \ log n）\）时间。其次，对于有界树宽图，我们提出了一种算法，该算法在时间\（O（n \ cdot \ log（n / \ epsilon））\）中近似于\（1+ \ epsilon \）范围内的平均收益值。与需要二次时间的一般图上的经典精确算法相比。第三，对于比率属性，我们提出一种算法，该算法对于有界树宽图在时间\（O（n \ cdot \ log（| a \ cdot b |））= O（n \ cdot \ log（n \ cdot W） ）\），当输出为\（\ frac {a} {b} \）时，与运行时间为\（O（n ^ 2 \ cdot \ log（n \ cdot W））\）。我们已经实现了一些算法，并表明它们在标准基准测试中具有显着的加速作用。