An Efficient Algorithm for the Computation of Optimum Paths in Weighted Graphs

The goal of this paper is to identify a more efficient algorithm for the computation of the path of minimum ratio (i.e. the quotient of the weight divided by the length) in a weighted graph. The main application of this technique is to improve the efficiency of reachability analysis for flat counter automata with Difference Bound Matrix constraints on transitions. A previous result showed that these paths could be defined using Presburger arithmetic. However, this method is costly, as the complexity of deciding satisfiability of Presburger formulae has a double exponential lower bound. Our solution avoids the use of Presburger arithmetic, by computing the function between any n ∈ N and the weight of the minimal path of length n. The function is computed iteratively, by computing the minimal fixed point of a system of set constraints, involving semilinear sets. This requires a min operator on linear sets, which is implemented using rewrite rules.