Min/Max-Poly Weighting Schemes and the NL vs UL Problem∗

For a graph G(V,E) (|V | = n) and a vertex s ∈ V , a weighting scheme (w : E → N) is called a min-unique (resp. max-unique) weighting scheme, if for any vertex v of the graph G, there is a unique path of minimum (resp. maximum) weight from s to v. Instead, if the number of paths of minimum (resp. maximum) weight is bounded by n for some constant c, then the weighting scheme is called a min-poly (resp. max-poly) weighting scheme. In this paper, we propose an unambiguous non-deterministic log-space (UL) algorithm for the problem of testing reachability graphs augmented with a min-poly weighting scheme. This improves the result due to Allender and Reinhardt[12] where a UL algorithm was given for the case when the weighting scheme is min-unique. Our main technique involves triple inductive counting, and generalizes the techniques of [8, 13] and [12], combined with a hashing technique due to [6] (also used in [7]). We combine this with a complementary unambiguous verification method, to give the desired UL algorithm. At the other end of the spectrum, we propose a UL algorithm for testing reachability in layered DAGs augmented with max-poly weighting schemes. To achieve this, we first reduce reachability in layered DAGs to the longest path problem for DAGs with a unique source, such that the reduction also preserves the max-unique and max-poly properties of the graph. Using our techniques, we generalize the double inductive counting method in [9] where UL algorithm was given for the longest path problem on DAGs with a unique sink and augmented with a max-unique weighting scheme. An important consequence of our results is that, to show NL = UL, it suffices to design log-space computable min-poly (or max-poly) weighing schemes for layered DAGs.