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

Abstract

For a graph G ( V , E ) (| V | = n ) and a vertex s ∈ V , a weighting scheme ( W : E ↦ Z + ) 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 , where weight of a path is the sum of the weights assigned to the edges. Instead, if the number of paths of minimum (resp. maximum) weight is bounded by n c for some constant c , then the weighting scheme is called a min-poly (resp. max-poly ) weighting scheme. In this article, we propose an unambiguous nondeterministic log-space (UL) algorithm for the problem of testing reachability graphs augmented with a min-poly weighting scheme. This improves the result in Reinhardt and Allender [2000], in which 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 Immerman [1988], Szelepcsényi [1988], and Reinhardt and Allender [2000], combined with a hashing technique due to Fredman et al. [1984] (also used in Garvin et al. [2014]). 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 Limaye et al. [2009], in which the 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 ) weighting schemes for layered DAGs.