The Distribution Function of the Longest Path Length in Constant Treewidth DAGs with Random Edge Length
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This paper is about the length X_ MAX of the longest path in directed acyclic graph (DAG) G=(V,E) that have random edge lengths, where |V|=n and |E|=m. Especially, when the edge lengths are mutually independent and uniformly distributed, the problem of computing the distribution function [X_ MAX< x] is known to be #P-hard even in case G is a directed path. This is because [X_ MAX< x] is equal to the volume of the knapsack polytope, an m-dimensional unit hypercube truncated by a halfspace. In this paper, we show that there is a deterministic fully polynomial time approximation scheme (FPTAS) for computing [X_ MAX< x] in case the treewidth of G is bounded by a constant k; where there may be exponentially many source-terminal paths in G. The running time of our algorithm is O(n(2mn/ϵ)^k^2+4k) to achieve a multiplicative approximation ratio 1+ϵ. On the way to show our FPTAS, we show a fundamental formula that represents [X_ MAX< x] by n-1 repetition of definite integrals. This also leads us to more results. In case the edge lengths obey the mutually independent standard exponential distribution, we show that there exists an (2kmn)^O(k) time exact algorithm. We also show, for random edge lengths satisfying certain conditions, that computing [X_ MAX< x] is fixed parameter tractable if we choose treewidth k, the additive error ϵ' and x as the parameters by using the Taylor approximation.
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