Counting approximately-shortest paths in directed acyclic graphs

TL;DR

This paper introduces a fully-polynomial time approximation scheme for counting s-t paths within a length threshold in directed acyclic graphs, extending to multiple instances with different weights.

Contribution

It presents a novel approximation algorithm for counting constrained shortest paths in DAGs, including a method for handling multiple graph instances with different weights.

Findings

Efficient approximation of path counts within length thresholds in DAGs.

Extension of the algorithm to multiple graph instances with different weights.

Potential applications in approximate solutions of related optimization problems.

Abstract

Given a directed acyclic graph with positive edge-weights, two vertices s and t, and a threshold-weight L, we present a fully-polynomial time approximation-scheme for the problem of counting the s-t paths of length at most L. We extend the algorithm for the case of two (or more) instances of the same problem. That is, given two graphs that have the same vertices and edges and differ only in edge-weights, and given two threshold-weights L_1 and L_2, we show how to approximately count the s-t paths that have length at most L_1 in the first graph and length at most L_2 in the second graph. We believe that our algorithms should find application in counting approximate solutions of related optimization problems, where finding an (optimum) solution can be reduced to the computation of a shortest path in a purpose-built auxiliary graph.