Approximately Counting Approximately-Shortest Paths in Directed Acyclic Graphs

Abstract

Given a directed acyclic graph with positive edge-weights, two vertices s and t, and a threshold-weight L, we present a fullypolynomial 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 L1 and L2, we show how to approximately count the s-t paths that have length at most L1 in the first graph and length not much larger than L2 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.

Faster FPTASes for Counting and Random Generation of Knapsack Solutions

“…In [2], the technique of Štefankovič et al [6] was extended to counting 0/1 Knapsack solutions on a DAG, leading to an FPTAS running in time O (mn 3 log(n)ε −1 ).…”

“…In [2], the 0/1 Knapsack problem was extended to a directed acyclic graph (DAG) with nonnegative arc weights, in connection to various applications in biological sequence analysis (see the references in [2]). Given two vertices s and t, we have to count the number of s, t-paths of total weight at most C .…”

“…That is, given an input for 0/1 Knapsack, we need to generate one solution uniformly at random among all solutions. It follows from general results such as [3] that for a large class of problems (including 0/1 Knapsack) counting and random generation are inter-reducible. As such, also random generation of 0/1 Knapsack solutions is hard and we are thus interested in faster algorithms that generate a solution with near-uniform probability.…”

“…Our assumption that additions and comparisons of O (log C )-bit numbers take unit time is also made in[7,9,6]. 3 Our notion of near-uniform sampling is the same as in e.g.,[3, Section 6].…”

Exact Counting and Sampling of Optima for the Knapsack Problem

“…Many counting problems are #P-complete [21] and often approximations of the number of such solutions, especially approximations on the number of optimal solutions are sought [5]. Further examples include counting the number of shortest paths between two nodes in graphs [13] or exact counting of minimum-spanning-trees [1]. For the knapsack problem (KP), the problem of counting the number of feasible solutions, i.e.…”