Two-phase algorithms for the parametric shortest path problem

TL;DR

This paper introduces two-phase algorithms for the parametric shortest path problem, enabling faster solutions by preprocessing a parametric graph to create advice used in quick instantiation phases.

Contribution

It develops novel two-phase algorithms for parametric shortest paths, especially for linear weight functions, improving efficiency over traditional methods.

Findings

Preprocessing creates advice data structures for rapid instantiation.

Algorithms significantly reduce computation time for specific instances.

Focus on linear function weights for single source shortest path.

Abstract

A {\em parametric weighted graph} is a graph whose edges are labeled with continuous real functions of a single common variable. For any instantiation of the variable, one obtains a standard edge-weighted graph. Parametric weighted graph problems are generalizations of weighted graph problems, and arise in various natural scenarios. Parametric weighted graph algorithms consist of two phases. A {\em preprocessing phase} whose input is a parametric weighted graph, and whose output is a data structure, the advice, that is later used by the {\em instantiation phase}, where a specific value for the variable is given. The instantiation phase outputs the solution to the (standard) weighted graph problem that arises from the instantiation. The goal is to have the running time of the instantiation phase supersede the running time of any algorithm that solves the weighted graph problem from scratch, by taking advantage of the advice. In this paper we construct several parametric algorithms for the shortest path problem. For the case of linear function weights we present an algorithm for the single source shortest path problem.