Generalized Parametric Path Problems

TL;DR

This paper introduces a generalized framework for parametric path problems across various domains, analyzing their complexity and tractability under different assumptions and extensions.

Contribution

It formulates a broad parametric path problem, studies its complexity, and identifies conditions for tractability and NP-hardness in linear and non-linear cases.

Findings

Linear parametric weights allow polynomial-time algorithms.

Non-linear weights lead to NP-hard problems.

Multi-dimensional parameterization also results in NP-hardness.

Abstract

Parametric path problems arise independently in diverse domains, ranging from transportation to finance, where they are studied under various assumptions. We formulate a general path problem with relaxed assumptions, and describe how this formulation is applicable in these domains. We study the complexity of the general problem, and a variant of it where preprocessing is allowed. We show that when the parametric weights are linear functions, algorithms remain tractable even under our relaxed assumptions. Furthermore, we show that if the weights are allowed to be non-linear, the problem becomes NP-hard. We also study the mutli-dimensional version of the problem where the weight functions are parameterized by multiple parameters. We show that even with two parameters, the problem is NP-hard.