Logarithmic Approximation for Road Pricing on Grids

TL;DR

This paper presents a polynomial-time logarithmic approximation algorithm for maximizing revenue in a grid graph road pricing problem, introducing a novel 'assume-implement dynamic programming' technique.

Contribution

The paper introduces a new approximation algorithm for grid graph road pricing and a novel 'assume-implement' dynamic programming technique for such problems.

Findings

Achieves an $O(\log |E|)$ approximation ratio.

Employs a novel 'assume-implement' dynamic programming method.

Provides polynomial-time solution for grid graph pricing problem.

Abstract

Consider a graph $G = (V, E)$ and some commuters, each specified by a tuple $(u, v, b)$ consisting of two nodes in the graph $u, v \in V$ and a non-negative real number $b$, specifying their budget. The goal is to find a pricing function $p$ of the edges of $G$ that maximizes the revenue generated by the commuters. Here, each commuter $(u, v, b)$ either pays the lowest-cost of a $u$-$v$ path under the pricing $p$, or 0, if this exceeds their budget $b$. We study this problem for the case where $G$ is a bounded-width grid graph and give a polynomial-time approximation algorithm with approximation ratio $O(\log |E|)$. Our approach combines existing ideas with new insights. Most notably, we employ a rather seldom-encountered technique that we coin under the name 'assume-implement dynamic programming.' This technique involves dynamic programming where some information about the future decisions of the dynamic program is guessed in advance and 'assumed' to hold, and then subsequent decisions are forced to 'implement' the guess. This enables computing the cost of the current transition by using information that would normally only be available in the future.