Logarithmic Approximation for Road Pricing on Grids

TL;DR

The paper tackles pricing on graphs with budgets, where drivers pay the cheapest u–v path cost under edge prices if that cost does not exceed their budget. It proves that for complete grids of fixed width ω, one can achieve a polynomial-time O(log |E|)-approximation to the maximum revenue by decomposing the grid into blocks, discretizing prices, and solving rooted-subproblem instances via an innovative assume-implement dynamic programming approach complemented by a grid-compression lemma. The core contributions include the assume-implement DP technique, a constant-factor solution for rooted instances, a hierarchical grid decomposition with independent subproblems, and a block-level analysis that yields the overall logarithmic approximation. This advances pricing-for-envy-free-revenue for grid-like networks and lays groundwork for extensions to bounded-pathwidth or bounded-treewidth graphs and more realistic modeling of traveler behavior and congestion.

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.

Figures (4)

• Figure 1: Dynamic programming states for row $i+1$ (left) and $i$ (right). For reference, here $i = 2$. Black dotted edges represent the graph $G_i$. Black vertices marked with asterisks are non-root endpoints for the drivers in $B_{R_i}$ -- those are exactly the drivers from the sum in \ref{['eq:dp']}.
• Figure 2: Example of two paths in a block $H$ desired by two different drivers (red and blue) and their partition into sections. Note that the lower section of the red path is empty, because the corresponding endpoint lies on the middle row, so, by definition, $y = v$.
• Figure 3: A scheme of the partition of edges of $H_{\mathrm{ext}}$. The middle row is marked with a triangle. Black dotted edges are priced at $\infty$ in all $S^i_{s, t}$. The red edges are used to optimize for the lower section ($S^{\mathrm{low}}_{s,t}$), the blue edges for the upper section ($S^{\mathrm{up}}_{s,t}$), and the green edges for the middle section ($S^{\mathrm{mid}}_{s,t}$). Whenever one of those groups is used to optimize revenue, the other two are priced to $0$.
• Figure 4: An example of a compressed $2$-layer, where vertices in $U \cup D$ are connected using the newly created paths. Note that not all vertices in the border rows belong to $U$ or $D$, but only those that have edges going outside of the $2$-layer.

Theorems & Definitions (22)

• theorem 0
• definition 1
• lemma 2
• lemma 3
• proof
• definition 4
• definition 5
• lemma 6
• proof
• remark 7