Robust Reward Placement under Uncertainty

TL;DR

This work studies Robust Reward Placement (RRP), a problem of placing reward states on a network to maximize the worst-case performance across multiple Markov Mobility Models (MMM) under a knapsack budget. The authors prove $\mathbf{NP}$-hardness even for a single MMM, and show $\mathbf{inapprox}$ results for multiple MMMs via a Hitting Set reduction. They introduce $\\Psi$-Saturate, a pseudo-polynomial-time bicriteria algorithm that achieves an $\\epsilon$-additive approximation by allowing a budget overrun of $\\mathcal{O}(\\ln|\\Pi|/\\epsilon)$. They also provide several heuristics, including a dynamic-programming-based approach (DP-RRP) and four greedy variants, and validate the methods with experiments on synthetic graphs and real movement data. The results demonstrate the practical effectiveness of $\\Psi$-Saturate and DP-RRP in achieving robust reward placement under mobility uncertainty, with insights into when each method performs best.

Abstract

We consider a problem of placing generators of rewards to be collected by randomly moving agents in a network. In many settings, the precise mobility pattern may be one of several possible, based on parameters outside our control, such as weather conditions. The placement should be robust to this uncertainty, to gain a competent total reward across possible networks. To study such scenarios, we introduce the Robust Reward Placement problem (RRP). Agents move randomly by a Markovian Mobility Model with a predetermined set of locations whose connectivity is chosen adversarially from a known set $Π$ of candidates. We aim to select a set of reward states within a budget that maximizes the minimum ratio, among all candidates in $Π$, of the collected total reward over the optimal collectable reward under the same candidate. We prove that RRP is NP-hard and inapproximable, and develop $Ψ$-Saturate, a pseudo-polynomial time algorithm that achieves an $ε$-additive approximation by exceeding the budget constraint by a factor that scales as $O(\ln |Π|/ε)$. In addition, we present several heuristics, most prominently one inspired by a dynamic programming algorithm for the max-min 0-1 KNAPSACK problem. We corroborate our theoretical analysis with an experimental evaluation on synthetic and real data.

Figures (5)

• Figure 1: Moving agent under two settings; sunny and rainy; tables show numbers of steps and initial probabilities.
• Figure 2: Hitting Set (left) and RRP reduction (right).
• Figure 3: Preprocessing and running time vs. $n$, $L$ for Erdős-Rényi (left) and Scale-Free (right) datasets.
• Figure 4: Reward placement robustness scores on Erdős-Rényi (top) and Scale-Free (bottom) datasets.
• Figure 5: Time and robustness score vs. $L$ on the Xuanwu dataset.

Theorems & Definitions (17)

• Theorem 1
• proof
• Theorem 2
• proof
• Lemma 3
• proof
• Lemma 3
• proof
• Lemma 3
• Lemma 3