Online Budgeted Matching with General Bids

TL;DR

This paper establishes an upper bound of 1-\kappa on the competitive ratio for any deterministic online algorithm and proposes a novel meta algorithm, called MetaAd, which reduces to different algorithms with first known provable competitive ratios parameterized by the maximum bid-to-budget ratio.

Abstract

Online Budgeted Matching (OBM) is a classic problem with important applications in online advertising, online service matching, revenue management, and beyond. Traditional online algorithms typically assume a small bid setting, where the maximum bid-to-budget ratio (κ) is infinitesimally small. While recent algorithms have tried to address scenarios with non-small or general bids, they often rely on the Fractional Last Matching (FLM) assumption, which allows for accepting partial bids when the remaining budget is insufficient. This assumption, however, does not hold for many applications with indivisible bids. In this paper, we remove the FLM assumption and tackle the open problem of OBM with general bids. We first establish an upper bound of 1-κon the competitive ratio for any deterministic online algorithm. We then propose a novel meta algorithm, called MetaAd, which reduces to different algorithms with first known provable competitive ratios parameterized by the maximum bid-to-budget ratio κ\in [0, 1]. As a by-product, we extend MetaAd to the FLM setting and get provable competitive algorithms. Finally, we apply our competitive analysis to the design learning-augmented algorithms.

Figures (6)

• Figure 1: Competitive ratio without FLM. MetaAd (Exp) represents the MetaAd with $\varphi(x)=C(e^{\theta x}-1)$ and MetaAd (Quad) represents the MetaAd with $\varphi(x)=Cx^2)$.
• Figure 2: Competitive ratio with FLM. MetaAd (Exp) represents MetaAd with $\varphi(x)=C(e^{\theta x}-1)$ and BJN2007 represents the algorithm in primal_dual_adwards_buchbinder2007online.
• Figure 3: Illustration of the scoring strategies in MetaAd and LOBM. The example has 3 offline nodes ($u_1,u_2,u_3$). The algorithms select the offline node with the largest score.
• Figure 4: (a) Worst-case and average reward of MetaAd with different choices of $\theta$. (b) Worst-case reward of LOBM with different choices of $\theta$ and $\lambda$. (c)Average reward of LOBM with different choices of $\theta$ and $\lambda$.
• Figure 5: Reward (normalized by the offline optimal reward) at high percentiles (95% - 100%). $\theta$ is chosen as 1.

Theorems & Definitions (18)

• Definition 1: Bid-budget ratio
• Proposition 4.1
• Theorem 4.2
• Lemma 1: Conditions for competitive ratio
• Corollary 4.2.1
• Corollary 4.2.2
• Corollary 4.2.3
• Theorem 4.3
• Theorem 5.1
• proof