Online Budgeted Matching with General Bids

TL;DR

This paper addresses the online budgeted matching problem with general bids, removing the fractional last matching assumption, establishing upper bounds, and proposing a versatile meta algorithm with provable competitive ratios.

Contribution

It introduces MetaAd, a novel meta algorithm that handles general bids without the FLM assumption and extends to learning-augmented settings.

Findings

Upper bound of 1-k on competitive ratio for any deterministic algorithm

MetaAd reduces to different algorithms with provable ratios based on k

Extended MetaAd to FLM setting with competitive guarantees

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 (\kappa) 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-\kappa on 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 \kappa \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.