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A New Impossibility Result for Online Bipartite Matching Problems

Flavio Chierichetti, Mirko Giacchini, Alessandro Panconesi, Andrea Vattani

TL;DR

The authors address online bipartite matching with random arrivals by constructing a simple irregular cuckoo hashing instance with $n$ users and $n$ ads, where $\Pr[D=d] = \frac{1}{d(d-1)}$ for $d\ge 2$. They prove that no online algorithm can achieve a competitive ratio exceeding $1 - \frac{e}{e^e}$, approximately $0.82062$, across several problem variants, while the constructed instance admits a quasi-complete offline maximum matching of size $(1-o(1))\,n$. The analysis combines a configuration-model coupling with a refined Karp–Sipser argument extended to unbounded degrees, and an extremality result showing the construction is worst-case within a natural family. They further generalize the approach to non-equibipartite graphs and demonstrate that the worst case occurs at unit user-to-ads ratio, $u=1$, within the generalized class. The work raises open questions about tightness and whether the bound is achievable in broader settings, while providing a clean analytic bound with a near-unity gap to prior results and strong theoretical insight into the problem structure.

Abstract

Online Bipartite Matching with random user arrival is a fundamental problem in the online advertisement ecosystem. Over the last 30 years, many algorithms and impossibility results have been developed for this problem. In particular, the latest impossibility result was established by Manshadi, Oveis Gharan and Saberi in 2011. Since then, several algorithms have been published in an effort to narrow the gap between the upper and the lower bounds on the competitive ratio. In this paper we show that no algorithm can achieve a competitive ratio better than $1- \frac e{e^e} = 0.82062\ldots$, improving upon the $0.823$ upper bound presented in (Manshadi, Oveis Gharan and Saberi, SODA 2011). Our construction is simple to state, accompanied by a fully analytic proof, and yields a competitive ratio bound intriguingly similar to $1 - \frac1e$, the optimal competitive ratio for the fully adversarial Online Bipartite Matching problem. Although the tightness of our upper bound remains an open question, we show that our construction is extremal in a natural class of instances.

A New Impossibility Result for Online Bipartite Matching Problems

TL;DR

The authors address online bipartite matching with random arrivals by constructing a simple irregular cuckoo hashing instance with users and ads, where for . They prove that no online algorithm can achieve a competitive ratio exceeding , approximately , across several problem variants, while the constructed instance admits a quasi-complete offline maximum matching of size . The analysis combines a configuration-model coupling with a refined Karp–Sipser argument extended to unbounded degrees, and an extremality result showing the construction is worst-case within a natural family. They further generalize the approach to non-equibipartite graphs and demonstrate that the worst case occurs at unit user-to-ads ratio, , within the generalized class. The work raises open questions about tightness and whether the bound is achievable in broader settings, while providing a clean analytic bound with a near-unity gap to prior results and strong theoretical insight into the problem structure.

Abstract

Online Bipartite Matching with random user arrival is a fundamental problem in the online advertisement ecosystem. Over the last 30 years, many algorithms and impossibility results have been developed for this problem. In particular, the latest impossibility result was established by Manshadi, Oveis Gharan and Saberi in 2011. Since then, several algorithms have been published in an effort to narrow the gap between the upper and the lower bounds on the competitive ratio. In this paper we show that no algorithm can achieve a competitive ratio better than , improving upon the upper bound presented in (Manshadi, Oveis Gharan and Saberi, SODA 2011). Our construction is simple to state, accompanied by a fully analytic proof, and yields a competitive ratio bound intriguingly similar to , the optimal competitive ratio for the fully adversarial Online Bipartite Matching problem. Although the tightness of our upper bound remains an open question, we show that our construction is extremal in a natural class of instances.

Paper Structure

This paper contains 12 sections, 10 theorems, 71 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Overview
  3. Related Work
  4. Preliminaries
  5. Adaptation of the Analysis of Balister and Gerke to Unbounded Degree Distributions
  6. Impossibility Result
  7. Online Matching
  8. Maximum Matching
  9. Reduction to the Configuration Model
  10. Extremality
  11. Generalization to Non-Equibipartite Graphs
  12. Conclusion

Key Result

theorem 1

Let $\rho\in(0,1]$ be any constant. Let $f(x)=\sum_{i=0}^\infty z_i\cdot x^i$, and $\hat{f}(x) = \sum_{i=0}^\infty \hat{z}_i\cdot x^i$, where $(z_i)_{i=0}^\infty, (\hat{z}_i)_{i=0}^\infty$ are non-negative sequences such that $f'(1)=\hat{f}'(1)=\mu < \infty$ and $f'(2), \hat{f}'(2) < \infty$. Consid Moreover, with probability $1-o(1)$, after the first phase of Karp-Sipser, at least $(\hat{f}(w_2)

Figures (1)

  • Figure 1: The Competitive Ratios induced by the $D_u$ instances, obtained numerically for $u > 0$. The plot shows how $u = 1$ (which corresponds to the case of unbounded user-degree) gives the strongest upper bound. In Theorem \ref{['thm:optimal_u']}, we analytically establish the optimality of $u = 1$.

Theorems & Definitions (25)

  • definition 1: Irregular Cuckoo Hashing Model
  • definition 2: Configuration Model
  • theorem 1
  • lemma 1: bg15, Lemma 3.4
  • lemma 2: bg15, Lemma 3.5
  • lemma 3: Adaptation of bg15, Theorem 3.6
  • proof
  • lemma 4: bg15, proof of Theorem 3.7
  • lemma 5: Adaptation of bg15, Theorem 3.7
  • proof
  • ...and 15 more