Online bipartite matching with imperfect advice

TL;DR

This work investigates online unweighted bipartite matching with $n$ offline and $n$ online vertices under adversarial and random-order arrival models. It proves that no learning-augmented method can be both $1$-consistent and strictly surpass the $1/2$-robust baseline in the adversarial model, contrasting with the $1-1/e$ benchmark for advice-free algorithms. In the Random Order setting, it introduces TestAndMatch, a meta-algorithm that achieves $1$-consistency and a robust ratio approaching the best advice-free bound, while gracefully degrading with advice quality via a sublinear distribution-testing phase that estimates the $L_1$ distance between realized and predicted type-count distributions. The approach leverages distribution-testing techniques (Poissonization, MinimaxTest) to quantify advice quality using only a small initial sample, enabling practical extensions like remapping and coarsening of advice and patching for imperfect advice. Overall, the paper provides a principled, testable framework for incorporating imperfect predictive information into online matching, with theoretical guarantees that interpolate between baseline performance and optimality depending on advice fidelity.

Abstract

We study the problem of online unweighted bipartite matching with $n$ offline vertices and $n$ online vertices where one wishes to be competitive against the optimal offline algorithm. While the classic RANKING algorithm of Karp et al. [1990] provably attains competitive ratio of $1-1/e > 1/2$, we show that no learning-augmented method can be both 1-consistent and strictly better than $1/2$-robust under the adversarial arrival model. Meanwhile, under the random arrival model, we show how one can utilize methods from distribution testing to design an algorithm that takes in external advice about the online vertices and provably achieves competitive ratio interpolating between any ratio attainable by advice-free methods and the optimal ratio of 1, depending on the advice quality.

Figures (7)

• Figure 1: Gadget for $n=2$. Red edges observed when $v_2$ arrives.
• Figure 2: Illustration of $\mathcal{G}_1$ and $\mathcal{G}_2$ for \ref{['thm:hardness']}
• Figure 3: For $n=4$, there may be $2^4 = 16$ possible types but at most $n = 4$ of them can ever be non-zero. Here, $c^*(\{ u_1, u_3 \}) = 1, c^*(\{ u_2, u_3 \}) = 1$ and $c^*(\{ u_1, u_2, u_4\}) = 2$. We see that type $\{u_1, u_2, u_4\}$ appears twice in $c^*$ and $|T^*|=3$.
• Figure 4: A (conservative) competitive ratio plot for $\frac{\hat{n}}{n} > \beta$. If MinimaxTest (\ref{['alg:minimaxtest']}) succeeds, we have $L_1(p^*, q) < 2 \left( \frac{\hat{n}}{n} - \beta \right) - 2 \varepsilon$ whenever $\hat{L}_1 < 2 \left( \frac{\hat{n}}{n} - \beta \right) - \varepsilon$. Observe that there is a smooth interpolation between the achieveable competitive ratio as $L_1(p^*, q)$ degrades whilst paying only $o(1)$ for robustness.
• Figure 5: Consider $\hat{\mathcal{G}}$ made by taking the union of two complete bipartite graphs ($\hat{\mathcal{G}}'$) and adding the red dashed edges. By connecting $v_i$ to $u_{(i + n/2) \mathrm{mod\ } n}$, $|\hat{T}| = r = n$. Meanwhile, if we coarsen $\hat{c}$ into $\hat{c}'$ by ignoring the red dashed edges, $\hat{\mathcal{G}}'$ still has a maximum matching of size $\hat{n}' = n$ while $|\hat{T}'| = r' = 2$, thus requiring significantly less samples to test since $s_{\hat{r}',\varepsilon,\delta} \ll s_{\hat{r},\varepsilon,\delta}$. Furthermore, if $\mathcal{G}^* = \hat{G}'$, then $L_1(c^*, \hat{c}) = 2n$ and we will reject the advice $\hat{c}$ if we do not coarsen it first.

Theorems & Definitions (18)

• Theorem 2.1: adapted from JHW18
• Lemma 2.2
• Theorem 3.1
• proof
• Theorem 4.1
• Lemma 4.2
• proof
• Lemma 4.3
• proof
• Lemma 4.4