Online Bipartite Matching with Advice: Tight Robustness-Consistency Tradeoffs for the Two-Stage Model

TL;DR

The tight tradeoff between Consistency and Robustness is characterized for four settings of two-stage matching: unweighted, vertex-weighted, edge-weighted, edge-weighted, and fractional budgeted allocation.

Abstract

Two-stage bipartite matching is a fundamental problem of optimization under uncertainty introduced by Feng, Niazadeh, and Saberi (2021), who study it under the stochastic and adversarial paradigms of uncertainty. We propose a method to interpolate between these paradigms, using the Algorithms with Predictions (ALPS) framework. To elaborate, given some form of information (e.g. a distributional prediction) about the uncertainty, we consider the optimal decision assuming that information is correct to be some "advice", whose accuracy is unknown. In the ALPS framework, we define Consistency to be an algorithm's performance relative to the advice, and Robustness to be an algorithm's performance relative to the hindsight-optimal decision. We characterize the tight tradeoff between Consistency and Robustness for four settings of two-stage matching: unweighted, vertex-weighted, edge-weighted, and fractional budgeted allocation. Additionally, we show our algorithm achieves state-of-the-art performance in both synthetic and real-data simulations.

Figures (14)

• Figure 1: Robustness-Consistency tradeoffs of various algorithms
• Figure 2: Comparison with the guarantees of Mahdian et al.
• Figure 3: Illustration of the hardness instance for unweighted matching. $S$ is on the left and $D$ is on the right. The first arrival neighbors both vertices of $S$. The second arrival neighbors exactly one vertex of $S$, but it could be either vertex. The advice is to match the green edge $(1,1)$.
• Figure 4: Plots of $f_{\mathsf{L}}$ and $f_{\mathsf{U}}$ for three values of $R$. Left: $R = 0$. Middle: $R = \frac{5}{9}$. Right: $R = \frac{3}{4}$. The consistencies achieved are $C=1, C=\frac{8}{9}, C=\frac{3}{4}$, respectively. In the right plot, the green dashed line indicates the penalty function used by feng2021two.
• Figure 5: Example to illustrate the features of the algorithm. $S$ is on the left and $D_1$ is on the right. The vertices in $S$ are $i=1,2,3,4$ and the vertices in $D_1$ are $j=1,2$ (labelling goes from top to bottom). The number next to $j \in S$ is its weight $w_j$. The green edges are suggested by the advice, and our algorithm's decisions are illustrated for the case where $R=5/9$.

Theorems & Definitions (38)

• Definition 1: Robustness and Consistency
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Lemma 1: Structural Lemma