Optimal Rounding for Two-Stage Bipartite Matching
Tristan Pollner, Amin Saberi, Anders Wikum
TL;DR
This work studies two-stage bipartite matching where edges are revealed in two batches, deriving polynomial-time algorithms that approximate the optimum online policy within the integrality gaps of natural LP relaxations. The Round-Augment framework rounds a two-stage fractional solution by dependent rounding of the first stage and a second-stage max-weight augmentation, leveraging negative association of offline node availabilities to bound the second-stage gain. It achieves a tight $7/8$-approximation for vertex-weighted graphs and a $2\\sqrt{2}-2\\approx 0.828$-approximation for edge-weighted graphs under arbitrary second-stage distributions, matching the relaxations' integrality gaps. The paper also provides sample-based bounds, showing that polynomially many samples from the second-stage distribution suffice to preserve these guarantees, thereby enabling practical deployment when the distribution is unknown or large.
Abstract
We study two-stage bipartite matching, in which the edges of a bipartite graph on vertices $(B_1 \cup B_2, I)$ are revealed in two batches. In stage one, a matching must be selected from among revealed edges $E \subseteq B_1 \times I$. In stage two, edges $E^θ\subseteq B_2 \times I$ are sampled from a known distribution, and a second matching must be selected between $B_2$ and unmatched vertices in $I$. The objective is to maximize the total weight of the combined matching. We design polynomial-time approximations to the optimum online algorithm, achieving guarantees of $7/8$ for vertex-weighted graphs and $2\sqrt{2}-2 \approx 0.828$ for edge-weighted graphs under arbitrary distributions. Both approximation ratios match known upper bounds on the integrality gap of the natural fractional relaxation, improving upon the best-known approximation of 0.767 by Feng, Niazadeh, and Saberi for unweighted graphs whose second batch consists of independently arriving nodes. Our results are obtained via an algorithm that rounds a fractional matching revealed in two stages, aiming to match offline nodes (respectively, edges) with probability proportional to their fractional weights, up to a constant-factor loss. We leverage negative association (NA) among offline node availabilities -- a property induced by dependent rounding -- to derive new lower bounds on the expected size of the maximum weight matching in random graphs where one side is realized via NA binary random variables. Moreover, we extend these results to settings where we have only sample access to the distribution. In particular, $\text{poly}(n,ε^{-1})$ samples suffice to obtain an additive loss of $ε$ in the approximation ratio for the vertex-weighted problem; a similar bound holds for the edge-weighted problem with an additional (unavoidable) dependence on the scale of edge weights.