Online Stochastic Matching: Beating 1-1/e (0905.4100v1)

Abstract: We study the online stochastic bipartite matching problem, in a form motivated by display ad allocation on the Internet. In the online, but adversarial case, the celebrated result of Karp, Vazirani and Vazirani gives an approximation ratio of $1-1/e$. In the online, stochastic case when nodes are drawn repeatedly from a known distribution, the greedy algorithm matches this approximation ratio, but still, no algorithm is known that beats the $1 - 1/e$ bound. Our main result is a 0.67-approximation online algorithm for stochastic bipartite matching, breaking this $1 - {1/e}$ barrier. Furthermore, we show that no online algorithm can produce a $1-\epsilon$ approximation for an arbitrarily small $\epsilon$ for this problem. We employ a novel application of the idea of the power of two choices from load balancing: we compute two disjoint solutions to the expected instance, and use both of them in the online algorithm in a prescribed preference order. To identify these two disjoint solutions, we solve a max flow problem in a boosted flow graph, and then carefully decompose this maximum flow to two edge-disjoint (near-)matchings. These two offline solutions are used to characterize an upper bound for the optimum in any scenario. This is done by identifying a cut whose value we can bound under the arrival distribution.

• The paper’s main contribution is a novel online algorithm that achieves a 0.67 approximation factor by leveraging an offline optimal solution for stochastic bipartite matching.
• It employs a two-phase strategy by decomposing a max-flow solution into two edge-disjoint matchings using the power of two choices for efficient online allocation.
• The algorithm outperforms the classical 1-1/e bound, offering practical improvements in ad allocation systems by maximizing matched impressions and revenue.

Overview of the Paper on Online Stochastic Matching

The paper investigates the problem of online stochastic bipartite matching, a field relevant for display ad allocation on the internet. The authors aim to improve the approximation ratio for stochastic bipartite matching beyond the known limit set by adversarial models. The seminal work by Karp, Vazirani, and Vazirani achieved a competitive ratio of $1 - 1/e \approx 0.632$ for adversarial online matching, a bound often encountered across various online algorithms and shown to be tight. The contribution of this paper is a novel approximation algorithm that achieves a $0.67$ approximation factor, surpassing the $1 - 1/e$ threshold previously observed under stochastic conditions.

Approach and Algorithms

The authors base their approach on leveraging an offline optimal solution derived from the expected instance of the graph to guide online allocation. The novelty lies in utilizing two disjoint sets of solutions computed offline for guiding the online decision-making process. The expected instance's optimal solution is utilized to form a max-flow problem in a modified flow graph. This flow is then carefully decomposed into two edge-disjoint matchings employed during the online phase. The use of the power of two choices, adapted from concepts in load balancing, is crucial to this methodology.

Key Results and Claims

The paper’s primary result is the introduction of an algorithm achieving an approximation factor of $\frac{1 - e^{-2}}{4 - 2/e} \approx 0.67$, which notably beats the prior performance bound of $1 - 1/e$ in a stochastic context. Additionally, the authors rigorously prove that no online algorithm can consistently achieve a $1 - o(1)$ approximation for arbitrarily small $\epsilon$, reinforcing the strengths of their proposed algorithm. The algorithm’s robustness is enhanced by confirming it avoids over-focusing on input scenarios susceptible to adversarial weaknesses.

Theoretical and Practical Implications

The work advances the theoretical boundary of online bipartite matching by proving it is possible to exceed the $1 - 1/e$ barrier in stochastic models. Practically, this has substantial implications for applications in ad allocation where ads need to be matched to impressions efficiently. By outperforming traditional limits, the approach could significantly enhance ad allocation systems, maximizing the number of matched impressions and, subsequently, potential revenues.

Future Prospects and Challenges

The advancement prompts further exploration into multi-choice stochastic algorithms and their broader applications in combinatorial optimization problems with stochastic input models. One open question involves extending the two-matching scheme to generalize $k$-matchings for potentially better approximation ratios. Additionally, solving the problem in cases with non-integer impression arrival rates remains a theoretical challenge that would further affirm the robustness and applicability of these techniques.

In conclusion, this paper enriches the field of online matching by challenging existing theoretical assumptions within stochastic frameworks, enabling more effective algorithms for real-world applications like online ad delivery. The elegance of merging offline computation with online adaptability sets a precedent for approaching complex, uncertain environments in computational theory.