On Conceptually Simple Algorithms for Variants of Online Bipartite Matching

Abstract

We present a series of results regarding conceptually simple algorithms for bipartite matching in various online and related models. We first consider a deterministic adversarial model. The best approximation ratio possible for a single-pass deterministic online algorithm is 1/2, which is achieved by any greedy algorithm. Dürr et al. [15] recently presented a 2-pass algorithm called Category-Advice that achieves approximation ratio 3/5. We extend their algorithm to multiple passes. We prove the exact approximation ratio for the k-pass Category-Advice algorithm for all $$k \ge 1$$ , and show that the approximation ratio converges to the inverse of the golden ratio $$2/(1+\sqrt{5}) \approx 0.618$$ as k goes to infinity. The convergence is extremely fast—the 5-pass Category-Advice algorithm is already within $$0.01\%$$ of the inverse of the golden ratio. We then consider two natural greedy algorithms—MinDegree and MinRanking. We analyze MinDegree in the online IID model and show that its approximation ratio is exactly $$1-1/e$$ . We analyze MinRanking in the priority model and show that this natural algorithm cannot obtain the approximation of the Ranking algorithm in the ROM model.