Max-Min Greedy Matching Problem: Hardness for the Adversary and Fractional Variant

T.-H. Hubert Chan, Zhihao Gavin Tang, Quan Xue

Abstract: Eden, Feige, and Feldman considered the max-min greedy matching problem can be viewed as a game between an algorithm and an adversary. A bipartite graph between items and players is given to both parties upfront. The algorithm first chooses a priority order on the items, and then depending on the algorithm’s choice, the adversary chooses a priority order on the players. Then, the two priority orders are used in a greedy process to produce a matching between the items and the players; specifically, when it is a player’s turn, the highest priority item among its still available neighbors will be matched. The goal of the algorithm is to maximize the size of the resulting matching, while the goal of the adversary is to minimize its size. The previous work shows that the algorithm has a polynomial-time strategy to ensure a competitive ratio of strictly greater than $$\frac{1}{2}$$ . In this work, we show that from the adversary’s perspective, the adversarial order minimum matching problem is NP-hard to approximate with a ratio better than $$\frac{6}{5}$$ , assuming the small set expansion (SSE) hypothesis. On the other hand, we propose a fractional variant of the problem and examine the interplay between the algorithm and the adversary when one or both parties may use fractional permutations. An interesting result is that if the algorithm uses only integral item permutations, then an optimal response for the adversary can also be an integral player permutation. Moreover, we also show that in a fractional variant, the algorithm can use a round-robin strategy to achieve a competitive ratio of at least $$1-\frac{1}{e}$$ for input graphs with large enough granularity parameter m. Furthermore, we show that the analysis for the round-robin strategy is tight even for regular graphs.