Abstract
We consider a two-sided market under incomplete preference lists with ties, where the goal is to find a maximum size stable matching. The problem is APX-hard, and a 3/2-approximation was given by McDermid [1]. This algorithm has a non-linear running time, and, more importantly needs global knowledge of all preference lists. We present a very natural, economically reasonable, local, linear time algorithm with the same ratio, using some ideas of Paluch [2]. In this algorithm every person make decisions using only their own list, and some information asked from members of these lists (as in the case of the famous algorithm of Gale and Shapley). Some consequences to the Hospitals/Residents problem are also discussed.
Highlights
In 1962, Gale and Shapley [3] gave their famous simple deferred acceptance algorithm, which always finds stable matching in a two-sided market
If incomplete lists and ties are allowed in the preference lists, their algorithm still works, and gives some stable matching
McDermid [1] gave the first 3/2-approximation using our previous 3/2-approximation for the case, when ties are only on one side of the market [4]
Summary
In 1962, Gale and Shapley [3] gave their famous simple deferred acceptance algorithm, which always finds stable matching in a two-sided market. We proposed a simple linear time 5/3-approximating algorithm (called GSA2) for the general case of this problem, first presented at the first MATCH-UP workshop (see in [4,14,15]), and at the same time, an even simpler 3/2-approximation (called GSA1) was given for the special case where ties are allowed on one side only For this algorithm, the proof was very short, see Section 2. It was (partially) answered by McDermid [1], who gave the first 3/2-approximation for the general case He used GSA1 (and not GSA2), but not with simple repetitions, at some points he stopped the main algorithm, constructed an auxiliary graph, and solved a maximum matching problem on it.
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