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An Optimal Randomized Online Algorithm for Reordering Buffer Management In the reordering buffer management problem an input sequence of colored items arrives online, and has to be rescheduled in a permuted output sequence of the same items, with the help of a buffer that can hold $k$ items. The items enter the buffer in their order of arrival. When the buffer is full, one item must be removed and scheduled in the output sequence, making room for a new input item to enter the buffer. The objective is to minimize the total number of color changes between consecutive items in the output schedule.

We give an $O(\log\log k)$-competitive randomized online algorithm for reordering buffer management, where $k$ is the buffer size. Our bound matches the lower bound of Adamaszek et al. (STOC 2011). Our algorithm has two stages which are executed online in parallel. The first stage computes deterministically a feasible fractional solution to an LP relaxation for reordering buffer management. The second stage "rounds" using randomness the fractional solution. The first stage is based on the online primal-dual schema, combined with a dual fitting charging scheme. As primal-dual steps and dual fitting steps are interleaved and in some sense conflicting, combining them is challenging. We also note that we apply the primal-dual schema to a relaxation with mixed packing and covering constraints. The first stage produces a fractional LP solution with cost within a factor of $O(\log\log k)$ of the optimal LP cost. The second stage is an online algorithm that converts any LP solution to an integral solution, while increasing the cost by a constant factor. This stage generalizes recent results that gave a similar approximation guarantee using an offline rounding algorithm.

This is a joint work with Yuval Rabani.