A Constant Factor Approximation Algorithm for Reordering Buffer Management

TL;DR

This paper introduces the first constant factor approximation algorithm for the reordering buffer management problem, improving upon previous bounds and demonstrating an offline approach that outperforms online algorithms.

Contribution

It presents the first constant factor approximation for RBM using LP relaxation rounding, establishing a new theoretical bound and surpassing prior online algorithms.

Findings

First constant factor approximation for RBM

Improves previous $O(\sqrt{ ext{log }k})$ bound

Establishes a constant upper bound on LP relaxation integrality gap

Abstract

In the reordering buffer management problem (RBM) a sequence of $n$ colored items enters a buffer with limited capacity $k$. When the buffer is full, one item is removed to the output sequence, making room for the next input item. This step is repeated until the input sequence is exhausted and the buffer is empty. The objective is to find a sequence of removals that minimizes the total number of color changes in the output sequence. The problem formalizes numerous applications in computer and production systems, and is known to be NP-hard. We give the first constant factor approximation guarantee for RBM. Our algorithm is based on an intricate "rounding" of the solution to an LP relaxation for RBM, so it also establishes a constant upper bound on the integrality gap of this relaxation. Our results improve upon the best previous bound of $O(\sqrt{\log k})$ of Adamaszek et al. (STOC 2011) that used different methods and gave an online algorithm. Our constant factor approximation beats the super-constant lower bounds on the competitive ratio given by Adamaszek et al. This is the first demonstration of an offline algorithm for RBM that is provably better than any online algorithm.