On Completing Latin Squares

Abstract

We present a (2/3-Ɛ)-approximation algorithm for the partial latin square extension (PLSE) problem. This improves the current best bound of 1- 1/e due to Gomes, Regis, and Shmoys [5]. We also show that PLSE is APX-hard. We then consider two new and natural variants of PLSE. In the first, there is an added restriction that at most k colors are to be used in the extension; for this problem, we prove a tight approximation threshold of 1 - 1/e. In the second, the goal is to find the largest partial Latin square embedded in the given partial Latin square that can be extended to completion; we obtain a 1/4-approximation algorithm in this case.