On completing latin squares

Authors:
Iman Hajirasouliha;Hossein Jowhari;Ravi Kumar;Ravi Sundaram
Affiliations:
Simon Fraser University, Burnaby, BC, Canada;Simon Fraser University, Burnaby, BC, Canada;Yahoo! Research, Sunnyvale, CA;Northeastern University, Boston, MA
Venue:
STACS'07 Proceedings of the 24th annual conference on Theoretical aspects of computer science
Year:
2007

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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.