Abstract
We present a \((\frac{2}{3}-\epsilon)\)-approximation algorithm for the partial latin square extension (PLSE) problem. This improves the current best bound of \(1 - \frac{1}{e}\) 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 \(1-\frac{1}{e}\). 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 \(\frac{1}{4}\)-approximation algorithm in this case.
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Hajirasouliha, I., Jowhari, H., Kumar, R., Sundaram, R. (2007). On Completing Latin Squares. In: Thomas, W., Weil, P. (eds) STACS 2007. STACS 2007. Lecture Notes in Computer Science, vol 4393. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70918-3_45
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DOI: https://doi.org/10.1007/978-3-540-70918-3_45
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