Hall's Condition for Partial Latin Squares

TL;DR

This paper explores the conditions under which partial Latin squares can be completed, extending known theorems and analyzing computational complexity for specific cases.

Contribution

It extends Ryser's Theorem to a broader class of partial Latin squares and proves NP-hardness for deciding completability under Hall's Condition.

Findings

Hall's Condition is necessary and sufficient for certain partial Latin squares.

Deciding completability with Hall's Condition is NP-hard.

Extension of Ryser's Theorem to new cases.

Abstract

Hall's Condition is a necessary condition for a partial latin square to be completable. Hilton and Johnson showed that for a partial latin square whose filled cells form a rectangle, Hall's Condition is equivalent to Ryser's Condition, which is a necessary and sufficient condition for completability. We give what could be regarded as an extension of Ryser's Theorem, by showing that for a partial latin square whose filled cells form a rectangle, where there is at most one empty cell in each column of the rectangle, Hall's Condition is a necessary and sufficient condition for completability. It is well-known that the problem of deciding whether a partial latin square is completable is NP-complete. We show that the problem of deciding whether a partial latin square that is promised to satisfy Hall's Condition is completable is NP-hard.