Equivalence of L0 and L1 Minimizations in Sudoku Problem

TL;DR

This paper investigates the conditions under which L0 and L1 minimizations are equivalent in solving Sudoku puzzles formulated as sparse linear systems, revealing differences based on puzzle types and specific cell configurations.

Contribution

It provides an analysis of the equivalence between L0 and L1 minimizations in Sudoku puzzles, identifying puzzle types and cell conditions affecting this equivalence.

Findings

L1 minimization yields unique solutions for type-I puzzles.

L1 minimization solutions are non-unique for type-II puzzles.

Certain cells influence the equivalence of L0 and L1 minimizations.

Abstract

Sudoku puzzles can be formulated and solved as a sparse linear system of equations. This problem is a very useful example for the Compressive Sensing (CS) theoretical study. In this study, the equivalence of Sudoku puzzles L0 and L1 minimizations is analyzed. In particular, 17-clue (smallest number of clues) uniquely completable puzzles with sparse optimization algorithms are studied and divided into two types, namely, type-I and -II puzzles. The solution of L1 minimization for the type-I puzzles is unique, and the sparse optimization algorithms can solve all of them exactly. By contrast, the solution of L1 minimization is not unique for the type-II puzzles, and the results of algorithms are incorrect for all these puzzles. Each empty cell for all type-II puzzles is examined. Results show that some cells can change the equivalence of L0 and L1 minimizations. These results may be helpful for the study of equivalence of L0 and L1 norm minimization in CS.