Multiple Constraint Satisfaction by Belief Propagation: An Example Using Sudoku

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Multiple Constraint Satisfaction by Belief Propagation: An Example Using Sudoku Todd K. Moon and Jacob H. Gunther Utah State University Abstract? The popular Sudoku puzzle bears structural resemblance to the problem of decoding linear error correction codes: solution is over a discrete set, and several constraints apply. We express the constraint satisfaction using a Tanner graph. The belief propagation algorithm is applied to this graph. Unlike conventional computer-based solvers, which rely on humanly specied tricks for solution, belief propagation is generally applicable, and requires no human insight to solve a problem. The presence of short cycles in the graph creates biases so that not every puzzle is solved by this method. However, all puzzles are at least partly solved by this method. The Sudoku application thus demonstrates the potential effectiveness of BP algorithms on a general class of constraint satisfaction problems. I. INTRODUCTION The belief propagation (BP) paradigm (also known as message passing (MP)) realizes Bayesian inference on graphs without cycles [1], [2], [3], and performs nearly Bayesian belief propagation on graphs with cy- cles [4], [5]. BP algorithms can be used to describe a variety of algorithms, including fast Hadamard trans- forms, the Kalman filter, fast Fourier transforms, MAP decoding algorithms (including decoding algorithms for turbo codes), and the Viterbi algorithm.

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Multiple Constraint Satisfaction by Belief Propagation: An Example Using Sudoku

Todd K. Moon and Jacob H. Gunther Utah State University

Abstractpopular Sudoku puzzle bears structural— The resemblance to the problem of decoding linear error correction codes: solution is over a discrete set, and several constraints apply. We express the constraint satisfaction using a Tanner graph. The belief propagation algorithm is appliedtothisgraph.Unlikeconventionalcomputer-based solvers, which rely on humanly speciﬁed tricks for solution, belief propagation is generally applicable, and requires no human insight to solve a problem. The presence of short cycles in the graph creates biases so that not every puzzle is solved by this method. However, all puzzles are at least partly solved by this method. The Sudoku application thus demonstrates the potential effectiveness of BP algorithms on a general class of constraint satisfaction problems.

I. INTRODUCTION The belief propagation (BP) paradigm (also known as message passing (MP)) realizes Bayesian inference on graphs without cycles [1], [2], [3], and performs nearly Bayesian belief propagation on graphs with cy-cles [4], [5]. BP algorithms can be used to describe a variety of algorithms, including fast Hadamard trans-forms, the Kalman ﬁlter, fast Fourier transforms, MAP decoding algorithms (including decoding algorithms for turbo codes), and the Viterbi algorithm. Most relevant to the current consideration, BP algorithms are also the means by which low density parity check (LDPC) codes are decoded [6]. In LDPC decoding, information about received bits that is implied collectively by the set of parity constraints is combined together in a (nearly) Bayesian way with information from the received data to provide information about the bits that were originally transmitted. In this paper, we demonstrate how the BP paradigm — borrowed closely from LDPC decoding — can be applied to a problem with multiple local combinatorial constraints, namely, the popular Sudoku puzzle. While there are other methods of solving Sudoku puzzles, the BP method is very general and does not require any human insight or tricks, nor does it require building solution trees. It is thus potentially applicable to a broad variety of problems as a general tool. Easy Sudoku puzzles can be solved by simple elimina-tion.DifﬁcultSudokupuzzlesareactuallyNP-complete [7].Thesepuzzlesarethuswell-scopedexamplesworthy ofstudyingthegeneralclassofNP-completeproblems. 1-4244-0166-6/06/$20.00(c)2006IEEE

The error correction decoding problem is also NP-complete[8],butveryeffectivesub-optimaldecoding algorithms exist. Even for codes with not particular structure, message passing algorithms can be somewhat effective, but not perfect [9]. In this paper, we present progress toward a solution, but do not provide a solution that works in all cases.

Interestingly, the algorithm works from general rules of inference, not from particular special cases and tricks, which are probably employed in the Sudoku puzzle posers and solvers which are widespread. It is thus applicable to a wide variety of constraint satisfaction problems.

In applying BP methods, the problem is mapped to a graph and messages representing Bayes probabilities are passed among the nodes of the graph. For the Sudoku puzzle, evidence about the contents of cells of the Sudoku puzzle is learned in a soft (probabilistic) way from the degree to which the cell contents satisfy the constraints of the puzzle. This application thus demon-strates application of BP to problems with multiple local constraints.

BP is Bayesian optimal for graphs without cycles, but suffers from biases for graphs with cycles. The graph associated with Sudoku, like the graphs for LDPC codes, does have cycles. Every node in the Sudoku graph lies on two cycles of length four. The biases introduced by these short cycles cause failure of the BP method for more difﬁcult puzzles.

Unlike the LDPC case, in which the combinatorial complexity of the marginalizations can be efﬁciently treated using a lattice structure, the marginalization as-sociated with Sudoku still retains combinatorial com-plexity. However, it is localized only to individual con-straints, so that a search over the global set of constraints is avoided.

Despite these shortcomings, the application reveals the possibility of applying BP techniques to multiconstraint problems, at least to eliminate many of the possibilities, perhaps leaving the problem sufﬁciently small that a global search may be possible.

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