  Recent research has proposed neural architectures for solving combinatorial
problems in structured output spaces. In many such problems, there may exist
multiple solutions for a given input, e.g. a partially filled Sudoku puzzle may
have many completions satisfying all constraints. Further, we are often
interested in finding any one of the possible solutions, without any preference
between them. Existing approaches completely ignore this solution multiplicity.
In this paper, we argue that being oblivious to the presence of multiple
solutions can severely hamper their training ability. Our contribution is two
fold. First, we formally define the task of learning one-of-many solutions for
combinatorial problems in structured output spaces, which is applicable for
solving several problems of interest such as N-Queens, and Sudoku. Second, we
present a generic learning framework that adapts an existing prediction network
for a combinatorial problem to handle solution multiplicity. Our framework uses
a selection module, whose goal is to dynamically determine, for every input,
the solution that is most effective for training the network parameters in any
given learning iteration. We propose an RL based approach to jointly train the
selection module with the prediction network. Experiments on three different
domains, and using two different prediction networks, demonstrate that our
framework significantly improves the accuracy in our setting, obtaining up to
21 pt gain over the baselines.
