Recently many neural models have been proposed to solve combinatorial puzzles
by implicitly learning underlying constraints using their solved instances,
such as sudoku or graph coloring (GCP). One drawback of the proposed
architectures, which are often based on Graph Neural Networks (GNN), is that
they cannot generalize across the size of the output space from which variables
are assigned a value, for example, set of colors in a GCP, or board-size in
sudoku. We call the output space for the variables as 'value-set'. While many
works have demonstrated generalization of GNNs across graph size, there has
been no study on how to design a GNN for achieving value-set invariance for
problems that come from the same domain. For example, learning to solve 16 x 16
sudoku after being trained on only 9 x 9 sudokus. In this work, we propose
novel methods to extend GNN based architectures to achieve value-set
invariance. Specifically, our model builds on recently proposed Recurrent
Relational Networks. Our first approach exploits the graph-size invariance of
GNNs by converting a multi-class node classification problem into a binary node
classification problem. Our second approach works directly with multiple
classes by adding multiple nodes corresponding to the values in the value-set,
and then connecting variable nodes to value nodes depending on the problem
initialization. Our experimental evaluation on three different combinatorial
problems demonstrates that both our models perform well on our novel problem,
compared to a generic neural reasoner. Between two of our models, we observe an
inherent trade-off: while the binarized model gives better performance when
trained on smaller value-sets, multi-valued model is much more memory
efficient, resulting in improved performance when trained on larger value-sets,
where binarized model fails to train.

本文介绍了一种基于递归关系网络 (RNN) 的方法来实现神经模型对于输出空间的不变性，进而解决在求解组合问题中出现的空间大小变化问题。我们提出的两种方法对于不同规模的问题都表现优异，并且存在一个权衡：二元模型在训练小范围的值集时表现更好，多元模型的内存效率更高，在训练大范围的值集时表现更好。

组合问题中输出空间不变性的神经模型

Neural Models for Output-Space Invariance in Combinatorial Problems

Subjective expected utility theory assumes that decision-makers possess
unlimited computational resources to reason about their choices; however,
virtually all decisions in everyday life are made under resource constraints -
i.e. decision-makers are bounded in their rationality. Here we experimentally
tested the predictions made by a formalization of bounded rationality based on
ideas from statistical mechanics and information-theory. We systematically
tested human subjects in their ability to solve combinatorial puzzles under
different time limitations. We found that our bounded-rational model accounts
well for the data. The decomposition of the fitted model parameter into the
subjects' expected utility function and resource parameter provide interesting
insight into the subjects' information capacity limits. Our results confirm
that humans gradually fall back on their learned prior choice patterns when
confronted with increasing resource limitations.

通过统计力学和信息论的思想，实验测试了资源限制下有限理性决策的正式化预测，并将该模型参数分解为受试者预期效用函数和资源参数，结果显示人类在面对资源限制时逐渐回退到他们学习的先前选择模式。

有限时间下的人类决策

Human Decision-Making under Limited Time

The author presents two tricks to accelerate depth-first search algorithms
for a class of combinatorial puzzle problems, such as tiling a tray by a fixed
set of polyominoes. The first trick is to implement each assumption of the
search with reversible local operations on doubly linked lists. By this trick,
every step of the search affects the data incrementally.
The second trick is to add a ghost square that represents the identity of
each polyomino. Thus puts the rule that each polyomino be used once on the same
footing as the rule that each square be covered once. The coding simplifies to
a more abstract form which is equivalent to 0-1 integer programming. More
significantly for the total computation time, the search can naturally switch
between placing a fixed polyomino or covering a fixed square at different
stages, according to a combined heuristic.
Finally the author reports excellent performance for his algorithm for some
familiar puzzles. These include tiling a hexagon by 19 hexiamonds and the N
queens problem for N up to 18.

介绍了用于加速一类组合拼图问题的深度优先搜索算法的两个技巧，一是在双向链表上实现可逆局部操作，二是添加一个代表多米诺骨牌标识的幽灵正方形，因此算法的性能得到优化。