差异与平等的软约束
Soft Constraints of Difference and Equality
摘要：
在许多组合问题中，可能需要对解决方案中分配的多样性或相似性进行建模。例如，人们可能希望最大化或最小化解决方案中不同值的数量。为了公式化这种类型的问题，我们可以使用众所周知的AllDifferent和AllEqual约束的软变体。我们提出了一个分类的六个软全球约束，通过结合后两个和两个标准的成本函数，这是最大化或最小化。我们证明了在这些约束条件下实现弧和边界一致性的复杂性，解决了NP-硬度既没有被证明也没有被证伪的情况。特别是，我们深入探讨的约束，确保至少k对变量有一个共同的值。我们发现，实现弧一致性是NP难的，但实现边界的一致性可以通过动态规划在多项式时间内完成。此外，我们表明，最大数量的对相等的变量可以近似为一个因素1/2的线性时间贪心算法。最后，我们提供了一个固定的参数易于处理的算法相对于出现在两个以上的不同域的值的数量。有趣的是，这种分类表明，强制平等比强制差异更难。
英文摘要：
In many combinatorial problems one may need to model the diversity or similarity of assignments in a solution. For example, one may wish to maximise or minimise the number of distinct values in a solution. To formulate problems of this type, we can use soft variants of the well known AllDifferent and AllEqual constraints. We present a taxonomy of six soft global constraints, generated by combining the two latter ones and the two standard cost functions, which are either maximised or minimised. We characterise the complexity of achieving arc and bounds consistency on these constraints, resolving those cases for which NP-hardness was neither proven nor disproven. In particular, we explore in depth the constraint ensuring that at least k pairs of variables have a common value. We show that achieving arc consistency is NP-hard, however achieving bounds consistency can be done in polynomial time through dynamic programming. Moreover, we show that the maximum number of pairs of equal variables can be approximated by a factor 1/2 with a linear time greedy algorithm. Finally, we provide a fixed parameter tractable algorithm with respect to the number of values appearing in more than two distinct domains. Interestingly, this taxonomy shows that enforcing equality is harder than enforcing difference.
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