如何为该匹配问题构造np约简? [英] how do I construct np-reduction for this matching problem?
问题描述
我有一个匹配的问题,我认为这很困难:
I have a matching problem, which I think is np-hard:
We need to arrange a dinner for a group of n people, some of the people are friends with eachother, some are not. We need to sit everyone at a table, but they should never sit with someone that they are not friends with. There are *k* tables K = r1+r2+···+rk=n.
**Input**
input is formatted as one first line k, then follow one line of k numbers, where each number represents a table and it's capacity. Then comes n lines of people, where we can see friendships of person i. All friendships are mutual.
**Output**
Output the formations of people that can be seated together, without having to sit with someone that they are not friends with
example:
Input:
2
3, 3
Alice: Bob, Claire, Eric, Juliet, Romeo
Bob: Alice, Claire, Juliet, Romeo
Claire: Alice, Bob, Juliet
Eric: Alice, Romeo
Juliet: Alice, Bob, Claire
Romeo: Alice, Bob, Eric
Output:
Romeo, Eric, Alice
Bob, Claire, Juliet
我相当确定这个问题是np完全的,但是我在寻找适当的减少量方面遇到了一些问题.该示例对应于以下(严重绘制)的图形:
Im fairly certain that this problem is np-complete, but I am having some problems finding a proper reduction. The example corresponds to the following (badly drawn)graph:
我对使用免费图表简化为独立集合有一个宽松的想法.但对于解决方案的任何想法,我将不胜感激
I have a loose idea around using a complimentary graph to reduce to independent set. but i would be very gratefull for any ideas for solutions
推荐答案
减少派系问题
首先,请注意NP是一类决策问题,因此我们将问题调整为是否有表格安排?"而不是输出表布置".在实践中,当然没有真正的区别.
Clique problem reduction
First off, note that NP is a class for decision problems, so we'll adjust the question to "is there a table arrangement?" instead of "output the table arrangement". In practice there is of course no real difference.
现在,给定一个图,假设我们想知道是否存在至少大小为 k
的集团.这是(决定) clique问题,它是著名的NP完全问题之一.
Now, given a graph, let's say we want to know if there is a clique of at least size k
. This is the (decision) clique problem, which is one of the famous NP-complete problems.
当且仅当您的匹配问题对同一个图具有大小为 k
的表有解决方案时,此图才会具有至少一个大小为 k
的集团.所有其他座位的座位应该不受限制,因此我们有 n-k
个单人桌.
This graph will have at least one clique of size k
if and only if your matching problem has a solution for the same graph, with a table of size k
. The seating for all the others should be unconstrained, so we have n-k
one-seat tables.
因此,我们可以创建一个匹配问题的实例,该实例等效于一个已知NP完全问题的实例.此实例的大小大致相同(没有指数爆炸),因此这意味着减少,证明了匹配是NP难的.因为它在NP中也(很明显?),它也是NP完整的.
Thus, we can create an instance of the matching problem that is equivalent to any instance of a known NP-complete problem. This instance is roughly the same size (no exponential blow-up), so this constitutes a reduction, proving that the matching is NP-hard. As it is also (clearly?) in NP, it is also NP-complete.
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