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Given a weighted directed acyclic graph G = (V, D, W) and a set of arches RE & # 39; of rewhere the weights of W They are at the vertices. The problem is partitioning Sun on a minimum number of paths separated from vertices that cover all vertices of Sun subject to restrictions that:

1. the weight of each route is maximum k.
2. each route must include at least one edge of D & # 39;

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## Longer trajectory in an acyclic graph directed with restrictions

Given a directed weighted acyclic graph G = (V, D, W) and a subset of edges RE & # 39; of re. The problem is finding the longest path in Sun that goes through exactly one edge of RE & # 39;.

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I have a directed acyclic graph that has uniform edge weights. I would like to extract from this graph an expansion tree (an arborescence) with the property that the total distance between all pairs of leaf nodes (the sum of the combined distances of the closest common ancestor of each pair) is minimized.

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## Wi-Fi: Do modern operating systems still send requests for targeted or directed Wi-Fi probes that contain SSID?

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Other sources, such as this or this one, seem to confirm that specific polling requests are no longer sent in the latest Android versions, at least.

If this is true, and perhaps also for iOS (and some desktop OS), are there any press releases, bug tracker entries, security reports or code confirmations that confirm it?

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## np complete – HAMCYCLE NOT DIRECTED to HAMPATH reduction

I will define the problems

UHAMPATH

Input: A non-directed graph G and 2 nodes, syt

Question: Is there a Hamiltonian path from s to t in G?

UHAMCYCLE

Input: a non-directed graph G

Question: Is there a Hamiltonian cycle in G?

$$UHAMCYCLE leq_p UHAMPATH$$

my reduction is as follows $$(G) a (G & # 39 ;, s, t)$$

``````function(G)

for each e = (u, v) in E(G)
G' = G
add nodes u' and v' to G'
add edges (u', u) and (v, v')
s = u'
t = v'
if there is a hamilton path from s to t:
return (G', s, t)
``````

Basically, if there is a hamilton cycle in G, then some edge in $$G$$ will form a Hamilton road in $$G & # 39;$$.

An example to further illustrate the reduction:

`````` Graph G

(a) ---- --(d)
|         |
|         |
|         |
|         |
|         |
|         |
|         |      |
(b) ------ (c)

Has a clear Hamilton cycle (a, b, c, d).

If we choose edges (a, d)

Graph G'

(s)--(a) ---- --(d)
|         |
|         |
|         |
|         |
|         |
|         |
|         |
(b) ------ (c)--(t)
``````

It doesn't have a hamilton path from s to t. However, if you choose any other advantage such as (a, d), then

``````(s)--(a) ---- --(d)--(t)
|         |
|         |
|         |
|         |
|         |
|         |
|         |
(b) ------ (c)
``````

$$(s a a to b to c to d to t)$$. Wait.

I am confused if I can use this line or not:

``````if there is a hamilton path from s to t:
return (G', s, t)
``````

Checking if a graph has a hamilton path is NP complete, but as we are trying to reduce it, I think we can.