Question

For each of the two questions below, give one of the following answers: "Yes", "No", or "Unknown, because it would resolve the question of whether P = NP". Give a brief justification of your answer. Suppose we define the decision version of the Shortest Path Problem as follows: Given a weighted graph $G = (V, E)$, two nodes $s, t \in V$, and a bound $k$, does there exist a path from $s$ to $t$ in $G$ of weight at most $k$? (a) Question: Is it the case that Vertex Cover $\leq_p$ Shortest Path? (b) Question: Is it the case that Shortest Path $\leq_p$ Independent Set?

          For each of the two questions below, give one of the following answers: "Yes", "No",
or "Unknown, because it would resolve the question of whether P = NP". Give a brief
justification of your answer.
Suppose we define the decision version of the Shortest Path Problem as follows: Given
a weighted graph $G = (V, E)$, two nodes $s, t \in V$, and a bound $k$, does there exist a
path from $s$ to $t$ in $G$ of weight at most $k$?
(a) Question: Is it the case that Vertex Cover $\leq_p$ Shortest Path?
(b) Question: Is it the case that Shortest Path $\leq_p$ Independent Set?

For each of the two questions below, give one of the following answers: "Yes", "No",
or "Unknown, because it would resolve the question of whether P = NP". Give a brief
justification of your answer.
Suppose we define the decision version of the Shortest Path Problem as follows: Given
a weighted graph G = (V, E), two nodes s, t ∈ V, and a bound k, does there exist a
path from s to t in G of weight at most k?
(a) Question: Is it the case that Vertex Cover Shortest Path?
(b) Question: Is it the case that Shortest Path Independent Set?

Added by April A.

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