Final answer:
The correct answer is A. If X can be solved in polynomial time, then so can Y.
Step-by-step explanation:
The correct answer is A. If X can be solved in polynomial time, then so can Y.
A problem X reducing to a problem Y in polynomial time means that any instance of problem X can be transformed into an instance of problem Y in polynomial time. Since Y is NP-complete, all other NP-complete problems can be reduced to Y as well. If X can be solved in polynomial time, it means that any instance of problem X can be solved in polynomial time. Since Y is also reducible to X in polynomial time, it implies that Y can also be solved in polynomial time. Therefore, option A is true.
The relationship between problems X and Y given that Y is NP-complete and X reduces to Y in polynomial time would indicate that if X is solvable in polynomial time (X is in P), then Y would also be solvable in polynomial time. However, the correct answer here is B. X is NP-complete. This inference is drawn because since Y is NP-complete and X reduces to Y in polynomial time, it means that X is at least as hard as Y.
Therefore, if Y is NP-complete, then X must also be NP-complete given that there is a polynomial-time reduction from X to Y. This satisfies the condition for a problem to be NP-complete, which requires that all problems in NP can be reduced to it in polynomial time. Since X is reducible to Y and Y is in NP-complete, X meets this condition.