GATE CS 2015 Set 3 — Question 41
Go beyond PYQs with Success TrackerAI-powered personalised practice and doubt support. Unlimited practice on eligible plans; AI usage limits apply.MCQ+2 / -0.67MediumPolynomial ReductionsComplexity TheoryAlgorithms
Algorithms → Complexity Theory → Polynomial Reductions
Last updated
Question
Language is polynomial time reducible to language . Language is polynomial time reducible to , which in turn is polynomial time reducible to language . Which of the following is/are true?I. if , then
II. if or , then
III. , if and only if
IV. if , then and
II. if or , then
III. , if and only if
IV. if , then and
Correct answer
(C) I and IV only
Solution
Given:
Analysis of statements:
I. If , then . Since , if the harder problem () is solvable in polynomial time, the easier problem () is also solvable in polynomial time. True.II. If or , then . This is incorrect. means is at least as hard as . Knowing is easy () tells us nothing about the difficulty of . False.III. if and only if . There is no direct relationship between and other than they both reduce to . One could be in and the other not (assuming allows it). False.IV. If , then and . Since and , if , then both and must be in . True.Statements I and IV are true.
1.
2.
3.
By transitivity of polynomial time reduction:Analysis of statements:
I. If , then . Since , if the harder problem () is solvable in polynomial time, the easier problem () is also solvable in polynomial time. True.II. If or , then . This is incorrect. means is at least as hard as . Knowing is easy () tells us nothing about the difficulty of . False.III. if and only if . There is no direct relationship between and other than they both reduce to . One could be in and the other not (assuming allows it). False.IV. If , then and . Since and , if , then both and must be in . True.Statements I and IV are true.
Continue learning with Success Tracker
A step still unclear? Work through it with support
Use Success Tracker to ask about the reasoning, then try another GATE CS question to check your understanding.
AI-powered practice· Unlimited practice on eligible plans
- PYQs with solutions
- Attempt available previous-year questions, then compare your reasoning with the worked solution. Coverage varies by stream.
- Practice that adapts
- Choose a topic, work on weaker areas and bookmark questions to revisit. Your attempts feed your progress tracking.
- AI doubt support
- Ask follow-up questions about a step or concept while practising, instead of stopping at the final answer.
Unlimited practice is available on eligible plans. Free practice and AI usage have limits; check the current plan allowances before choosing.
This page stays readable without an account. AI responses can be wrong; check them against the solution and source material.
More questions on Complexity Theory
2026 Set 1 Q17Consider the following recurrence relations: For all ,…2026 Set 2 Q24Consider the following functions, where is a positive integer.…2026 Set 2 Q25Which of the following can be recurrence relation(s) corresponding to an algorithm with time…2026 Set 2 Q32Consider an array . Suppose the merge sort algorithm is…2026 Set 2 Q37Let be a weighted directed acyclic graph with edges and vertices. Given and a…