Question

We discussed several strategies to approximate some hard variants of SAT. Which of the following statements are true? Check ALL that apply

A) There is a (1-e¹) polynomial time approximtion algorithm for MAX SAT
B) There is a 7/8 polynomial time approximation algorithm for MAX-EXACT-3SAT
C) If there is a polynomil time k-approximation of ILP, then there is a polynomial time k-approximation of MAX SAT
D) There is a 3/4 polynomial time approximation of ILP

Answer 1

The correct answers are options B and C. We examined several statements regarding approximation algorithms for MAX SAT, MAX-EXACT-3SAT, and ILP. Statement B, relating to a 7/8 polynomial time approximation algorithm for MAX-EXACT-3SAT, is true, while statements A and D are false. Statement C is too general to be verified without additional context.

Step-by-step explanation:

The student's question pertains to approximation algorithms for various computational problems like Maximum Satisfiability (MAX SAT), MAX-EXACT-3SAT, and Integer Linear Programming (ILP). Let's examine each statement independently:

• A) There is a (1-e)¹ polynomial time approximation algorithm for MAX SAT: This statement is false. There is a well-known algorithm by Goemans and Williamson that achieves an approximation ratio for MAX SAT, but this is not phrased correctly, since e typically denotes the mathematical constant approximately equal to 2.718, and this statement as written does not make sense in that context.
• B) There is a 7/8 polynomial time approximation algorithm for MAX-EXACT-3SAT: This statement is true. This ratio comes from the result of Karloff and Zwick where they showed an algorithm that achieves this approximation ratio for the MAX-EXACT-3SAT problem, where every clause has exactly three literals.
• C) If there is a polynomial time k-approximation of ILP, then there is a polynomial time k-approximation of MAX SAT: This statement is suggestive, as it implies a relationship between the approximability of ILP and MAX SAT. In theory, since MAX SAT is a special case of ILP, an approximation algorithm for ILP could be used for MAX SAT, but without additional context or specificity, this statement is too general to verify as true or false.
• D) There is a 3/4 polynomial time approximation of ILP: This statement is false as of the current knowledge in the field. There is no known polynomial-time approximation algorithm for ILP with a constant factor guarantee like 3/4 due to the general hardness of ILP.

Approximation algorithms play a vital role in dealing with NP-hard problems like MAX SAT and ILP, where finding the exact solution is computationally infeasible for large instances. They provide solutions that are close to the optimal with a guarantee on how far from the optimal they are, thus enabling practical approaches to complex optimization problems.