
Recent Posts
 Tree Transducer Language Membership July 22, 2021
 ContextSensitive Language Membership May 26, 2021
 ETOL Language Membership May 12, 2021
 ContextFree Programmed Language Membership April 28, 2021
 ETOL Grammar NonEmptiness April 14, 2021
Recent Comments
 Sean T. McCulloch on Satisfiability
 Jonathan Buss on Satisfiability
 Sean T. McCulloch on Pokemon
 Tim Knittel on Pokemon
 Sean T. McCulloch on Exact Cover by 3Sets
Archives
 July 2021
 May 2021
 April 2021
 March 2021
 February 2021
 January 2021
 December 2020
 November 2020
 October 2020
 September 2020
 August 2020
 March 2020
 February 2020
 January 2020
 December 2019
 November 2019
 October 2019
 September 2019
 August 2019
 July 2019
 June 2019
 May 2019
 April 2019
 March 2019
 February 2019
 January 2019
 December 2018
 November 2018
 October 2018
 September 2018
 August 2018
 July 2018
 June 2018
 May 2018
 April 2018
 March 2018
 February 2018
 January 2018
 December 2017
 November 2017
 October 2017
 September 2017
 August 2017
 July 2017
 June 2017
 May 2017
 April 2017
 March 2017
 February 2017
 January 2017
 December 2016
 November 2016
 October 2016
 September 2016
 August 2016
 July 2016
 June 2016
 May 2016
 April 2016
 March 2016
 February 2016
 January 2016
 December 2015
 November 2015
 October 2015
 September 2015
 August 2015
 July 2015
 June 2015
 May 2015
 April 2015
 March 2015
 February 2015
 January 2015
 December 2014
 November 2014
 October 2014
 September 2014
 August 2014
 July 2014
 June 2014
Categories
 Algebra and Number Theory
 Appendix Algebra and Number Theory
 Appendix Automata and Language Theory
 Appendix Games and Puzzles
 Appendix Mathematical Programming
 Appendix Network Design
 Appendix Program Optimization
 Appendix Sets and Partitions
 AppendixGraph Theory
 AppendixLogic
 Appendix: Sequencing and Scheduling
 Appendix: Storage and Retrieval
 Chapter 3 Exercises
 Core Problems
 Overview
 Problems not in appendix
 Uncategorized
Meta
Tag Archives: NAE3SAT
Protected: OneInThree 3SAT
Enter your password to view comments.
Posted in AppendixLogic
Tagged 3sat, Difficulty 7, LO4, NAE3SAT, OneInThree 3SAT, uncited reduction
Protected: Bipartite Subgraph on Cubic Graphs with no triangles
Enter your password to view comments.
Posted in Problems not in appendix
Tagged Bipartite Subgraph, Bipartite Subgraph on Cubic Graphs with no triangles, Difficulty 9, NAE3SAT
Protected: Parttion Into Perfect Matchings
Enter your password to view comments.
Posted in AppendixGraph Theory
Tagged 3DM, Difficulty 4, Difficulty 9, GT16, NAE3SAT, uncited reduction
Protected: Monochromatic Triangle
Enter your password to view comments.
Posted in AppendixGraph Theory
Tagged 3sat, Difficulty 7, GT6, Monochromatic Triangle, NAE3SAT, reductions
Protected: Set Splitting
Enter your password to view comments.
Posted in Chapter 3 Exercises
Tagged 3sat, Difficulty 5, Hypergraph 2colorability, NAE3SAT, reductions, Set Splitting, SP4, uncited reduction
Not All Equal 3SAT
This is jumping ahead a little, but this is a variant of 3SAT that will be helpful in the future (actually, for the next problem)
The problem: NotAllEqual3SAT (NAE3SAT). This is problem LO3 in the appendix.
The description: Just like 3SAT, except instead of requiring at least one of the literals in each clause to be true, we’re requiring at least one literal to be true and at least one literal to be false. So we’re removing the case where all three literals can be true.
Notice that if we have an assignment of variables that is a NAE3SAT solution, then if we negate all of the variables, we still have a NAE3SAT solution
The reduction: From regular 3SAT. We’re given a formula that is a collection of 3literal clauses. For each clause (x_{1} ∨ x_{2} ∨ x_{3}) we create a new variable c_{i} (for clause i) and add 2 clauses to our NAE3SAT instance:
(x_{1}∨ x_{2} ∨ c_{i}) and (x_{3} ∨ ~c_{i} ∨ F) (Where “F” is the constant value false. We’ll talk more about this later)
The idea is that c_{i} is there to “fix” the clause. If the original clause was satisfiable because of x_{1} or x_{2}, then c_{i} can be false, and its negation can make the second clause true. If the original clause was satisfiable because of x_{3}, we can make c_{i} true. If the original clause had all three literals satisfiable, we can make c_{i} false, and become an acceptable NAE3SAT solution.
Note that because the definition of NAE3SAT (and, for that matter, regular 3SAT) is defined in terms of collections of 3 variables, the constant value false we used above isn’t strictly legal. We’ve really just shown “NAE3SAT with some constants replacing variables” is NPHard. To show “normal” NAE3SAT is NPHard, we need to reduce the fixed constant version into the normal version:
Given an instance of NAE3SAT with fixed constants, we create 2 new variables x_{T} and x_{F}. We replace all instances of the constant value true with x_{T}, and all instances of the constant value false with x_{F}. We also add one additional clause:
(x_{T} ∨ x_{T} ∨ x_{F}).
Note that this new clause is only NAESatisfiable if x_{T} != x_{F}. We can’t directly bind x_{T} to true and x_{F} to false because if you take a NAE3SAT solution and negate all literals, you get another NAE3SAT solution. But because of that, this is NAESatisfiable if and only if the original formula is.
Difficulty: It depends on what you give them. If you make them do everything I did above, I’d say it’s an 8. If you let them stop at the NAE3SAT with fixed constants, then maybe a 7.
Posted in AppendixLogic
Tagged 3sat, Difficulty 8, LO3, NAE3SAT, reductions, uncited reduction